Optimal choice of order statistics under confidence region estimation - PowerPoint PPT Presentation

Optimal choice of order statistics under confidence region estimation in case of large samples cko 1 Aleksander Zaigrajew 2 Magdalena Alama-Bu´ 1 Instytut Matematyki i Fizyki Uniwersytet Technologiczno-Przyrodniczy w Bydgoszczy 2 Wydział Matematyki i Informatyki Uniwersytet Mikołaja Kopernika w Toruniu B˛ edlewo, 01.12.2016 B˛ edlewo, 01.12.2016 1 / 27

Outline Introduction 1 L-statistics 2 Main results 3 Examples 4 References 5 B˛ edlewo, 01.12.2016 2 / 27

X = ( X 1 , X 2 , . . . , X n ) is a sample from the distribution F θ ∈ F , � � � u − θ 1 � , u ∈ R 1 , θ 1 ∈ R 1 , θ 2 ∈ R 1 F = F θ : F θ ( u ) = F ( 0 , 1 ) + θ 2 F ( 0 , 1 ) is a distribution function of the standard distribution, while f is the density function 1 − α is a given confidence level B˛ edlewo, 01.12.2016 3 / 27

X = ( X 1 , X 2 , . . . , X n ) is a sample from the distribution F θ ∈ F , � � � u − θ 1 � , u ∈ R 1 , θ 1 ∈ R 1 , θ 2 ∈ R 1 F = F θ : F θ ( u ) = F ( 0 , 1 ) + θ 2 F ( 0 , 1 ) is a distribution function of the standard distribution, while f is the density function 1 − α is a given confidence level A strong confidence region of level 1 − α for θ B ( X ) : R n → B 2 such that P θ ( θ ∈ B ( X )) = 1 − α B˛ edlewo, 01.12.2016 3 / 27

X = ( X 1 , X 2 , . . . , X n ) is a sample from the distribution F θ ∈ F , � � � u − θ 1 � , u ∈ R 1 , θ 1 ∈ R 1 , θ 2 ∈ R 1 F = F θ : F θ ( u ) = F ( 0 , 1 ) + θ 2 F ( 0 , 1 ) is a distribution function of the standard distribution, while f is the density function 1 − α is a given confidence level A strong confidence region of level 1 − α for θ B ( X ) : R n → B 2 such that P θ ( θ ∈ B ( X )) = 1 − α Risk function R ( θ, B ) = E θ λ 2 ( B ( X )) → min B˛ edlewo, 01.12.2016 3 / 27

Consider the vector � θ 1 − t 1 ( X ) � θ 2 T = T ( X , θ ) = t 2 ( X ) − 1 , . t 2 ( X ) B˛ edlewo, 01.12.2016 4 / 27

Consider the vector � θ 1 − t 1 ( X ) � θ 2 T = T ( X , θ ) = t 2 ( X ) − 1 , . t 2 ( X ) If A ∈ B 2 is such that � � P θ T ( X , θ ) ∈ A = 1 − α, then B ( X ) = ( t 1 ( X ) , t 2 ( X )) + t 2 ( X ) A . B˛ edlewo, 01.12.2016 4 / 27

Consider the vector � θ 1 − t 1 ( X ) � θ 2 T = T ( X , θ ) = t 2 ( X ) − 1 , . t 2 ( X ) If A ∈ B 2 is such that � � P θ T ( X , θ ) ∈ A = 1 − α, then B ( X ) = ( t 1 ( X ) , t 2 ( X )) + t 2 ( X ) A . If for any b > 0, a ∈ R : t 1 ( bX + a 1 n ) = bt 1 ( X ) + a , t 2 ( bX + a 1 n ) = bt 2 ( X ) , (1) B˛ edlewo, 01.12.2016 4 / 27

Consider the vector � θ 1 − t 1 ( X ) � θ 2 T = T ( X , θ ) = t 2 ( X ) − 1 , . t 2 ( X ) If A ∈ B 2 is such that � � P θ T ( X , θ ) ∈ A = 1 − α, then B ( X ) = ( t 1 ( X ) , t 2 ( X )) + t 2 ( X ) A . If for any b > 0, a ∈ R : t 1 ( bX + a 1 n ) = bt 1 ( X ) + a , t 2 ( bX + a 1 n ) = bt 2 ( X ) , (1) then � � � � P θ T ( X , θ ) ∈ A = P ( 0 , 1 ) T ( X , ( 0 , 1 )) ∈ A = 1 − α, and distribution of the vector T doesn’t depend on θ ( vector T is a pivot ). B˛ edlewo, 01.12.2016 4 / 27

