Sharp bounds on the expectations of linear combinations of k th - PowerPoint PPT Presentation

Sharp bounds on the expectations of linear combinations of k th records expressed in the Gini mean difference units 1/14 Sharp bounds on the expectations of linear combinations of k th records expressed in the Gini mean difference units Pawe� l Marcin Kozyra and Tomasz Rychlik Institute of Matematics, Polish Academy of Sciences ´ Sniadeckich 8, 00 656 Warsaw, Poland e-mails: pawel m kozyra@wp.pl and trychlik@impan.pl XLII Conference on Mathematical Statistics B¸ edlewo November 28 - December 2, 2016

Sharp bounds on the expectations of linear combinations of k th records expressed in the Gini mean difference units 2/14 k th records: X 1 , X 2 , . . . — i.i.d. with some continuous common distribution function F and finite mean µ . R 1 , k , R 2 , k , . . . – k th record values based on X 1 , X 2 , . . . def = increasing subsequence of non-decreasing sequence of k th maxima in X 1 , X 2 , . . . .

Sharp bounds on the expectations of linear combinations of k th records expressed in the Gini mean difference units 3/14 Problem: Provide sharp bounds for �� n �� i =1 c i ( R i , k − µ E ∆ where c = ( c 1 , . . . , c n ) ∈ R n — arbitrarily fixed vector of combination coefficients, and ∆ = E | X 1 − X 2 | = E ( X 2:2 − X 1:2 ) the Gini mean difference of F .

Sharp bounds on the expectations of linear combinations of k th records expressed in the Gini mean difference units 4/14 Solution: �� n �� i =1 c i ( R i , k − µ 0 < u < 1 Ψ c , k ( u ) ≤ E inf ≤ sup Ψ c , k ( u ) , ∆ 0 < u < 1 where   n − 1 n n  [ − k ln(1 − u )] i Ψ c , k ( u ) = 1 2 u (1 − u ) k − 1 � � � c j − c j .  i ! i =0 j = i +1 j =1

Sharp bounds on the expectations of linear combinations of k th records expressed in the Gini mean difference units 5/14 Idea of proof: G n , k ( F ( x )) — distribution function of R n , k , where n − 1 [ − k ln(1 − u )] i � G n , k ( u ) = 1 − (1 − u ) k , 0 < u < 1 . i ! i =0

Sharp bounds on the expectations of linear combinations of k th records expressed in the Gini mean difference units 6/14 Expectation of k th record spacing: � ∞ � ∞ E ( R i +1 , k − R i , k ) = x G i +1 , k ( F ( dx )) − x G i , k ( F ( dx )) −∞ −∞ � ∞ = x (( G i +1 , k − G i , k ) ◦ F ) ( dx ) −∞ ∞ �� � = x G i +1 , k ( F ( x )) − G i , k ( F ( x )) � � −∞ � ∞ − [ G i +1 , k ( F ( x )) − G i , k ( F ( x ))] dx −∞ � ∞ � i � − k ln(1 − F ( x )) � k � = 1 − F ( x ) dx i ! −∞ � i � − k ln(1 − F ( x )) � � k � = 1 − F ( x ) dx . i ! 0 < F ( x ) < 1

Sharp bounds on the expectations of linear combinations of k th records expressed in the Gini mean difference units 7/14 Expectation of standard spacing + useful observations: Similarly, � k � � � k − l dx . F l ( x ) � E ( X l +1: k − X l : k ) = 1 − F ( x ) l 0 < F ( x ) < 1

Sharp bounds on the expectations of linear combinations of k th records expressed in the Gini mean difference units 7/14 Expectation of standard spacing + useful observations: Similarly, � k � � � k − l dx . F l ( x ) � E ( X l +1: k − X l : k ) = 1 − F ( x ) l 0 < F ( x ) < 1 In particular � ∆ = E ( X 2:2 − X 1:2 ) = 2 F ( x )[1 − F ( x )] dx . 0 < F ( x ) < 1

Sharp bounds on the expectations of linear combinations of k th records expressed in the Gini mean difference units 7/14 Expectation of standard spacing + useful observations: Similarly, � k � � � k − l dx . F l ( x ) � E ( X l +1: k − X l : k ) = 1 − F ( x ) l 0 < F ( x ) < 1 In particular � ∆ = E ( X 2:2 − X 1:2 ) = 2 F ( x )[1 − F ( x )] dx . 0 < F ( x ) < 1 � � � k 1 Also, R 1 , k = X 1: k and µ = E j =1 X j : k . k

Sharp bounds on the expectations of linear combinations of k th records expressed in the Gini mean difference units 8/14 Spacing representations:   k  R n , k − R 1 , k − 1 � E ( R n , k − µ ) = ( X j : k − X 1: k ) E  k j =1   n − 1 j − 1 k − 1 � � � � � = E R i +1 , k − R i , k ( X l +1: k − X l : k )   k i =1 j =2 l =1 � n − 1 k − 1 � k − l � � � � = R i +1 , k − R i , k − ( X l +1: k − X l : k ) , E k i =1 l =1 and

Sharp bounds on the expectations of linear combinations of k th records expressed in the Gini mean difference units 8/14 Spacing representations:   k  R n , k − R 1 , k − 1 � E ( R n , k − µ ) = ( X j : k − X 1: k ) E  k j =1   n − 1 j − 1 k − 1 � � � � � = E R i +1 , k − R i , k ( X l +1: k − X l : k )   k i =1 j =2 l =1 � n − 1 k − 1 � k − l � � � � = R i +1 , k − R i , k − ( X l +1: k − X l : k ) , E k i =1 l =1 and n n � n − 1 k − 1 � k − l � � � � � � c i ( R n , k − µ )= E R i +1 , k − R i , k − ( X l +1: k − X l : k ) . E c i k i =1 i =1 i =1 l =1

Sharp bounds on the expectations of linear combinations of k th records expressed in the Gini mean difference units 9/14 Bounds: By the integral representations of spacing expectations n � � c i ( R n , k − µ ) = Ψ c , k ( F ( x ))2 F ( x )[1 − F ( x )] dx E 0 < F ( x ) < 1 i =1 ≤ sup Ψ c , k ( u ) · ∆ 0 < u = F ( x ) < 1 � � ≥ 0 < u = F ( x ) < 1 Ψ c , k ( u ) · ∆ inf .

