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QUASI-LIKELIHOOD ESTIMATION OF THE PARAMETERS

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					    PARAMETER ESTIMATION FOR WEIBULL EXTENSION
   MODEL, BASED ON PROGRESSIVELY TYPE II CENSORED
                      SAMPLES

                                   ELSHAHAT, M.A.T.
                Department of Statistics, University of Zagazig, Egypt.

                                      ABSTRACT
      In this paper, the estimation of parameters based on a progressively type II
Censored samples from the Weibull extension model (WEN) is studied. The likelihood
equations, the maximum likelihood and Bayes estimators, as well as the estimators for
some lifetime parameters (Reliability and hazard function) are derived. The observed
Fisher information matrix, as well as the asymptotic variance – covariance matrix of the
MLE is derived. Approximate confidence intervals for the parameters are constructed
based on the S-normal approximation to the asymptotic distribution of MLE. The
Bayes estimators are obtained using both the symmetric squared error (SEL) loss
function and asymmetric LINEX and General Entropy (GE) loss functions. This was
done with respect to the non informative priors for the parameters. A two practical
examples consisting of data reported by Aarset (1987) and Salman et al. (1999) were
used for illustration and comparison. Some numerical results using simulation study
concerning different sample sizes, and progressive censoring schemes were reported.
KEY WORDS:
     Bayes estimates, Weibull Extension model, maximum likelihood, progressively
censored sample, symmetric and asymmetric loss functions, Monte Carlo simulation,
s-normal approximation, Non informative prior.

                                   1. INTRODUCTION
      A three – Parameter Weibull extension model (WEM) was originally proposed by
Tang et al. (2003) as an extension of the Weibull distribution. This life time distribution
is useful for modeling lifetime data with bathtub-shape or increasing hazard rate
function, which are common in reliability analysis. Tang et. al. (2003) presented a
detailed analysis of this model. Also, they studied shapes of the density and failure rate
function of WEM. Elshahat (2007a, 2007b) estimated the parameters of WEM using the
methods of Quasi–maximum likelihood, Bayesian and Quasi–Bayesian estimations.
      The WEM Probability density function (pdf) and cumulative distribution function
(cdf) respectively are

                               -1                     x /
     f(x;x /    exp[x / +  (1-e        )]                    (1)

                                      x /
     F(x, 1 – exp[ - (e        -1)] .                                  (2)
Where  and  are the scale parameters and  is shape parameter.
     The reliability function R(t) and hazard function H(t), at some t are given,
respectively by
                                         t /
         R (t) = 1 – F (t) = exp [ (1-e        )]         ,            t>0      (3)
                                -1                
         H (t) =   (t /            exp [t / ]             ,         t>0       (4)
When   1, the failure rate function is an increasing function; when 0 << 1, the failure
rate function has bathtub – shape property ( Xie et. al. (2000) and Tang et , al. (2003)).
The change point of the bathtub curve and the corresponding minimal failure rate at the
change point are:
                              1/
           t* = (1/-1)
and
                                    1-1/
         H (t*) =  (1/ - 1)              exp [1/ - 1]                           (5)

      In many life testing and reliability studies, the experimenter may not always obtain
complete information on failure times for all experimental units. For examples, units
may break accidentally in an industrial experiment, individuals in a clinical trial may
dropout of the study, or the study may have to be terminated for lack of funds. For
more detail see for example Nelson (1982) & Viveros & Balakrishnan (1994). There
are also situations where in the removal of units prior to failure is pre-planned in order
to reduce the cost and time associated with testing. Data obtained from such
experiments are celled censored data.

     The most common censoring schemes are type- I & Type II censoring, but the
conventional type I & Type II censoring schemes do not have flexibility of allowing
removal of units at point other than the terminal point of the experiment. For this
reason, we consider a more general censoring scheme called progressive type II right
censoring. Some early work on progressive censoring can be found in Cohen (1963),
(1966), Mann (1971) and Tomas and Wilson (1972). A book dedicated completely to
progressive censoring has been prepared recently by Balakrishnan and Aggarwala
(2000).

       A Type II progressively censored life test is conducted as follows. We design an
experiment in which n units are placed on a life test beginning at the same time, and the
test can be terminated at the time of any failure. In addition, one or more surviving
units may be removed randomly from the test (censored) at the time of each failure
occurring prior to the termination of the experiment; in other words, at the time of the
first failure X1,n ,R1 units are randomly removed from the remaining n-1 surviving
units.     At the second failure X2,n ,R2 units from the remaining n-2-R1 units are
                                                                    th
randomly removed. The test continues until the m failure. At this time, all remaining
Rm  n – m - R1- R2- . . . . - Rm-1 units are removed. Prior to the experiment, a number
m < n is determined and the censoring scheme (R1, R2,…, Rm) with Rj > 0 and
 m
 Σ R + m = n is specified. The resulting m ordered values, which are obtained as a
j 1 j
consequence of this type of censoring, are appropriately referred to as progressively
type II censored order statistics. Note that if, R1 = R2 = ……= Rm-1 = 0, so that Rm=
n – m, this scheme reduces to conventional type II one stage right censoring scheme.


                                                       2
Also note that if R1 = R2 = … = Rm = 0, so that m = n, the progressively type II
censoring scheme reduces to the case of complete sample (the case of no censoring).

      Many authors have discussed inference under progressive type II censored using
different life time distributions. For a comprehensive recent review of progressively
censoring, see Aggarwala (2000) and Balakrishnan & Rao (2001).

     This paper considers progressively type II censored data from a WEM. Section II
discusses the MLE of the parameters. In section III, The Bayes estimators are obtained
using both the symmetric squared error loss function (SEL), and asymmetric (LINEX
and General Entropy, GE) loss function. Section IV provides results of a practical
examples consisting of data reported by Aarset (1987) and Salman et al. (1999) and
some numerical results using simulation study concerning different sample sizes and
progressive censoring schemes.

2. Maximum Likelihood Estimators (MLE)
     Suppose that x = (x1,m,n, x2,m,n, … , xm ,m ,n) is a progressively Type-II censored
sample from a life test on n items whose lifetimes have a WEM with density shown in
(1), and R1, R2, … , Rm denote the corresponding numbers of units removed (with-
drawn) from the test. The likelihood function based on the progressively type- II
censored sample, see Balakrishnan and Aggrwala (2000), is given by

                       m                                                R
     L (, ,|x) = C f (xi , m ,n ; , ,) [1-F(xi ,m,n ; , ,)] i              (6)
                       i1
                                               m
Where C = n(n-1-R1)(n-2-R1-R2) … (n- Σ (Ri+1)); and f(x) and F(x) are given
                                             j1
respectively by (1), and (2). Substituting (1) and (2) into (6), the likelihood function is
given by
                                          m
                                m         Σ z               m      z
                                  -1
     L(, ,|x) = C mm(
                                
                                w
                                  i
                                      )e                         
                                         i1 i exp[n -  (R +1)e i ]
                                                                i                       (7)
                            i1                            i 1

                              
Where wi = (xi | ), zi = w     and xi  xi, m, n.
                              i
The natural logarithm of likelihood function is given by

     ℓ (, ,|x) = log L (, ,/x)

                                     m            m             m  z
 =log C+ m log + m log+(-1) Σ log wi+ Σ zi+n -  Σ (Ri + 1) e i. (8)
                                    i1         i1             i1

Upon differentiating (8) with respect to , and  and equating each results to zero,

                                                                ˆ ˆ       ˆ
 three equations must be simultaneously satisfied to obtain MLE  ,  and  . These
equations are given by


                                              3
        L         m (  1)               m          m                     z
              =                      + n-    Σ zi -  Σ (Ri + 1) (1 - zi) e i.
                                          i1       i1

        L         m                 m       z
              =         + n -  Σ (Ri + 1) e i.
                                  i1
        L        1                             z
              =        [m + [1- zi ((Ri + 1) e i -1)] log zi]                                         (9)
                

      The equations (9) have to be solved numerically. Note that WEM has Weibull
distribution as a special case, and for Weibull distribution, The MLE does not have a
closed form solution.