B ( X ) = ( t 1 ( X ) , t 2 ( X )) + t 2 ( X ) A . R ( θ, B ) = E θ λ 2 ( B ( X )) = E θ t 2 2 ( X ) · λ 2 ( A ) → min Since vector T is a pivot, then � � � P ( 0 , 1 ) T ( X , ( 0 , 1 )) ∈ A = 1 − α ⇔ p ( u ) d u = 1 − α A Einmahl, Mason 1992 � A opt = { u : p ( u ) � z α } , gdzie p ( u ) d u = 1 − α A opt B˛ edlewo, 01.12.2016 5 / 27

Asymptotics of A opt p ( u ) - density function of the vector � − t 1 ( X ) 1 � t 2 ( X ) , t 2 ( X ) − 1 Alama-Bu´ cko, Nagajew, Zaigrajew 2006 If for n → ∞ density p ( u ) is two-dimentional normal distribution N 2 (( 0 , 0 ) , W ) , then for n → ∞ the set A opt = { u : p ( u ) � z α } can be approximated by ellipse 1 − α A 0 = { u ∈ R 2 : ϕ W ( u ) � z ′ z ′ α } , α = √ 2 π det W Then as n → ∞ √ λ 2 ( A opt ) → λ 2 ( A 0 ) = 2 π ( − ln ( 1 − α )) det W . B˛ edlewo, 01.12.2016 6 / 27

case k = 2 Zaigraev A., Alama-Bu´ cko M. (2013). On optimal choice of order statistics in large samples for the construction of confidence regions for the location and scale. Metrika 76(4), 577-593 t 1 ( X ) = X k : n F − 1 ( q ) − X m : n F − 1 ( p ) X m : n − X k : n t 2 ( X ) = (2) , F − 1 ( q ) − F − 1 ( p ) , F − 1 ( q ) − F − 1 ( p ) where X k : n and X m : n are central order statistics, that is k m n → p , n → q , 0 < p < q < 1 for example k = [ np ] + 1 , m = [ nq ] + 1 . while F − 1 ( p ) = inf { t ∈ R : F ( t ) � p } , 0 < p < 1 . B˛ edlewo, 01.12.2016 7 / 27

case any (fixed) k Consider the case, where t 1 and t 2 are L -statistics depending on k order statistics. Let 1 < r 1 < r 2 < ... < r k < n be the numbers od ordered statistics such that r i n → p i , 0 < p 1 < p 2 < ... < p k < 1 and k k � � t 1 ( X ) = a i X r i : n , t 2 ( X ) = b i X r i : n . i = 1 i = 1 Questions: ( a 1 , a 2 , ..., a k ) ( b 1 , b 2 , ..., b k ) p 1 < p 2 < ... < p k to minimise risk function as n → ∞ . B˛ edlewo, 01.12.2016 8 / 27

k k � � t 1 ( X ) = a i X r i : n , t 2 ( X ) = b i X r i : n . i = 1 i = 1 t 1 i t 2 satisfy (1) when k k � � a i = 1 , b i = 0 . i = 1 i = 1 t 1 i t 2 are asymptotically unbiased estimators for θ 1 i θ 2 if k k � a i F − 1 ( p i ) = 0 , � b i F − 1 ( p i ) = 1 . i = 1 i = 1 B˛ edlewo, 01.12.2016 9 / 27

k k � � t 1 ( X ) = a i X r i : n , t 2 ( X ) = b i X r i : n . i = 1 i = 1 t 1 i t 2 satisfy (1) when k k � � a i = 1 , b i = 0 . i = 1 i = 1 t 1 i t 2 are asymptotically unbiased estimators for θ 1 i θ 2 if k k � a i F − 1 ( p i ) = 0 , � b i F − 1 ( p i ) = 1 . i = 1 i = 1 Idea: choose t 1 and t 2 as ABLUE . B˛ edlewo, 01.12.2016 9 / 27

