Asymptotic normality of numbers of observations near order - PowerPoint PPT Presentation

Asymptotic normality of numbers of observations near order statistics from stationary processes Krzysztof Jasiński Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń December 1, 2016 Krzysztof Jasiński Asymptotic normality of numbers of observations near order

Notation: X = ( X n , n ≥ 1 ) – sequence of random variables (rv’s) with cdf F . X 1 : n ≤ . . . ≤ X n : n – order statistics based on ( X 1 , . . . , X n ) . supp( F ) – support of cdf F . We set γ 0 := inf supp ( F ) = inf { x ∈ R : F ( x ) > 0 } , γ 1 := sup supp ( F ) = sup { x ∈ R : F ( x ) < 1 } . By γ λ we denote the unique λ th quantile of F where λ ∈ ( 0 , 1 ) . Krzysztof Jasiński Asymptotic normality of numbers of observations near order

Notation: X = ( X n , n ≥ 1 ) – sequence of random variables (rv’s) with cdf F . X 1 : n ≤ . . . ≤ X n : n – order statistics based on ( X 1 , . . . , X n ) . supp( F ) – support of cdf F . We set γ 0 := inf supp ( F ) = inf { x ∈ R : F ( x ) > 0 } , γ 1 := sup supp ( F ) = sup { x ∈ R : F ( x ) < 1 } . By γ λ we denote the unique λ th quantile of F where λ ∈ ( 0 , 1 ) . Krzysztof Jasiński Asymptotic normality of numbers of observations near order

Notation: X = ( X n , n ≥ 1 ) – sequence of random variables (rv’s) with cdf F . X 1 : n ≤ . . . ≤ X n : n – order statistics based on ( X 1 , . . . , X n ) . supp( F ) – support of cdf F . We set γ 0 := inf supp ( F ) = inf { x ∈ R : F ( x ) > 0 } , γ 1 := sup supp ( F ) = sup { x ∈ R : F ( x ) < 1 } . By γ λ we denote the unique λ th quantile of F where λ ∈ ( 0 , 1 ) . Krzysztof Jasiński Asymptotic normality of numbers of observations near order

For a > 0 and 1 ≤ k ≤ n we define the following counting rv’s K k : n ( − a , 0 ) := # { j ∈ { 1 , . . . , n } : X j ∈ ( X k : n − a , X k : n ) } and K k : n ( 0 , a ) := # { j ∈ { 1 , . . . , n } : X j ∈ ( X k : n , X k : n + a ) } . Interpretation These two rv’s provide information on how many observations fall into the open left and right a -vincity of the order statistic X k : n . Their properties have been studied since 1997 - Pakes and Steutel discussed the behavior of K n : n ( − a , 0 ) . Krzysztof Jasiński Asymptotic normality of numbers of observations near order

For a > 0 and 1 ≤ k ≤ n we define the following counting rv’s K k : n ( − a , 0 ) := # { j ∈ { 1 , . . . , n } : X j ∈ ( X k : n − a , X k : n ) } and K k : n ( 0 , a ) := # { j ∈ { 1 , . . . , n } : X j ∈ ( X k : n , X k : n + a ) } . Interpretation These two rv’s provide information on how many observations fall into the open left and right a -vincity of the order statistic X k : n . Their properties have been studied since 1997 - Pakes and Steutel discussed the behavior of K n : n ( − a , 0 ) . Krzysztof Jasiński Asymptotic normality of numbers of observations near order

For a > 0 and 1 ≤ k ≤ n we define the following counting rv’s K k : n ( − a , 0 ) := # { j ∈ { 1 , . . . , n } : X j ∈ ( X k : n − a , X k : n ) } and K k : n ( 0 , a ) := # { j ∈ { 1 , . . . , n } : X j ∈ ( X k : n , X k : n + a ) } . Interpretation These two rv’s provide information on how many observations fall into the open left and right a -vincity of the order statistic X k : n . Their properties have been studied since 1997 - Pakes and Steutel discussed the behavior of K n : n ( − a , 0 ) . Krzysztof Jasiński Asymptotic normality of numbers of observations near order

We consider the extended version of these rv’s K k n : n ( A ) := # { j ∈ { 1 , . . . , n } : X k n − X j ∈ A } , 1 ≤ k n ≤ n , where A is a Borel subset of real numbers and k n / n → λ ∈ [ 0 , 1 ] . In the literature three cases are discussed: 1 the case of central order statistics, when λ ∈ ( 0 , 1 ) , 2 the case of extreme order statistics, when k n or n − k n is fixed, 3 the case of intermediate order statistics, when λ ∈ { 0 , 1 } and both k n and n − k n → ∞ . Krzysztof Jasiński Asymptotic normality of numbers of observations near order

We consider the extended version of these rv’s K k n : n ( A ) := # { j ∈ { 1 , . . . , n } : X k n − X j ∈ A } , 1 ≤ k n ≤ n , where A is a Borel subset of real numbers and k n / n → λ ∈ [ 0 , 1 ] . In the literature three cases are discussed: 1 the case of central order statistics, when λ ∈ ( 0 , 1 ) , 2 the case of extreme order statistics, when k n or n − k n is fixed, 3 the case of intermediate order statistics, when λ ∈ { 0 , 1 } and both k n and n − k n → ∞ . Krzysztof Jasiński Asymptotic normality of numbers of observations near order