      The corresponding MLE of the reliability function R(t) and hazard rate function
                                                                  ˆ ˆ       ˆ
H(t) are given by (3) and (4) after replacing ,  and with MLE  ,  and  .

      The asymptotic variances and covariance's of the MLE for parameter ,  and
are given by the elements of the inverse of the Fisher information matrix.

                          2 
       Iij = E [                      ]    ;        i,j    = 1,2,3
                       
                                                                                    
The exact mathematical expressions for the above expectations are very difficult to
obtain. Therefore, we give the approximate (observed) asymptotic variance – covariance
matrix for the MLE, which is obtained by dropping the expectation operator E.

                                               1
        2              2         
                                         
                               
                                                           var(  )
                                                                  ˆ             ˆ ˆ
                                                                           cov(  ,  )        ˆ ˆ
                                                                                          cov(  ,  ) 
                                         
 1 =                                                                                    
I0                                                               ˆ ˆ
                                                        =  cov(  ,  )          ˆ
                                                                             var(  )          ˆ ˆ
                                                                                          cov(  ,  )  (10)
                                                                                         
                                                               ˆ ˆ
                                                           cov(  ,  )
                                                                               ˆ ˆ
                                                                           cov(  ,  )          ˆ
                                                                                            var(  )  
                              
                                      
                                   
                                                ˆ ˆ ˆ
                                                ,,
    With
 2              m (  1)          (  1) m      m                         z
         =-                      -               Σz -    Σ (R + 1) [1-(z + 1)]z e i
                                                    i        i            i      i
 2                   2               2      i1    i1

 2              1             m                 z
          =      [m - Σ zi [  (Ri + 1) (1 + zi) e i -1] log2zi]
  2         2      i1




                                                             4
 2        m
         =
  2        2

 2        m          1 m                     m                      1           z
         =        +        Σ zi (1+log zi) -  Σ (Ri + 1) zi[1+(1+ zi - ) log zi ]e i
                    i1                  i1                     

 2        m                z
      = Σ (Ri + 1) [1 – zi] e i -n
    i1

 2         m               z
         =      Σ (Ri + 1) zi e i log zi                                                (11)
          i1

                                                                          ˆ ˆ ˆ
     The asymptotic confidence intervals for ,  and  found by taking (  ,  ,  ) to
                                                                                  
be approximately normal distributed with mean ( ,  , ) and covariance matrix I 01 .
                                                                  ˆ             ˆ
Therefore,(1-  )100% confidence intervals for ,  and  become  ± z  var (  ) ,
                                                                                2
      ˆ   ˆ ˆ ˆ
where  =  ,  ,  and is the standard S-normal variate.

      Finally, in order to construct the asymptotic confidence interval of reliability
function R (t), Xie et al. (2003) approximated the variance of R (t) with

                                                                ˆ                        ˆ
     var ( R (t)) = (R(t) / )2 var(  ) + (R(t) / )2 var(  ) + (R(t) / )2 var(  )
           ˆ                           ˆ

                                                           ˆ ˆ
                        + 2 (R(t) / ) (R(t) / ) cov(  ,  )

                                                           ˆ ˆ
                        + 2 (R(t) / ) (R(t) / ) cov(  ,  )

                                                           ˆ ˆ
                        + 2 (R(t) / ) (R(t) / ) cov(  ,  )                      (12)

Where R(t) /  =  R(t) [1-ez + z ez]

                            
        R(t) /  =               R(t) (log z)zez
                              

        R(t) /  =  R(t) [1 – ez]                                                    (13)

        Z= (t/ )
                    

3. Bayes Estimation:
     Most of the Bayesian inference procedures have been developed under the usual
squared error loss function (quadratic loss), which is symmetrical, and associates equal


                                                     5
importance to the losses due to overestimation and under estimation of equal magnitude.
However, such a restriction may be impractical. For example, in the estimation of
reliability and failure rate function, an overestimate is usually much more serious than
an underestimate; in this case the use of a symmetrical loss function might be
inappropriate, as has been recognized by Cacciari and Montanari (1987), Basu and
Ebrahimi (1991) and Soliman (2005). An example of an asymmetrical loss function
stated by Feynman (1987) that in the disaster of a space shuttle, the management
overestimated the average life or reliability of the solid fuel rocket booster. A useful
asymmetric loss known linear exponential (LINEX) loss function was introduced in
Zellner (1986) and was widely used in several papers. Another useful asymmetric loss
function is the General Entropy (GE) loss. This loss function was used in several
papers, as an example see Dey et al. (1987), Dey and Lin (1992) and Soliman (2000,
2002, and 2005). In the following subsections, the Bayes estimators are obtained using
squared error loss function (SEL), LINEX loss function and GE loss function.

      Consider independent non-informative (or vague) type of priors for the parameters
,  and  as

                   1
     g1() =                             ,                0<<q                  (14)
                   q

                   1
     g2() =                        ,                     0<<A                  (15)
                   A

                   1
     g3() =                        ,  > 0               (16)
                   

That is we have

                1
g (,,) =                          ,                0<<q,0<<A, >0           (17)
               Aq

     Combining (14) with equation (7) and using Bayes theorem, the joint trivariate
posterior distribution is derived as follows

                           m  m 1D 0 e D3
     W (,, |x) =                                                            (18)
                                  J1

Where

               m
     D0 =  w 1
              i                                   ,
          i1

                   m
        D1 =       Σ zi
                   i1




                                                  6
            m         z
     D2 = Σ (Ri + 1) e i                        ,
           i1

     D3 = D1 +   (n – D2) ,

            A q
     J1 =    mm-1 D0eD3 d d d 
           0 0 0

      Now, marginal posterior of any parameter is obtained by integrating the joint
posterior distribution with respect to other parameters. The posterior pdf of  can be
written, after simplification, as

     W1 ( | x) = J2/J1             ,                          0< <q              (19)

Where

           qA
     J2 =   mm-1 D0eD3 d d 
           0 0

      Similarly integrating W(, , | x) with respect to  and  , the marginal posterior
of can be obtained as

     W2 (| x) = m J3/J1               ,                      0< <A              (20)

Where
            q
     J3 =   m-1 D0eD3 d d 
           0 0

     And finally, the marginal posterior of  can be obtained as

                     m-1
     W3 (| x) =          J4/J1            ,              0< <∞               (21)

Where
          qA
     J4 =   m D0eD3 d d 
          0 0

3.1 Bayes estimator under squared error loss function (SEL).

      Under squared error loss function (symmetric), the usual estimator of a parameters
(or a given function of the parameters) is the posterior mean. Thus, Bayes estimators of
the parameters, reliability function and hazard rate functions are obtained by using the
posterior densities (18), (19) , (20) & (21).

