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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) i1 m Where C = n(n-1-R1)(n-2-R1-R2) … (n- Σ (Ri+1)); and f(x) and F(x) are given j1 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 mm( w i )e i1 i exp[n - (R +1)e i ] i (7) i1 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) i1 i1 i1 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. i1 i1 L m m z = + n - Σ (Ri + 1) e i. i1 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 i1 i1 2 1 m z = [m - Σ zi [ (Ri + 1) (1 + zi) e i -1] log2zi] 2 2 i1 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 i1 i1 2 m z = Σ (Ri + 1) [1 – zi] e i -n i1 2 m z = Σ (Ri + 1) zi e i log zi (11) i1 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 , i1 m D1 = Σ zi i1 6 m z D2 = Σ (Ri + 1) e i , i1 D3 = D1 + (n – D2) , A q J1 = mm-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 = mm-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 = mm-1 D0eD3 d d d 0 0 0 Aq J6 = m+1m-1 D0eD3 d d d 0 0 0 qA J7 = mm 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 dd 0 0 0 A q J9 = m+1 m D0 D5exp [D3 + z] d dd 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 + vD - 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 jm-i1 * * * 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-i1 * * 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 3.4610 2.4910 (0.489, 1.345) (0.436,0.488) (0.00017,0.0062) 0.369 0.491 0.011 -5 -4 [2] 2.178 1.710 1.7210 (0.04,0.532) (0381,0.501) (0.007,0.042) 0.304 0.476 0.011 -4 -5 [3] 1.492 7.5910 9.5710 (0.056,0.421) (0.432,0.491) (0.007,0.072) 1.032 0.809 0.00689 -4 [4] 1.213 0.196 3.2110 (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 3.6710 3.7910 (0.72,1.716) (0.391,0.545) (0.0003,0.0084) 0.276 0.615 0.054 -3 [6] 1.513 0.014 1.4210 (0.158,0.345) (0.428,0.684) (0.034,0.072) 1.07 0.451 0.0085 -5 [7] 2.103 0.0024 5.610 (0.64,1.34) (0.321,0.568) (0.0012,0.032) 1.71 0.663 0.012 -4 [8] 0.044 0.026 1.210 (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 1.1410 (1.057,1.69) (0.439,0.544) (0.00033,0.0007) 1.2 0.405 0.00036 -3 -7 [10] 20 0.09 8.9910 4.0810 (0.813,1.59) (0.325,0.485) (0.0002,0.00053) 0.07 0.594 0.026 -3 -4 [11] 2.046 8.910 6.310 (0.05,0.12) 0.402,0.683) (0.007,0.032) 0.223 0.426 0.0038 -3 -6 [12] 1.631 5.510 8.1110 (0.216,0.23) (0.424,0.428) (0.0036,0.0041) 1.5 0.502 0.028 -6 -4 [13] 2.088 6.010 7.0510 (1.32,1.67) (0.421,0.632) (0.013,0.042) 1.33 0.522 0.00058 -4 [14] 2.51 0.0032 2.7310 (0.732,2.27) (0.48,0.56) (0.00014,0.0011) 2.196 0.483 0.00049 -4 -7 [15] 0.484 3.010 2.610 (1.66,2.7) (0.407,0.558) (0.00026,0.0007) 1.449 0.508 0.0013 -3 -4 [16] 0.332 3.810 2.1310 (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 9.2710 (-6.57,18.57) (0.386,1.214) (0.0,0.00009) 0.066 0.557 0.022 -3 -4 [18] 2.057 3.2910 4.5910 (0.034,0.085) (0.341,0.751) (0.008,0.32) 2.98 0.606 0.00103 -10 [19] 2.191 0.11 8.8710 (0.528,6.1) (0.528,0.685) (-0.00004,0.0024) 2.016 0.516 0.00041 -7 [20] 0.266 0.011 3.4910 (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 6.6510 (0.94,0.945) (0.643,0.650) (0.00107,0.0011) 0.087 0.543 0.017 -2 -4 [22] 1.997 1.8510 2.7210 (0.071,0.127) (0.481,0.586) (0.006,0.021) 0.04 0.522 0.021 -4 -4 [23] 2.133 4.7210 3.910 (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 2.9410 7.7810 (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 3.4510 3.6710 2.5310 5.4710 [2] -10 -10 -11 0.0 0.0 0.0 1.0410 