Sharp bounds on the expectations of linear combinations of k th records expressed in the Gini mean difference units 10/14 Upper bound attainability: � Ψ c , k ( F ( x ))2 F ( x )[1 − F ( x )] dx = sup Ψ c , k ( u ) · ∆ 0 < F ( x ) < 1 0 < u = F ( x ) < 1 when u 0 = arg sup Ψ c , k ( u ) ∈ (0 , 1): F ( x ) = either 0 or 1 or u 0 , i.e. for two-point distributions F u 0 = u 0 δ a + (1 − u 0 ) δ b for any a < b .

Sharp bounds on the expectations of linear combinations of k th records expressed in the Gini mean difference units 10/14 Upper bound attainability: � Ψ c , k ( F ( x ))2 F ( x )[1 − F ( x )] dx = sup Ψ c , k ( u ) · ∆ 0 < F ( x ) < 1 0 < u = F ( x ) < 1 when u 0 = arg sup Ψ c , k ( u ) ∈ (0 , 1): F ( x ) = either 0 or 1 or u 0 , i.e. for two-point distributions F u 0 = u 0 δ a + (1 − u 0 ) δ b for any a < b . Bound attainability for continuous parent distributions: in the limit, by continuous approximation of two-point distribution.

Sharp bounds on the expectations of linear combinations of k th records expressed in the Gini mean difference units 10/14 Upper bound attainability: � Ψ c , k ( F ( x ))2 F ( x )[1 − F ( x )] dx = sup Ψ c , k ( u ) · ∆ 0 < F ( x ) < 1 0 < u = F ( x ) < 1 when u 0 = arg sup Ψ c , k ( u ) ∈ (0 , 1): F ( x ) = either 0 or 1 or u 0 , i.e. for two-point distributions F u 0 = u 0 δ a + (1 − u 0 ) δ b for any a < b . Bound attainability for continuous parent distributions: in the limit, by continuous approximation of two-point distribution. Cases sup Ψ c , k ( u ) = either lim u ց 0 Ψ c , k ( u ) or lim u ր 1 Ψ c , k ( u ): double limit: continuous approximation of two-point distribution F u , and u ց 0 ( u ր 1, respectively).

Sharp bounds on the expectations of linear combinations of k th records expressed in the Gini mean difference units 10/14 Upper bound attainability: � Ψ c , k ( F ( x ))2 F ( x )[1 − F ( x )] dx = sup Ψ c , k ( u ) · ∆ 0 < F ( x ) < 1 0 < u = F ( x ) < 1 when u 0 = arg sup Ψ c , k ( u ) ∈ (0 , 1): F ( x ) = either 0 or 1 or u 0 , i.e. for two-point distributions F u 0 = u 0 δ a + (1 − u 0 ) δ b for any a < b . Bound attainability for continuous parent distributions: in the limit, by continuous approximation of two-point distribution. Cases sup Ψ c , k ( u ) = either lim u ց 0 Ψ c , k ( u ) or lim u ր 1 Ψ c , k ( u ): double limit: continuous approximation of two-point distribution F u , and u ց 0 ( u ր 1, respectively). Similarly for lower bounds.

Sharp bounds on the expectations of linear combinations of k th records expressed in the Gini mean difference units 11/14 Special cases: single k th record R n , k : Bounds: extremes of � i � n − 1 � � − k ln(1 − u ) Ψ n , k ( u ) = 1 � (1 − u ) k − 1 − 1 . 2 u i ! i =0

Sharp bounds on the expectations of linear combinations of k th records expressed in the Gini mean difference units 11/14 Special cases: single k th record R n , k : Bounds: extremes of � i � n − 1 � � − k ln(1 − u ) Ψ n , k ( u ) = 1 � (1 − u ) k − 1 − 1 . 2 u i ! i =0 Upper bounds: k = 1 ⇒ Ψ n , k (1 − ) = + ∞ Ψ n , k (0+) = 1 k ≥ 2 and n = 2 ⇒ 2 , n , k ( u ) = 0) > 1 Ψ n , k ( unique solution to Ψ ′ k = 2 and n ≥ 3 ⇒ 2 , n , k ( u ) = 0) > 1 Ψ n , k ( smaller of 2 solutions to Ψ ′ k ≥ 3 and n ≥ 3 ⇒ 2 .

Sharp bounds on the expectations of linear combinations of k th records expressed in the Gini mean difference units 12/14 Special cases: single k th record R n , k cont.: Lower bounds: Ψ n , k (0+) = 1 k = 1 ⇒ 2 Ψ n , k (1 − ) = − 1 k = 2 and n ≥ 2 ⇒ 2 , n , k ( u ) = 0) < − 1 Ψ n , k ( unique solution to Ψ ′ k ≥ 3 and n = 2 ⇒ 2 , n , k ( u )=0) < − 1 Ψ n , k ( greater of 2 solutions to Ψ ′ k ≥ 3 and n ≥ 3 ⇒ 2 .