                          ~ ~       ~
     The Bayes estimators  ,  and  of parameters , and  are



                                                    7
      ~
       = J5 / J1

      ~
       = J /J                                                                     (22)
              6   1
      ~
       = J /J
             7    1

Where
             qA 
      J5 =    mm-1 D0eD3 d  d d
             0 0 0
             Aq
      J6 =    m+1m-1 D0eD3 d  d d
             0 0 0
              qA
      J7 =    mm D0eD3 d  d d
             0 0 0
                         ~     ~
The Bayes estimators R (t) and H (t) of the reliability R (t) and the hazard rate function
H (t), respectively are 
      ~
      R (t) = J8 / J1              ,                                                  (23)

and
      ~
      H (t) = J9 / J1                  ,                                              (24)

where
             A q
      J8 =    m m-1 D0 exp [D3 - (D4 -1)] d dd 
            0 0 0
             A q
      J9 =    m+1 m D0 D5exp [D3 + z] d dd
            0 0 0

        D4 = ez and D5 = (t /)-1

      numerical technique and computer facilities are used to evaluate the Bayes
estimators given by equations (22), (23) & (24).

3.2 Bayes estimator under LINEX loss function.
                                                           ~
      Under the assumption that the minimal loss occurs at ~ = u, the LINEX loss
                                                           u
function for u = u (,, ) can be expressed as

      L ()ea – a - 1     ; a≠0                                                  (25)
            ~        ~
Where  = ( ~ - u) , ~ is an estimate of u .
            u        u


                                            8
     The sign and magnitude of ' a ' represent the direction, and degree of symmetry,
respectively. (a > 0 means overestimation is more serious than underestimation, and
a < 0 means the opposite). For ' a ' closed to zero, the LINEX loss function is
approximately the squared error loss, and therefore almost symmetric. Following
                                    ~
Zellner (1986), the Bayes estimator ~ of u under LINEX loss function is
                                    u
      ~
      ~ = - 1 log (E [exp (- a u)])
      u                                                                               (26)
            a       u

Where Eu is equivalent to the posterior S-expectation with respect to the posterior
pdf (u). Provided that Eu [exp (- a u)] exist, and is finite.

                                                   ~
                                                   ~
    Now, if in (26) u = , then the Bayes estimate  of parameter  relative to the
LINEX loss function in (25) is
      ~
      ~    1
       =-   log (J10 / J1)               ,                                        (27)
           a
where
             q A
      J10 =    m m-1 D0 eD3-a d  d d ,
             0 0 0
                                            ~
                                            ~
Set u =  in (26), then the Bayes estimate,  , of parameters relative to the LINEX loss
function is
      ~
      ~      1
       = - log (J11 / J1)                                                        (28)
             a
where
            qA 
      J11 =    m m-1 D0 eD3-a d  d d ,
            0 0 0
                                            ~
                                            ~
Set u =  in (26), then the Bayes estimate,  , of parameters relative to the LINEX loss
function is
      ~
      ~      1
       = - log (J12 / J1)                                                        (29)
             a
where
              Aq
      J12 =    m m-1 D0 eD3-a d  d d ,
             0 0 0
                                                                            ~
                                                                            ~
Set u = R (t) in (26), where R (t) is given by (3), then the Bayes estimate R (t) is
      ~
      ~         1
      R (t) = - log (J / J )                                                         (30)
                a       13    1

where




                                              9
                 Aq
     J13 =    m m-1 D0 exp[- a e-(D3-1)] d  d d
                0 0 0
     Also, after setting u = H (t), in (26), where H (t) is given by (4), the Bayes
          ~
          ~
estimate H (t) of the hazard rate function of (3) relative to the LINEX loss function is
      ~
      ~         1
      H (t) = -    log (J14 / J1)                                                    (30)
                a
Where
                 Aq
     J        =    m m-1 D exp [ - a  D D ] d  d d ,
         14                    0               4 5
                0 0 0
     As mentional earlier, the integrals involved in (27) – (31) are not solvable
analytically and, therefore, a numerical technique and computer facilities are needed to
evaluate the Bayes estimators.

3.3 Bayes estimator under General Entropy (GE) loss function.
     Another useful asymmetric loss function is the General Entropy (GE) loss
                    ˆ
                    ~ v
                    u           ~
                                uˆ
          ˆ
     L2 ( ~ , u)  ( ) – V log ( ) -1
          u                                                                              (32)
                    u           u
                                   ˆ
      Whose minimum occurs at ~ = u .This loss function is generalization of the
                                   u
Entropy–loss used in several papers where V = 1. When V > 0 , a positive error
  ˆ                                                                                 ˆ
( ~ > u) causes more serious consequences than a negative error. The Bayes estimate ~
  u                                                                                 u
under GE loss (23) is
                                  1
      ~ = [ E (u-v) ] v
      ˆ
      u                                                                               (33)
                 u
                          -v
Provided that E (u ) exist, and is finite.
                      u
                                                                   ˆ ˆ ˆ ˆ
                                                                   ~ ~ ~ ~               ˆ
                                                                                         ~
Set u =  ,  , , R (t), H(t) , in (33), then the Bayes estimates  ,  ,  , R (t) and H (t)
of parameters, respectively , , R (t) and H(t) (where R (t) and H(t) are given by (3)
(4); respectively) relative to General Entropy (GE) loss function (32) are
      ˆ
      ~                          1
       =[J           /J ] v
                 15       1
         ˆ
         ~                       1
          =[J        /J ] v
               1 16
      ˆ
      ~         1
      =[J /J ] v                                                                        (34)
          17 1
      ˆ
      ~                              1
      R (t)= [ J          /J ] v
                     18       1
      ˆ
      ~                              1
      H (t)= [ J          /J ] v
                     19       1
Where
                 Aq                         D
                              -v m m-1
     J        =                   D e 3d d  d 
         15      0 0 0                     o



                                                   10
                           Aq                  D
                                   m-v m-1
            J        =                  D e 3d d  d 
                16        0 0 0               o
                           Aq                  D
                                   m m-v-1
            J        =                 D e 3d d  d 
                17        0 0 0                o
                           Aq
                                   m m-1
            J        =               exp [D + vD - 1)] d d  d 
                18         0 0 0                    3             5
                           Aq
            J        =   
                                   m-v m-v-1
                                                  D -vz
                                                  D5 v e 3 d d  d 
                19        0 0 0
Equations (34) could be evaluated numerically.