1.6510 9.410 [3] 0.247 0.13 -3 -14 0.0 -15 2.1710 1.5110 2.0110 [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 1.3410 -1.8810 -1.7110 [6] 0.118 0.016 -3 0.0 0.0 0.0 3.9910 [7] -7 -7 -8 -4 0.0 -15 1.4610 1.6210 7.8110 2.410 3.6410 [8] -3 -4 -5 -6 -7 -8 6.410 6.0510 7.4110 3.910 1.110 2.310 [9] 50 35 0.307 0.033 -3 0.019 -6 -7 9.0710 -3.7610 -1.6810 [10] 20 0.197 0.041 -5 0.011 -6 -7 8.2210 -5.410 -6.410 [11] -12 -12 -12 0.0 0.0 0.0 1.0310 1.310 1.7410 [12] -3 -4 -5 -6 -8 -8 3.4510 9.1710 9.8310 3.1310 5.3210 5.110 [13] -9 -9 -4 0.0 0.0 0.0 5.710 7.0310 9.7510 [14] 0.392 0.019 -4 -3 -5 -7 2.2210 8.9410 3.110 5.110 [15] 0.273 0.038 -4 0.01 -5 -6 1.1710 -2.110 -1.7810 [16] 0.457 0.04 -4 -4 -5 -6 1.510 9.810 -4.210 -2.910 [17] 50 10 6.415 0.211 -5 -4 -4 -5 2.4310 -2.9510 -2.9510 -1.16910 [18] -10 -10 -10 0.0 0.0 0.0 6.4410 8.02910 8.2210 [19] 1.589 0.04 -4 0.070 -3 -5 7.2310 1.0210 8.9310 [20] 0.261 0.017 -5 -4 -6 -7 7.7610 4.6110 5.61410 8.8410 [21] 20 10 -3 -4 -6 -7 -10 -10 1.3210 1.8610 3.6210 2.4610 5.5610 2.3410 [22] -9 -9 -9 0.0 0.0 0.0 8.6710 7.910 6.1210 [23] -13 -13 -13 0.0 0.0 0.0 5.710 9.510 1.3410 [24] 20 5 -8 -8 -8 0.0 0.0 0.0 2.9910 4.4610 2.7210 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 1.2910 2.6810 8.84510 [2] 0.951 0.892 0.985 0.124 -3 0.015 2.2510 [3] 0.98 0.931 0.999 0.036 -4 -3 3.2810 1.2110 [4] 0.858 0.781 0.901 0.396 -3 0.156 2.110 20 [5] 0.999 0.958 1.000 -4 -7 -7 8.0110 4.5510 3.4210 [6] 0.879 0.860 0.934 0.272 0.14 0.073 [7] 0.987 0.980 0.991 -4 0.012 -4 1.3410 3.7710 [8] 0.999 0.987 1.000 -4 -6 -6 2.5510 2.3510 1.2810 50 35 [9] 0.998 0.964 1.000 -4 -7 -7 6.4610 6.2710 5.4710 [10] 0.998 0.976 1.000 -3 -6 -7 4.13410 1.5110 9.4510 [11] 0.79 0.65 0.85 0.689 0.43 0.473 [12] 0.995 0.978 1.000 -4 -6 -5 4.610 8.710 1.0410 [13] 0.896 0.850 0.948 0.24 0.01 0.057 [14] 0.998 0.972 1.000 -4 -7 -7 8.1510 6.7610 3.2510 [15] 0.999 0.965 1.000 -4 -7 -7 7.0410 6.9910 4.6410 [16] 0.999 0.959 1.000 -4 -7 -7 9.6410 2.210 1.7810 50 10 [17] 0.998 0.978 1.000 -6 -6 -6 1.7710 3.3610 1.7710 [18] 0.87 0.86 0.94 0.358 0.016 0.127 [19] 0.998 0.761 1.000 -3 -8 -8 1.6110 3.1210 4.9310 [20] 0.999 0.986 1.000 -4 -6 -7 6.2910 1.2510 5.7310 20 10 [21] 0.998 0.975 1.000 -3 -9 -7 1.9310 1.0510 2.9310 [22] 0.938 0.874 0.968 0.134 -3 0.018 3.6310 [23] 0.835 0.782 0.998 0.509 0.027 0.258 [24] 0.971 0.962 0.996 0.058 -4 -3 7.1410 3.210 REFERENCES: 1 – Aarset, M.V. (1987). How to identify bathtub hazard rate. IEEE Trans. Reliab. R - 36, pp.106 – 108. 2 – Aggarwala, R. (2001).Progressive censoring: a review, " in Hand book of statistics, vol. 20,: Advances in Reliability.Balakrishnan,N. and Rao. Eds. 2001, pp. 373 – 429. 3 – Balakrishnan, N. and Sandhu, R.A. (1995). A simple simulational algorithm for generating progressive type II censored samples. The American Statistician, vol. 49, No. 2, pp. 229 – 230. 4 – Balakrishnan, N. and Aggarwala, R. (2000). Progressive censoring: Theory, method and applications. Boston: Birkhauser and publishers. 5 – Basu, A. P. and Ebrahimi, N. (1991). Bayesian approach to life testing and reliability estimation using asymmetric loss function. J. Statistical Planning & Inference. Vol. 29, pp. 21 – 31. 6 – Cacciari, M. and Montanari , G. C. (1987). A method to estimate the Weibull Parameters for progressively censored test. IEEE Trans. Reliab. Vol. R – 36, pp. 87- 93. 21 7 – Cohen, A.C. (1963). 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