Our purpose is to... . . . study limiting behavior of suitably centered and normed versions of rv’s K k n : n ( A ) := # { j ∈ { 1 , . . . , n } : X k n − X j ∈ A } , under some conditions. Krzysztof Jasiński Asymptotic normality of numbers of observations near order

Recent results of this problem With the assumption that X 1 , X 2 , . . . are i.i.d. rv’s ... ... and imposing some restrictions of the cdf F , Iliopoulos, Dembińska and Balakrishnan (2012) and Dembińska (2012) studied the case of central order statistics. All these restrictions require that F is continuous . Krzysztof Jasiński Asymptotic normality of numbers of observations near order

Dembińska (2014) studied the case of extreme order statistics when F is continuous with left bounded support. Moreover, there were obtained analogous result for discontinuous F , devoted to three cases: extreme, intermediate and central. Discontinuity of F here means that the corresponding λ th quantiles (including γ 0 , γ 1 ) are not accumulation points of its support. We extend the discontinuous results to strictly stationary and ergodic observations satisfying an extra condition which guarantees some local independence. Krzysztof Jasiński Asymptotic normality of numbers of observations near order

Dembińska (2014) studied the case of extreme order statistics when F is continuous with left bounded support. Moreover, there were obtained analogous result for discontinuous F , devoted to three cases: extreme, intermediate and central. Discontinuity of F here means that the corresponding λ th quantiles (including γ 0 , γ 1 ) are not accumulation points of its support. We extend the discontinuous results to strictly stationary and ergodic observations satisfying an extra condition which guarantees some local independence. Krzysztof Jasiński Asymptotic normality of numbers of observations near order

Lemma 1 Let X = ( X n , n ≥ 1 ) be a strictly stationary and ergodic sequence of rv’s with an arbitrary cdf F and 1 ≤ k n ≤ n such that k n / n → λ ∈ [ 0 , 1 ] . Then a . s . (a) X k n : n − → γ 0 as n → ∞ provided that λ = 0 and γ 0 > −∞ , a . s . (b) X k n : n − → γ 1 as n → ∞ provided that λ = 1 and γ 1 < ∞ , a . s . − → γ as n → ∞ if λ ∈ ( 0 , 1 ) . (c) X k n : n Krzysztof Jasiński Asymptotic normality of numbers of observations near order

Main result - discontinuous F Let X = ( X n , n ≥ 1 ) strictly stationary sequence with discontinuous F . Let � ∞ n = 1 α X ( n ) < ∞ and 1 ≤ k n ≤ n , k n / n → λ ∈ [ 0 , 1 ] . Moreover, assume that if λ = 0, then γ 0 > −∞ ; if λ = 1, then γ 1 < ∞ ; if λ ∈ ( 0 , 1 ) , then there exists a unique λ th quantile γ λ . Then, for any A ∈ B ( R ) , as n → ∞ , √ n � K k n : n ( A ) � d → N ( 0 , σ 2 − P ( X 1 ∈ γ λ − A ) − X ) , n provided that γ λ is not an accumulation point of support of F and σ 2 � P ( X 1 ∈ γ λ − A , X j ∈ γ λ − A ) − p 2 � X = p ( 1 − p ) + 2 � ∞ � = 0 , j = 2 where p = P ( X 1 ∈ γ λ − A ) . Krzysztof Jasiński Asymptotic normality of numbers of observations near order

Sketch of proof � ∞ n = 1 α X ( n ) < ∞ implies that α X ( n ) → 0. So X is an α -mixing process. Lemma 2 Under the assumptions of Theorem, the random sequence Y = ( Y n , n ≥ 1 ) , where Y n = I ( X n ∈ γ λ − A ) − P ( X 1 ∈ γ λ − A ) , n ≥ 1, satisfies the following conditions: (a) E ( Y 1 ) = 0 , P ( | Y 1 | ≤ C ) = 1 for some constant C , (b) Y is an α -mixing strictly stationary process , ∞ � α Y ( n ) < ∞ , where α Y ( n ) , n ≥ 1 , (c) n = 1 are the α -mixing coefficients of the process Y . Krzysztof Jasiński Asymptotic normality of numbers of observations near order

Sketch of proof Combining Lemma 2, Theorem 10.3 of Bradley (2007) and Theorem 2 of Rio (1995) yield σ 2 X > 0. Let n U n := √ n � K k n : n ( A ) � Y i � √ n . − P ( X 1 ∈ γ λ − A ) , W n := n i = 1 Using central limit theorem (Theorem 10.3 of Bradley (2007)) d → N ( 0 , σ 2 we get W n − X ) . X = ( X n , n ≥ 1 ) is strictly stationary and α -mixing, so is ergodic too. By Lemma 1, discontinuity of F implies P ( X k n : n = γ λ for all large n ) = 1 . Krzysztof Jasiński Asymptotic normality of numbers of observations near order