4. Illustrative Examples and Simulation Study.
A. A Numerical Examples.

      To illustrate the usefulness of the proposed estimators obtained in sections 2 and 3
with real situations, we considered here two real data sets as follows : ( i ) first initially
reported by Aarset (1987) to identify the bathtub hazard rate contains life time of 50
devices. For the purpose of illustrating the methods presented in this example, a Type II
progressive censored sample of size m = 35 has been randomly generated from the n =
50 observations recorded in Aarset (1987) Table (1). The observations and the Five
stage removal pattern applied are reported in Table (1) and have been used earlier in Ng
(2005) . For This example, 15 failure times are censored; 35 are observed.

Table (1) . Progressively Type - II censored Sample (m = 15, n = 50) From Aarset
             (1987)
    i    1       2    3     4       5     6    7     8     9     10     11  12
x
  i,m,n 0.1     0.2   1     1       1     1    1     2     3      6      7  11
    R                0        0       0       3         0     0       0    0    0    0    3    0
        i
    i                13      14      15      16         17   18       19   20   21   22   23   24
x                    18      18      18      18         21   32       36   45   47   50   55   60
 i,m,n
    R                0        0       0       0         0    3        0    0    0    0    0    0
        i
    i                25      26      27      28         29   30       31   32   33   34   35
x                    63      63      67      67         75   79       82   84   84   85   86
 i,m,n
    R                3        0       0       0         0    0        0    3    0    0    0
        i

                                                                      ˆ ˆ       ˆ
     Assuming that the data have come from the WEM in (1), The MLE  ,  and  of
parameters ,  and  are then computed from the solution of equations (9) using
computer facilities. Tables 2, 3 provide the values of the estimates, their standard
deviation and their covariances. In addition, we also present (entries within
parentheses) the corresponding 95% approximate confidence intervals for the
parameters using the asymptotic s-normality of the MLE. The approximate MLE of



                                                             11
reliability R(t) with 95% confidence interval at corresponding time to failure are
obtained.

      The Bayes estimators (SEL, LINEX, GE) for the parameters ,  and , reliability
R (t) and failure rate H (t) function (t = 1) are computed using the (22) – (34) and are
given in Table 4. All of the results obtained in this article specialized to both the
complete sample case by taking ( m = n, R = 0, i = 1, 2, … , m ) and the Type II
                                                                           i
censored sample for (R = 0 , i = 1, 2, …, m -1, R = n – m).
                                  i                                               m

Table 2 . MLE's, variances & covariances and 95% confidence intervals for , 
         and  for Aarset data.
   
   ˆ           ˆ
                         ˆ     SD (  )
                                      ˆ        ˆ
                                          SD (  )      ˆ
                                                   SD (  )
                                                             COV    COV      COV
                                                             ˆ        ˆ
                                                               ˆ ) (, )
                                                            (,    ˆ        ˆ ˆ
                                                                            ( ,  )
    3.183              0.34               0.013                                     -4              -6
                                                             1.675                                        0.0291         - 0.0017     - 0.00004
(0.646, 5.719)     (0.29, 0.39)       (0.008, 0.019)                       5.4x10          7.3x10

                                                       Complete Sample
    3.235             0.348            0.014                                                        -6
                                                             0.573             0.0043                     0.0138         -.00095      - 0.000026
(1.751, 4.719)     (0.31, 0.39)   (0.0091, 0.018)                                          5.3x10

                                              Censored Type II Sample
    3.614              0.38               0.017
                                                             6.059         0.000948        1.5x10-5       0.0132         -0.0057      - 0.000092
(-1.12, 8.439)     (0.32, 0.44)       (0.01, 0.025)


                                     ˆ
Table 3. Estimations of reliability R (t) and its 95% confidence intervals for Aarset
          data .
         Progressive Sample        Complete Sample       Censored type II sample
Time
        ˆ      Lower Upper ˆ            Lower Upper        ˆ      Lower Upper
  t    R (t) limit               R (t) limit              R (t)
                       limit                       limit           limit   limit

   1        .96        .942            .978           .959       .952               .969         .948          .911            .985

   5        .91        .887            .934           .907           .90            .914         .876              .81         .942

  10        .867       .857            .876           .861       .838               .883         .809          .734            .884

  20        .793       .745             .84           .781       .734               .828         .693          .641            .745

  30        .727       .652            .802           .709       .642               .777            .59        .527            .652

  50        .606       .494            .719           .578           .48            .676            .41        .285            .535

  70        .498       .368            .629           .462       .345               .578         .267         .0138            .395

 100        .359       .231            .486           .314       .191               .437         .122         .0081            .202

 150        .187       .119            .255           .144       .049               .239         .021         .0054            .037

 200        .085       .038            .131           .054      .0013               .107       .00196              0.0        .0077

 225        .054      .0030            .104            .03           0.0            .066       .00045              0.0        .0024




                                                                     12
ii) The second real data for time between failures (thousands of hours) of secondary
    reactor pumps first initially reported by salman et al. (1999) and used by Bebbington
    et al. (2007). In this example, a type II progressive censored sample size m = 14 has
    been randomly generated form the n = 23 observation recorded in salman et al.
    (1999). The observations and the three stages are reported in table 5. For this
    example, 9 failure times are censored; 14 are observed. The results are summarized
    in Tables 6, 7, 8.

Table 4. Bayes Estimates of the parameters, (reliability and Hazard rate function
         for t =1) .

                                    LINEX                                     GE.

Parameter    SEL                      a                                       E

                      0.5       5          -0.5         -2      1       2           -2       -5

          6.797    4.799   1.131        9.547       12.712 3.695    0.645       7.532.   9.166

          0.366    0.366   0.364        0.366       0.367   0.364   0.363       0.367    0.369

          0.012    0.012   0.012        0.012       0.012   0.011   0.011       0.018    0.142

 R (t =1)   0.955    0.955   0.945        0.955       0.955   0.955   0.954       0.955    0.955

H( t = 1)   0.022    0.022   0.022        0.022       0.024   0.021   0.020       0.022    0.024
                                    Complete sample

          1.581    0.492   0.821        1.73        8.906   1.153   0.690       1.706    2.301

          0.38     0.38    0.379        0.381       0.381   0.387   0.377       0.382    0.383

          0.014    0.014   0.013        0.014       0.014   0.011   0.011       0.019    0.053

 R (t =1)   0.976    0.976   0.976        0.976       0.976   0.976   0.976       0.976    0.976

H( t = 1)   0.014    0.014   0.014        0.014       0.014   0.013   0.013       0.015    0.020
                              Censored type II sample

          9.405   6.185    1.195    3.614           13.493 5.257    0.828    10.065 11.264

          0.376   0.376    0.374    0.380           0.376   0.373   0.372    0.376       0.379

          0.012   0.012    0.011    0.017           0.013   0.011   0.011    0.012       0.184

R (t =1)    0.948   0.947    0.946    0.948           0.948   0.946   0.946    0.947       0.947

H( t = 1)   0.027   0.025    0.025    0.027           0.027   0.024   0.024    0.025       0.027




                                                 13
Table 5 . Progressively type II censored sample (m = 14, n = 23) from salman et
         al.(1999).
   i         1          2          3          4         5           6         7
xi,m,n    0.062       0.07       0.101      0.15      0.199       0.273     0.495
  Ri         0          0          0          3         0           0         0

      i             8              9                 10                   11            12                  13            14
 xi,m,n           0.605          0.614             0.746                0.954          1.06               1.359         1.921
   Ri                3               0                  0                3               0                   0               3

Table 6 . MLE's, variances & covariances and 95 % confidence intervals for , 
         and  for Salman et al. data.
          
