DS-optimal designs for RCR models with heteroscedastic errors - PowerPoint PPT Presentation

DS-optimal designs for RCR models with heteroscedastic errors Mateusz Wilk 1 , 2 , Aleksander Zaigrajew 1 1 WMiI UMK, 2 IP PAN 28 11 2016

Model y = β 1 + β 2 x + e ( x ) , x ∈ [ a , b ] ⊂ R β = [ β 1 β 2 ] T E [ β ] = β 0 cov [ β ] = ∆ E [ e ( x )] = 0 var [ e ( x )] = g ( x )

Estimator � − 1 F T V − 1 Y ˆ F T V − 1 F � β 0 =     1 x 1 y 1 1 x 2 y 2     F =  Y =  V = cov [ Y ]     . . . . . .       1 x n y n

Design � x ( 1 ) x ( 2 ) . . . x ( k ) � ζ ∼ ω 1 ω 2 . . . ω k k � ω 1 , ω 2 , . . . , ω k ∈ ( 0 , 1 ) , ω i = 1 i = 1 x ( 1 ) , x ( 2 ) , . . . , x ( k ) are distinct

Information matrix b � [ 1 x ] T [ 1 x ] / g ( x ) d ζ ( x ) M 0 ( ζ ) = a M ( ζ ) = ( M 0 ( ζ ) − 1 + n ∆) − 1 M ( ζ ) ≈ 1 = 1 � − 1 � cov [ ˆ nF T V − 1 F β 0 ] n

Lemma 1 . If ag ( b ) � bg ( a ) and g ( x ) � ( b − x ) g ( a ) + ( x − a ) g ( b ) b − a abg ( x ) � bx ( b − x ) g ( a ) + ax ( x − a ) g ( b ) b − a

or if ag ( b ) � bg ( a ) and g ( x ) � ( b − x ) g ( a ) + ( x − a ) g ( b ) b − a abg ( x ) � bx ( b − x ) g ( a ) + ax ( x − a ) g ( b ) b − a

then for any given k -point design ζ there exists 2-point design � � a b ζ ∗ = 1 − ω ∗ ω ∗ such that M 0 ( ζ ∗ ) � M 0 ( ζ ) , and M ( ζ ∗ ) � M ( ζ ) .

Example. If a , b > 0 then the function g ( x ) = cx , c > 0 , satisfies all conditions from Lemma 1.

DS -optimality A design ζ ∗ is said to be DS -optimal for the BLUE of β 0 , if � � � β 0 − ˆ β 0 ( ζ ∗ ) � < ǫ → max ∀ ε > 0 P

DS -optimality ◮ A design is DS -optimal if and only if it is simultaneously D - and E - optimal, i.e. | M ( ζ ) | → max and λ min [ M ( ζ )] → max. ◮ If a design is DS -optimal, then it is also A -optimal, i.e. tr M ( ζ ) − 1 → min.

Lemma 2 . For the FCR model ( ∆ = 0) condition ( 1 + a 2 ) / g ( a ) = ( 1 + b 2 ) / g ( b ) is necessary and sufficient for the existence of the DS -optimal design ζ ∗ among all 2-point designs with the support at endpoints. It has the form � a � b ζ ∗ = . 1 / 2 1 / 2

Example. The function g ( x ) = cx , c > 0 , satisfies the above condition if and only if ab = 1 .

Lemma 3 . For the RCR model ( ∆ = diag ( δ 1 , δ 2 ) ) the DS -optimal design ζ ∗ among all 2-point designs with the support at endpoints exists only for two cases: ( i ) a = − b and g ( b ) = g ( − b ) , � − b � b ζ ∗ = here 1 / 2 1 / 2

a 2 � = b 2 and ( ii ) n ( δ 1 − δ 2 ) = ( 1 + a 2 ) g ( b ) − ( 1 + b 2 ) g ( a ) , b 2 − a 2 here � � a b ζ ∗ = , 1 − ω ∗ ω ∗ where � ( 1 + a 2 ) / g ( a ) ω ∗ := . � � ( 1 + a 2 ) / g ( a ) + ( 1 + b 2 ) / g ( b )

Example. For the linear function g ( x ) = cx , c > 0 , (i) does not hold, while if a 2 � = b 2 then the DS -optimal design exists if and only if n ( δ 1 − δ 2 ) = c ( 1 − ab ) / ( a + b ) .

Literature ◮ Cheng, J., Yue R.-X., Liu, X. (2013), Optimal designs for random coefficient regression models with heteroscedastic errors , Communications in Statistics-Theory and Methods 42, 2798-2809. ◮ Liski, E.P., Luoma, A., Zaigraev, A. (1999), Distance optimality design criterion in linear models , Metrika 49, 193-211. ◮ Liski, E.P., Mandal, N.K., Shah, K.R., Sinha, B.K. (2002), Topics in Optimal Design , Lecture Notes in Statistics: 163. Springer-Verlag, New York.