# Illustration of how to derive formulas used for the calculation of by howardtheduck

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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
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