          ˆ               ˆ
                                             ˆ
                                                           var (  )
                                                                  ˆ             ˆ
                                                                          var (  )         ˆ
                                                                                      var (  )         Cov          Cov           Cov
                                                                                                       ( , )
                                                                                                        ˆ ˆ         ˆ ˆ
                                                                                                                   (,)          ˆ ˆ
                                                                                                                                 ( ,  )
     5.912              0.534              0.00717                                             -6                                          -5
                                                             17.646        0.0569     3.7x10            1.262      0.0007        -3.7x10
(-2.321,14.15)       (0.66,1.001)        (0.003,0.01)
                                                  Complete Sample
    8.722                0.422               0.04                                              -5                                         -5
                                                              18.598      0.01512     7.8x10            -0.332      -0.02        7.8x10
 (-2.69,17.2)        (0.181,0.663)       (0.022,0.057)
                                            Censored Type II Sample
    6.469                0.562               0.0073                                           -6                                         -5
                                                              13.256       0.0386      4x10             1.014     -0.00014        2x10
(-1.67,13.61)        (0.177,0.948)       (0.0035,0.011)

                                   ˆ
Table 7. Estimation of reliability R (t) and its 95% confidence intervals for Salman
et al. data
                 Progressive Sample                        Complete Sample                             Censored type II sample
Time




           ˆ             Lowe    Upper                ˆ         Lower      Upper                     ˆ         Lower       Upper
  t




           R (t)        r limit   limit              R (t)      Limit       Limit                    R (t)     Limit       Limit
 1             .98        .952      1.0              0.843      0.707      0.978                      .98       .972        .989
 5           0.939       0.916    0.961              0.658      0.507       0.81                    0.937      0.907       0.969
 10           0.89       0.857    0.922               0.52      0.369      0.671                    0.885      0.818       0.952
 20          0.782        0.48     0.86              0.337      0.128      0.546                    0.768      0.612       0.923
 30           0.66        0.32     0.75              0.218      0.045      0.481                    0.633      0.381       0.884
 50          0.397         0.0    0.505              0.086        0.0      0.342                    0.345      0.046       0.736
 70          0.175         0.0    0.334               0.03        0.0      0.197                    0.123        0.0       0.457
100          0.021         0.0    0.271             0.0047        0.0       0.58                    0.007        0.0       0.067
150                -6      0.0  0.00122                    -5     0.0      0.0025                         -8     0.0             -5
           8.8x10                                  9.3x10                                          6.9x10                 2.9x10

              From the results, we observe the following.
i– All of the results obtained in tables 2, 3, 4, 6, 7 and 8 specialized to both the complete
    sample case by taking (m = n, Ri = 0, i = 1, 2, , …, m )and the type II censoring
          sample for (Ri = 0, i = 1, …, m-1, Rm = n – m).

ii- Tables 4&8 shows that the Bayes estimators relative to asymmetric loss function
     (LINEX and GE) are sensitive to the value of parameters a and E.These parameters
     one the opportunity to estimate the unknown parameters with more flexibility. But
     the problem of choosing values of the parameters a and E of the selected loss
     function.




                                                                 14
 Table 8. Bayes Estimates of the parameters, (reliability and Hazard rate function
                          for t =1) for Salman et al data.
                                          LINEX                                         GE
                                          A                                             E
Parameter    SEL
                        0.5        5            -0.5        -2         1         2             -2        -5

          13.186    12.272    3.456         13.644      14.187    12.946    12.787         13.28     13.502

          0.247     0.246     0.236         0.249       0.252     0.228     0.219          0.257     0.286

          0.00912   0.00912   0.00912       0.00912     0.00918   0.0091    0.009          0.0092    0.00924

R (t =1)    0.918     0.918     0.917         0.918       0.918     0.917     0.917          0.918     0.918

H( t = 1)   0.026     0.026     0.026         0.026       0.026     0.025     0.025          0.026     0.027
                                        Censored Sample
          8.994     2.185     0.445         13.167      14.175    0.519     0.266          10.944    12.763

          0.506     0.401     0.337         0.42        0.462     0.33      0.301          0.453     0.545

          0.013     0.00947   0.00947       0.00947     0.0095    0.00944   0.00942        0.00949   0.0095

R (t =1)    0.956     0.94      0.938         0.941       0.942     0.942     0.94           0.941     0.943

H(t = 1)    0.27      0.031     0.031         0.025       0.029     0.03      0.03           0.031     0.033
                                Censored Type II Sample
          13.463    12.761    3.778         13.819      14.266    13.287    13.175         13.534    13.704

          0.236     0.235     0.208         0.237       0.241     0.217     0.208          0.245     0.274

          0.00923   0.00923   0.00923       0.00923     0.00923   0.00916   0.00923        0.00925   0.00932

R (t =1)    0.912     0.912     0.912         0.913       0.913     0.912     0.912          0.913     0.913

H(t = 1)    0.026     0.026     0.026         0.026       0.026     0.026     0.026          0.027     0.027
iii- As the same as Soliman (2005) notation, the analytical ease with which results can
     be obtained using asymmetric loss function makes them attractive for use in applied
     problems. Obviously, we do not expect much to conclude from this reanalysis,
     perhaps we are capable to show that the proposed estimators can be easily obtained
     in practical situations in spite of non-existence of their closed form solutions.
iv- The estimated values of Bayes estimators are not very far from the estimated values
    of maximum likelihood estimators unless the value of  estimator.
v – All Bayes estimators (SEL, LINEX, GE) for the parameters , , R(t) and H(t) are
    nearly to equal for all values of a and E were used.
vi- The asymmetric Bayes estimates (LINEX, GE) of parameter  is overestimates,
    while of R(t=1) is underestimates . Finally, the asymmetric estimates of parameters
     and  are largest (or smaller) them the MLE depending on the size of n and m.
    (the asymmetric Bayes estimate of parameter  is overestimates in Aarset data while


                                                     15
   underestimates in salman et al data & asymmetric Bayes estimate of parameter is
   underestimates using Aarset data but is over estimate using Salman et al. data .
vii- As anticipated all Bayes estimates relative to both LINEX loss and GE loss (for a
    closed to zero and E = -1) are the same as the symmetric Bayes squared error loss.
B. Simulation Results.
      We discuss results of a Monte Carlo Simulation study testing the performance of
MLE of the model Parameters for different sample sizes, and censoring schemes the
average MLE's, average variances & covariances, risks and 95% confidence intervals,
are also examined via simulation. Progressively censored type II samples from WEM
distribution with  = 1.5 ,  = 0.5 and  = 0.001 using the algorithm presented in
Balakrishnan and Sanhu (1995) as follows :

