Illustration of how to derive formulas used for the calculation of

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					Introduction
  Illustration of how to derive formulas used for
the calculation of exact least median of squares
(LMS) and exact least quantile of squares (LQS)
      estimators for all quantiles and all multiple
                         linear regression models


        Arzdar KİRACI
        Baskent University
Introduction
  Illustration of how to derive formulas used for
the calculation of exact least median of squares
(LMS) and exact least quantile of squares (LQS)
      estimators for all quantiles and all multiple
                          linear regression models
Introduction
  Illustration of how to derive formulas used for
the calculation of exact least median of squares
(LMS) and exact least quantile of squares (LQS)
      estimators for all quantiles and all multiple
                          linear regression models

yi   0  1 xi   i         Steele and Steiger 1986

         y a  yb               y a  yc     LQS ( xa  xc )
LQS
                      LQS
                                         1 
         xa  xb
1                       0
                                    2                 2
   Introduction
  Idea
                   (
                 r 
          s( ( yy1 1 1 11 xxx1
           1 y1 1 1
                   2 LQSLQS 2
                       LQS
                         LQS
                              )  r (   )
                            1 ))    sy(3y 1 1 1 x3 x3 )
                                    (3y3  1
                                       2     LQS LQS 2
                                               LQS
                                               LQS
                                                   3)
         Table 1: Data for the model yi=1xii
                              1       3 3


i 1     2 3    4   5  6   7   8
yi -118 62 122 -58 42 -55 -65 -100
xi -40 20 40 -20 13.3 -20 -20 200
Introduction
                             i   1    2   3    4     5      6   7   8

                             yi -118 62 122 -58 42          -55 -65 -100
                             xi -40   20 40    -20 13.3 -20 -20 200

Idea
         s1 ( y1  
                   1
                    LQS
                        x )  s3 ( y3  
                        1                     1
                                               LQS
                                                     x3 )
          s
          LQS
          sLQS
              y s
                s LQS
                    LQS
                    LQS
                       y y11  y73  118 55
                       y   y3  y 6 122   122
                                     118  65

 LQS
  LQS
        
         3
          1
          1    1
               3
               1   7
                   6
                   3    3
                        7
                         
                         6
                                             205
                                               3. 3
                                                 .95
 1
 1
          s
          s
          1
           LQS
           LQS
           3
           1  x s
               1
               1
               3s LQS
                   3
                   6
                   7
                    LQS
                    LQS
                       x x11  x73 40402040
                       x3
                        7
                        6
                           x3  x 6   
                                      40  20
Introduction
Idea             For the model yi=1xii

        sa ( y a  
                   1
                    LQS
                          xa )  sb ( yb     b
                                                LQS
                                                      xb )
              Two type of formulas give
       the candidates for the exact LQS estimator

                y a  yb                      y a  yb
       LQS
                                    LQS
                                            
                xa  xb                       xa  xb
       1                             1
   Theorethic work
   Lemma1: The relation that the (approximate) exact LQS estimator values satisfy: If LQSh has
a (local) global minimum at ( A)  LQS then the following equality (5) is satisfied for (y+z+1) data
{(x ihy ,y ihy ),…,(x ih ,y ih ),…,(x i ,y i )}  Dn, where y+z  d.
                                   h z       h z


LQSh(     ) = ri 2 (
         LQS
                h y
                           ) = ... = ri 2 ( LQS) = … = ri 2 ( LQS)
                         LQS
                                          h                h z




 Theorem (for exact LQS estimator): Let A={(x ih y ,y ihy ),…,(x ih ,y ih ),…,(x i          ,y i          )}Dn
                                                                                       h z          h z


be the data that satisfy (5) at ( A) LQS, where y+z  d. If  L is calculated for some {(xa,ya),
   (xb,yb), … }  A and suitable choices of signs as defined in (9) then ( A=L) LQS=L.

              sih yih  sih  y yih  y              sih xih  sih  y xih y 
      Ydx1 =           ...
                                          and Xdxd = 
                                                                 ...
                                                                                 
                                                                              
              si yi  si yi                          si xi  si xi 
              h h          h z   h z               h h          h z   h z 




           Y = X                LQS
                                                        LQS
                                                                  =X Y ­1
         Theorethic work
         Result


Model:
 Model:               x xi x
              yi=1xi11i1i ii
               Model:aK2 xi cxa 2i yy)ic 1 b2 ( xba2  ac  c c 2 (c 2 b yb a a
    Model: Model:+1( y y(i==x)iii ( y ) y x) x y  ( y)y )
                    yi= x                   y y
                                              b  xb          y
               1 
                  K      1
                                                            (x  x 2 )
                       x a1  ( x x 1y ( x2  x b2 ) a cb1  ( yca  x x) 1 x 1 2( x 2 xb )
Estimators: 1:Estimatorbabxc ( 2)K xcy1 )y xx2 (xb2 a 2) c x( cxa( ya 2xby )) 2
                                             b 
    Estimator: K  1K xK ya( y x a1 y)  c  y b1  y c )  a  ( cy  ay) b
 Estimators                        2
                                    1:
                          a1  b  y c1b xb1 ( cy y a yy xc1 1 a
                2   2 x a xa1xb ( xb 2  xc K) x a b1  axc  xba 2 )  xc1  ( xa 2b  xb 2 )
                    Estimator:c 2 ) x021 xc 2 xxba22 )  xc1  ( xa 2  xb 2 )
                                  
                       x a1  ( xb 2  x                    (x (
                                      x a 2  ( y c  yb )   2y( y a  y c )  xc 2  ( yb  y a )
                                                      b

                  y y  x d ( K
            y Estimator2: 1 ) x aK xc()x 2 a  2b)       yaa y bxdy2 (Ky  y ) ( x  x )
                                                                x
                                 a11 (11  c 2 )  xb1  ( xcc2  2x a a )  xc1  ( x aa2  xcb 2 )
                            K
              a K y d a (1 ca1
                                   x K x2  
                            x
      0  2: 0  
       K
                                                    x                    2 b
                                                                         
                                                                
 Estimators                                   b
                 2      2 K 2 x a1  ( yb2 y c x xb12x y c  yx a)  xb1  ( y a  yb )
                                                           ) a 2  (b      a  xc
                                                                                                2
                          2 
                 Y = X                                             ­1
                             LQSa1  ( xb 2 LQS  xb1  ( xc 2  xa 2 )  xc1  ( xa 2  xb 2 )
                                   x                 xc 2 )
                                                             =X Y
 Result Thank you



www.baskent.edu.tr/~arzdar

				
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