Estimators, (Sarhan & Greenberg, 1962) ABLUE for θ 1 and θ 2 are written by the formulas: t 1 ( X ) = K 2 Z 1 − K 3 Z 2 t 2 ( X ) = − K 3 Z 1 + K 1 Z 2 , , ∆ ∆ where � �� � f ( u j ) − f ( u j − 1 f ( u j ) X r i : n − f ( u j − 1 ) X r j − 1 : n k + 1 � Z 1 = p j − p j − 1 j = 1 � �� � f ( u j ) F − 1 ( p j ) − f ( u j − 1 ) F − 1 ( p j − 1 ) f ( u j ) X r i : n − f ( u j − 1 ) X r j − 1 : n k + 1 � Z 2 = p j − p j − 1 j = 1 k + 1 k + 1 ( f ( u j ) − f ( u j − 1 ) 2 ( f ( u j ) u j − f ( u j − 1 ) u j − 1 ) 2 � � K 1 = , K 2 = p j − p j − 1 p j − p j − 1 j = 1 j = 1 k + 1 ( f ( u j ) − f ( u j − 1 ))( f ( u j ) u j − f ( u j − 1 ) u j − 1 ) � ∆ = K 1 K 2 − K 2 K 3 = , 3 p j − p j − 1 j = 1 u i = F − 1 ( p i ) p 0 = 0 , p k + 1 = 1 , f ( u 0 ) = f ( u k + 1 ) = 0 , B˛ edlewo, 01.12.2016 10 / 27

√ d n ( t 1 ( X ) , t 2 ( X ) − 1 ) → N (( 0 , 0 ) , W ) where covariance matrix W can be written as K 2 − K 3   ∆ ∆   W =  .    − K 3 K 1 ∆ ∆ B˛ edlewo, 01.12.2016 11 / 27

√ d n ( t 1 ( X ) , t 2 ( X ) − 1 ) → N (( 0 , 0 ) , W ) where covariance matrix W can be written as K 2 − K 3   ∆ ∆   W =  .    − K 3 K 1 ∆ ∆ Lemma √ √ � − t 1 ( X ) 1 � d n t 2 ( X ) , t 2 ( X ) − 1 = n ( t 1 ( X ) , t 2 ( X ) − 1 ) B˛ edlewo, 01.12.2016 11 / 27

Then for n → ∞ R ( θ, B ) = 1 n · E θ t 2 2 ( X ) · λ 2 ( A opt ) B˛ edlewo, 01.12.2016 12 / 27

Then for n → ∞ R ( θ, B ) = 1 2 ( X ) · λ 2 ( A opt ) ∼ 1 n · E θ t 2 n · θ 2 2 · E ( 0 , 1 ) t 2 2 ( X ) · λ 2 ( A 0 ) B˛ edlewo, 01.12.2016 12 / 27

Then for n → ∞ R ( θ, B ) = 1 2 ( X ) · λ 2 ( A opt ) ∼ 1 n · E θ t 2 n · θ 2 2 · E ( 0 , 1 ) t 2 2 ( X ) · λ 2 ( A 0 ) √ ∼ 1 n · θ 2 2 · 2 π ( − ln ( 1 − α )) det W B˛ edlewo, 01.12.2016 12 / 27

det W = 1 ∆ ∆ = K 1 K 2 − K 2 3 Then ( p ∗ 1 , p ∗ 2 , ..., p ∗ k ) ∈ arg inf det W ⇔ ( p ∗ 1 , p ∗ 2 , ..., p ∗ k ) ∈ arg sup ∆( p 1 , p 2 , . . . , p k ) . B˛ edlewo, 01.12.2016 13 / 27

the formula of ∆ For any p 1 , p 2 , . . . , p k : k + 1 ( A i B j − A j B i ) 2 � ∆( p 1 , p 2 , . . . , p k ) = ( p i − p i − 1 )( p j − p j − 1 ) , i < j , i , j = 1 where A i = f ( u i ) − f ( u i − 1 ) , B i = u i f ( u i ) − u i − 1 f ( u i − 1 ) . u i = F − 1 ( p i ) p 0 = 0 , p k + 1 = 1 B˛ edlewo, 01.12.2016 14 / 27

the formula of ∆ For any p 1 , p 2 , . . . , p k : k + 1 ( A i B j − A j B i ) 2 � ∆( p 1 , p 2 , . . . , p k ) = ( p i − p i − 1 )( p j − p j − 1 ) , i < j , i , j = 1 where A i = f ( u i ) − f ( u i − 1 ) , B i = u i f ( u i ) − u i − 1 f ( u i − 1 ) . u i = F − 1 ( p i ) p 0 = 0 , p k + 1 = 1 Lemma Larger k ⇒ larger value of sup ∆ . B˛ edlewo, 01.12.2016 14 / 27

What about the behaviour of ∆ for p 1 → 0 p k → 1 ? or . it depends on the limit distribution of the properly normed X min and X max c n ( X max − d n ) , a n ( X min − b n ) Frechet distribution ( H 1 ), Weibull distr. ( H 2 ) i Gumbel distr. ( H 3 ) attraction to D ( H i ) , i = 1 , 2 , 3 : von Mises conditions we analyze behaviour of ∆ at the ends of the support by the von Mises conditions B˛ edlewo, 01.12.2016 15 / 27