     1 – Generate m independent uniform (0, 1) observations w* , w * . … , w m .
                                                             1     2
                                                                             *

     2 – Determine the value of the censored scheme Ri for i = 1, 2, … , m .
                                 m
     3 – Set E* = 1 / ( i +  Rj) for i = 1, 2, …, m.
              i            jm-i1

                          *
              *      * E1
     4 – Set V i = (w i )   for i = 1, 2, …, m.
                                       m
                    *
     5 – Set U i, m,n = U * = 1 -  V k   *
                                                    for i = 1, 2, …, m.
                          i      k m-i1
                       *          *
         Then U * , U 2 , … , U m is the required progressively Type II censoring
                 1
         sample from the Uniform (0,1) distribution .
     6 – Finally, for ( = 1.5 ,  = 0.5 and  = 0.001) , we set xi,m,n = xi = F-1 (u* )
                                                                                     i


                           log (1  u * ) (1/)
           =  [ log (1-              i
                                         )]      for i = 1, 2, …, m .
                               
          Then x1, x2, …, xm is the required progressive Type II censored sample
from the WEM ( = 1.5 ,  = 0.5 and  = 0.001) .
       For different choices of sample sizes & progressive censoring schemes, we
generate 2000 sets of data. Table 9 lists the different censoring schemes (Sc.) used in
the simulation study. For each set of simulated data, we computed the MLE from (8).
The MATHCAD (2001) program is employed to solve the nonlinear equations (9) with
starting values (0 = 5.5, 0 = 0.6 and 0 = 0.0001). We also computed the local
estimate of the variance – covariance matrix by inverting the observed Fisher
information matrix (10) and use that to construct two sided 95% confidence for the
parameters using the asymptotic s-normality of the MLE. Tables 10 – 12 provide the
                                                               2000
average values of the estimates (   ), their averages (  =    / 2000), their standard
                                                          ˆ
                                    i                         j 1 i
                                                          2000
                                                     ˆ          ˆ
deviation, their covariances and their risks (Risk (  ) =  ( -  0 )2 / 2000 ,  0 is the
                                                           i 1
true parameter). In addition, we also present (entries within parentheses) the



                                              16
corresponding 95% approximate confidence intervals for the parameters.              All the
averages were computed over 2000 repetitions.
                  ˆ         ˆ
      The MLE R (t) and H (t) of the reliability and hazard rate function were computed
                                                              ˆ ˆ        ˆ
at time t = 1 after replacing , ,  by their corresponding  ,  and  in (3) and (4),
                                                 ˆ          ˆ
respectively , Also, the estimated risks of R (t) and H (t) and 95% approximate
                                                                       ˆ
confidence interval for R (t = 1) using the asymptotic normality of R (t) with variance
estimated using (12) are computed using the value R (t = 1,  = 1.5,  = 0.5 &  =
0.001) = 0.998 and H ( t = 1,  = 1.5,  = 0.5 &  = 0.001) = 0.0014, as true values.
                                                           ˆ          ˆ
Table 12 summarize the average values of the estimates R (t = 1), H (t), the estimated
          ˆ           ˆ
risks of R (t = 1) & H (t) and 95 % approximate confidence interval for R (t=1). All the
averages were computed over 2000 repetitions.
C. Remarks:
     From the results which are given in Tables 10 – 12, we observe the following:
                                                          ˆ
1 – When n is large (n ≥ 50) the average value of the mle  are nearly equal with
                                                            ˆ
    censored schemes given in Table 9. Also, the value of  are much enclose to
    each other.
                                  ˆ                                          ˆ
2 – The average estimate risk of  is less than the average estimate risk of  which is
     less than the average estimate risk of  .
                                            ˆ
             ˆ                                    ˆ
3 – The mle  of  is more efficient than the mle  of  which is also more efficient
       than the mle  of  under are censoring schemes studied.
                     ˆ
                                    ˆ ˆ
4 – The average estimate of Cov (  ,  ) is less than average estimate of average estimate
                                      ˆ ˆ
     of average estimate of Cov (  ,  ), and average estimate of average estimate of
           ˆ ˆ                                               ˆ ˆ
     Cov (  ,  ), is less than average estimate of Cov (  ,  ) and when m/n increases
     the covariances are reduce significantly.
                        ˆ ˆ               ˆ ˆ
5 – The values of Cov (  ,  ) and Cov (  ,  ) are very small and converge to zero.
6 – In case of censoring scheme R1 = n – m, Ri = 0, i = 2, … , m , seem to provide the
                                     ˆ ˆ    ˆ
     smallest risk of the estimates  ,  &  as m/n increases, but in censoring scheme
                                           m m
     R m 1 = n – m , Ri = 0, i = 1, 2, …,   ,   + 2, … , m , the risks of estimators  ,
                                                                                      ˆ
        2                                  2 2
      ˆ    ˆ
      &  are increases when m/n increase.
                                                                         ˆ
7 – In censoring scheme R1 = n – m , Ri = 0  i  1, the average of R (t = 1) are
     approximately equal and average of its risks are also nearly to equal. The same
                  ˆ
     behavior for H (t = 1) and its risks.
                                                    n-m
8– In censoring scheme Rm = n – m & R1 = Rm =             and other Ri = 0 , the average
                                                      2
                       ˆ
     risk of estimator R (t = 1) reduces significantly when m/n in creases and in
     censoring scheme R m 1 = n – m, its risks are decreases when m/n were decreases.
                           2




                                            17
                                                      ˆ
9 – Finally, the behavior of average risk estimate of H (t = 1) (hazard function) seem to
     provide the following:
                                                                                       n-m
     - In censoring schemes R1 = Rm =                                                      and R m 1 = n – m, if m/n increases, its
                                                                                        2        2
          risks are increases.
                                                                      ˆ
     - In censoring scheme Rm = n –m , the risk of hazard rate H (t = 1) increases
         significantly as the effective sample proportion m/n decreases.
Table (9) : Progressive censored schemes used in the Monte Carlo simulation study

    n     m                   ( R , R , R ,……………………………..… , R )                                                           Scheme
                                    1            2               3                                              m
                                                                                                                            No.
    100   50   R = 50 , R = 0 , i = 2, … , 50                                                                               [1]
                   1            i
               R        = 50, R = 0 , i = 1, … , 49                                                                        [2]
                   50               i
               R =R            = 25, R = 0 , i = 2,3, … , 49                                                               [3]
                   1      50                             i
               R        = 50 , R = 1,2, … , 50 , i  26                                                                    [4]
                   26               i
          20   R = 80 , R = 0 , i = 2, … , 20                                                                              [5]
                   1            i
               R        = 80, R = 0 , i = 12, … , 19                                                                       [6]
                   20               i
               R        = 80 , R = 0 , i = 1,3, … , 20 , i  11                                                            [7]
                   11                   i
               R =R            = 40 R = 0 , i = 2, … , 19                                                                  [8]
                   1      20                         i
    50    35   R =R            =R                = R =R                            = 3 , R = 0, i  4 , 11, 18, 25 , 32    [9]
                   4      11                18                   25           32          i
          20   R        = 30, R = 0 , i = 2, 3, … , 20                                                                     [10]
                   20               i
               R        = 30 , R = 0 , i = 1,2, … , 19                                                                     [11]
                   20                   i
               R        = 30 , R = 0 , i = 1,2, … , 20 , i  11                                                            [12]
                   11                   i
               R =R            = 15 , R = 2 , 3 , … , 19                                                                   [13]
                   1      20                                 i
               R =R =R                          = 10, R = 0, i = 1, … , 20 , i  1 , 8, 15                                 [14]
                   1      8             15                            i
               R = 2 , i = 1, … ,15 , R = 0 , j = 16, … , 20                                                               [15]
                   i                                                      j
               R = 2 , i = 6 , 7 , … , 20 , R = 0 , j = 1, … , 5                                                           [16]
                   i                                                               j
    50    10   R = 40 , R = 0 , i = 2, … , 10                                                                              [17]
                   1            i
               R        = 40, R = 0 , i = 1, … , 9                                                                         [18]
                   10               i
               R =R            = 20, R = 0 , i = 2 , … , 9 ,                                                               [19]
                   1      10                             i
               R = 40, R = 0 , i = 1, 2, … , 10 , i  6                                                                    [20]
                   6                i
    20    10   R = 10 , R = 0 , i = 2, 3, … , 10                                                                           [21]
                   1                i
               R        = 10 , R = 0 , i = 1, 2 , … , 9                                                                    [22]
                   10                   i
               R        = 10 , R = 0 , i = 2 , … , 9 , 10 , i  6                                                          [23]
                   6                        i
          5    R = 15 , R = 0 , i = 2, … , 5                                                                               [24]
                   1                    i




                                                                                       18
Table (10) : Average MLE's , 95% confidence Intervals and Average Risks .

Sc.                                                                                 Risk
                       
                       ˆ                 ˆ
                                                          ˆ
                                                           
No     n     m
                                                                          
                                                                          ˆ         ˆ
                                                                                                  ˆ
                                                                                                   
                       1.024            0.444             0.00454                          -3             -5
[1]    100   50                                                         0.242    3.4610        2.4910
                  (0.489, 1.345)   (0.436,0.488)    (0.00017,0.0062)
                       0.369            0.491               0.011                         -5              -4
[2]                                                                     2.178    1.710         1.7210
                   (0.04,0.532)     (0381,0.501)       (0.007,0.042)
                       0.304            0.476               0.011                          -4             -5
[3]                                                                     1.492    7.5910        9.5710
                  (0.056,0.421)    (0.432,0.491)       (0.007,0.072)
                       1.032            0.809             0.00689                                         -4
[4]                                                                     1.213     0.196         3.2110
                    (0.04,1.65)     (0.49,0.898)      (0.0024,0.013)
                       1.086            0.444             0.00514                          -3             -5
[5]    100   20                                                          1.02    3.6710        3.7910
                   (0.72,1.716)    (0.391,0.545)     (0.0003,0.0084)
                       0.276            0.615               0.054                                         -3
[6]                                                                     1.513     0.014         1.4210
                  (0.158,0.345)    (0.428,0.684)       (0.034,0.072)
                        1.07            0.451              0.0085                                        -5
[7]                                                                     2.103     0.0024        5.610
                    (0.64,1.34)    (0.321,0.568)      (0.0012,0.032)
                        1.71            0.663               0.012                                        -4
[8]                                                                     0.044     0.026         1.210
                  (1.698,1.723)    (0.662,0.664)       (0.008,0.023)
                        1.28             0.48             0.00051                                         -4
[9]    50    35                                                         1.239     1.525         1.1410
                   (1.057,1.69)    (0.439,0.544)    (0.00033,0.0007)
                         1.2            0.405             0.00036                          -3             -7
[10]         20                                                          0.09    8.9910        4.0810
                   (0.813,1.59)    (0.325,0.485)    (0.0002,0.00053)
                        0.07            0.594               0.026                         -3             -4
[11]                                                                    2.046    8.910         6.310
                    (0.05,0.12)     0.402,0.683)       (0.007,0.032)
                       0.223            0.426              0.0038                         -3              -6
[12]                                                                    1.631    5.510         8.1110
                   (0.216,0.23)    (0.424,0.428)     (0.0036,0.0041)
                         1.5            0.502               0.028                         -6              -4
[13]                                                                    2.088    6.010         7.0510
                    (1.32,1.67)    (0.421,0.632)       (0.013,0.042)
                        1.33            0.522             0.00058                                         -4
[14]                                                                     2.51     0.0032        2.7310
                   (0.732,2.27)      (0.48,0.56)    (0.00014,0.0011)
                       2.196            0.483             0.00049                         -4             -7
[15]                                                                    0.484    3.010         2.610
                     (1.66,2.7)    (0.407,0.558)    (0.00026,0.0007)
                       1.449            0.508              0.0013                         -3              -4
[16]                                                                    0.332    3.810         2.1310
                    (1.21,1.81)     (0.43,0.586)      (0.0009,0.003)
                       5.999              0.8            0.000037                                         -7
[17]   50    10                                                         20.239     0.09         9.2710
                   (-6.57,18.57)   (0.386,1.214)       (0.0,0.00009)
                       0.066            0.557               0.022                          -3             -4
[18]                                                                    2.057    3.2910        4.5910
                  (0.034,0.085)    (0.341,0.751)        (0.008,0.32)
                        2.98            0.606             0.00103                                         -10
[19]                                                                    2.191      0.11         8.8710
                    (0.528,6.1)    (0.528,0.685)    (-0.00004,0.0024)
                       2.016            0.516             0.00041                                         -7
[20]                                                                    0.266     0.011         3.4910
                  (1.504,2.523)    (0.507,0.639)    (0.00026,0.0006)
                       0.942            0.644             0.00108                                         -9
[21]   20    10                                                         0.311     0.021         6.6510
                   (0.94,0.945)    (0.643,0.650)    (0.00107,0.0011)
                       0.087            0.543               0.017                          -2             -4
[22]                                                                    1.997    1.8510        2.7210
                  (0.071,0.127)    (0.481,0.586)       (0.006,0.021)
                        0.04            0.522               0.021                          -4            -4
[23]                                                                    2.133    4.7210        3.910
                  (0.021,0.063)    (0.482,0.601)       (0.008,0.032)
                       0.035            0.446              0.0098                          -3             -5
[24]   20    5                                                          2.145    2.9410        7.7810
                  (0.013,0.062)    (0.412, 0.502)     (0.0007,0.012)




                                              19
Table (11) : Average of Standard deviations & covariances.

Sc.                               Standard deviation                                       Covariance
No.    n     m
                      
                      ˆ                   ˆ
                                                             ˆ
                                                                             ˆ ˆ
                                                                            (  , )         ˆ ˆ
                                                                                            ( , )           ˆ ˆ
                                                                                                             ( ,  )
[1]    100   50     0.226               0.013                        -4               -3              -5               -6
                                                        3.4510            3.6710         2.5310          5.4710
[2]                         -10                 -10               -11         0.0             0.0              0.0
                  1.0410             1.6510           9.410
[3]                  0.247                0.13                       -3              -14      0.0                     -15
                                                        2.1710            1.5110                          2.0110
[4]                  0.997               0.317             0.17                0.0            0.0               0.0
[5]    100   20      0.148               0.039                       -4       0.011                   -5               -6
                                                        1.3410                            -1.8810         -1.7110
[6]                 0.118               0.016                        -3       0.0              0.0              0.0
                                                        3.9910
[7]                          -7                  -7                  -8              -4       0.0                     -15
                  1.4610             1.6210           7.8110             2.410                          3.6410
[8]                        -3                    -4                  -5              -6              -7               -8
                   6.410             6.0510           7.4110             3.910          1.110           2.310
[9]    50    35     0.307               0.033                        -3      0.019                    -6               -7
                                                        9.0710                            -3.7610         -1.6810
[10]         20     0.197               0.041                        -5      0.011                    -6               -7
                                                        8.2210                             -5.410          -6.410
[11]                        -12                -12                   -12      0.0              0.0              0.0
                  1.0310             1.310            1.7410
[12]                         -3                  -4                  -5               -6              -8              -8
                  3.4510             9.1710           9.8310            3.1310         5.3210           5.110
[13]                       -9                    -9                  -4       0.0             0.0               0.0
                   5.710             7.0310           9.7510
[14]                0.392               0.019                        -4               -3             -5               -7
                                                        2.2210            8.9410          3.110           5.110
[15]                0.273               0.038                        -4       0.01                    -5               -6
                                                        1.1710                             -2.110         -1.7810
[16]                0.457                0.04                        -4              -4               -5               -6
                                                            1.510          9.810          -4.210          -2.910
[17]   50    10     6.415               0.211                        -5               -4              -4                -5
                                                        2.4310            -2.9510        -2.9510         -1.16910
[18]                        -10                  -10                 -10       0.0             0.0              0.0
                  6.4410            8.02910           8.2210
[19]                 1.589               0.04                        -4      0.070                    -3               -5
                                                        7.2310                            1.0210          8.9310
[20]                0.261               0.017                        -5               -4               -6              -7
                                                        7.7610            4.6110         5.61410         8.8410
[21]   20    10              -3                  -4                  -6               -7             -10              -10
                  1.3210             1.8610           3.6210            2.4610         5.5610          2.3410
[22]                         -9                 -9                   -9       0.0              0.0              0.0
                  8.6710              7.910           6.1210
[23]                       -13                 -13                   -13      0.0             0.0              0.0
                  5.710              9.510            1.3410
[24]   20    5               -8                  -8                  -8       0.0             0.0              0.0
                  2.9910             4.4610           2.7210




                                                       20
Table (12) : Average MLE of R(t=1) & H(t=1) , Average estimated Risk of R(t) &
              H(t) and 95% confidence intervals for R(t=1).
                               Lower      Upper
                     ˆ
                     R(t  1) Limit of                                    Risk           Risk
 N    m    Sc. No.                       limit of    ˆ
                                                     H(t  1)           ˆ (t  1)
                                                                        R              ˆ
                                                                                       H(t  1)
                               ˆ
                              R(t  1)    ˆ
                                         R(t  1)
100   50     [1]     0.998   0.922       1.0                  -3                 -8                 -9
                                                    1.2910            2.6810        8.84510
             [2]     0.951   0.892       0.985      0.124                        -3   0.015
                                                                       2.2510
             [3]     0.98    0.931       0.999      0.036                        -4             -3
                                                                       3.2810        1.2110
             [4]     0.858   0.781       0.901      0.396                       -3    0.156
                                                                       2.110
      20     [5]     0.999   0.958       1.000                -4                 -7             -7
                                                    8.0110            4.5510        3.4210
             [6]     0.879   0.860       0.934      0.272              0.14           0.073
             [7]     0.987   0.980       0.991                -4       0.012                    -4
                                                    1.3410                           3.7710
             [8]     0.999   0.987       1.000                -4                 -6             -6
                                                    2.5510            2.3510        1.2810
50    35     [9]     0.998   0.964       1.000                -4                 -7             -7
                                                    6.4610            6.2710        5.4710
            [10]     0.998   0.976       1.000                    -3             -6             -7
                                                    4.13410           1.5110        9.4510
            [11]     0.79    0.65        0.85       0.689              0.43           0.473
            [12]     0.995   0.978       1.000               -4                 -6              -5
                                                    4.610             8.710         1.0410
            [13]     0.896   0.850       0.948      0.24               0.01           0.057
            [14]     0.998   0.972       1.000                -4                 -7             -7
                                                    8.1510            6.7610        3.2510
            [15]     0.999   0.965       1.000                -4                 -7             -7
                                                    7.0410            6.9910        4.6410
            [16]     0.999   0.959       1.000                -4                -7              -7
                                                    9.6410            2.210         1.7810
50    10    [17]     0.998   0.978       1.000                -6                 -6             -6
                                                    1.7710            3.3610        1.7710
            [18]     0.87    0.86        0.94       0.358              0.016          0.127
            [19]     0.998   0.761       1.000                -3                 -8             -8
                                                    1.6110            3.1210        4.9310
            [20]     0.999   0.986       1.000                -4                 -6             -7
                                                    6.2910            1.2510        5.7310
20    10    [21]     0.998   0.975       1.000                -3                 -9             -7
                                                    1.9310            1.0510        2.9310
            [22]     0.938   0.874       0.968      0.134                        -3   0.018
                                                                       3.6310
            [23]     0.835   0.782       0.998      0.509              0.027          0.258
            [24]     0.971   0.962       0.996      0.058                        -4            -3
                                                                       7.1410        3.210


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                                           22

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    the Rayleigh distribution. Commun. In Statist. Theor. Meth. Vol. 29, No.1, pp. 95 –
    107.
19 - ------------ (2002). Reliability estimation in a generalized life – model with
    application to Burr XII. IEEE Trans. Reliab. Vol. 51, No. 3, pp. 337 – 343.
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    data using Burr – XII model. IEEE Trans. reliab. Vol. 54, No. 1 , pp. 34 – 42.
21 - Tang, Y., Xie, M. and Goh, T. N. (2003). Statistical analysis of a Weibull extension
    model. Commun. In Statist. Theor. Meth. Vol. 33, pp. 913 – 928.
22 – Thomas, D.R. and Wilson, W.M. (1972). Linear order statistic estimation for the
    two – parameter Weibull and extreme value distributions form type II progressively
    censored samples. Technometrics, vol. 14, pp. 679 – 691.
23 – Viveros, R. and Balakrishnan, N. (1994). Interval estimation of parameters of life
    from progressively censored data. Technometrics, vol. 36, pp. 84 – 91.
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    Vol. 73, No. 2 pp. 279 – 285.
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                                           22