Uniqueness of characterization of distributions by regressions of - PowerPoint PPT Presentation

Introduction and some known results Equivalent condition Special cases for ℓ = 2 Uniqueness of characterization of distributions by regressions of generalized order statistics Mariusz Bieniek Institute of Mathematics University of Maria Curie–Skłodowska Lublin, Poland mariusz.bieniek@umcs.lublin.pl XLII Konferencja Statystyka Matematyczna Będlewo, 2016

Introduction and some known results Equivalent condition Special cases for ℓ = 2 Outline of the talk Introduction and some known results 1 Equivalent condition for uniqueness of characterization 2 Special cases for ℓ = 2 3

Introduction and some known results Equivalent condition Special cases for ℓ = 2 Generalized order statistics Fix n � 1 and parameters γ 1 , . . . , γ n > 0 Define c n − 1 = � n i = 1 γ i and m i = γ i − γ i + 1 − 1 , 1 � i � n − 1 . Assume that F is absolutely continuous cdf with density f and the support ( α, β ) Let ¯ F ( x ) = 1 − F ( x ) denote the survival function of F The random variables X ( r ) ∗ , 1 � r � n , are called generalized order statistics based on F , if their joint density is � ¯ � n − 1 � � ¯ � m i f ( x i ) f X ( 1 ) ∗ ,..., X ( n ) � γ n − 1 f ( x n ) , ∗ ( x 1 , . . . , x n ) = c n − 1 � F ( x i ) F ( x n ) i = 1 for −∞ < x 1 � . . . � x n < + ∞

Introduction and some known results Equivalent condition Special cases for ℓ = 2 Generalized order statistics Fix n � 1 and parameters γ 1 , . . . , γ n > 0 Define c n − 1 = � n i = 1 γ i and m i = γ i − γ i + 1 − 1 , 1 � i � n − 1 . Assume that F is absolutely continuous cdf with density f and the support ( α, β ) Let ¯ F ( x ) = 1 − F ( x ) denote the survival function of F The random variables X ( r ) ∗ , 1 � r � n , are called generalized order statistics based on F , if their joint density is � ¯ � n − 1 � � ¯ � m i f ( x i ) f X ( 1 ) ∗ ,..., X ( n ) � γ n − 1 f ( x n ) , ∗ ( x 1 , . . . , x n ) = c n − 1 � F ( x i ) F ( x n ) i = 1 for −∞ < x 1 � . . . � x n < + ∞

Introduction and some known results Equivalent condition Special cases for ℓ = 2 Generalized order statistics Fix n � 1 and parameters γ 1 , . . . , γ n > 0 Define c n − 1 = � n i = 1 γ i and m i = γ i − γ i + 1 − 1 , 1 � i � n − 1 . Assume that F is absolutely continuous cdf with density f and the support ( α, β ) Let ¯ F ( x ) = 1 − F ( x ) denote the survival function of F The random variables X ( r ) ∗ , 1 � r � n , are called generalized order statistics based on F , if their joint density is � ¯ � n − 1 � � ¯ � m i f ( x i ) f X ( 1 ) ∗ ,..., X ( n ) � γ n − 1 f ( x n ) , ∗ ( x 1 , . . . , x n ) = c n − 1 � F ( x i ) F ( x n ) i = 1 for −∞ < x 1 � . . . � x n < + ∞

Introduction and some known results Equivalent condition Special cases for ℓ = 2 Generalized order statistics Fix n � 1 and parameters γ 1 , . . . , γ n > 0 Define c n − 1 = � n i = 1 γ i and m i = γ i − γ i + 1 − 1 , 1 � i � n − 1 . Assume that F is absolutely continuous cdf with density f and the support ( α, β ) Let ¯ F ( x ) = 1 − F ( x ) denote the survival function of F The random variables X ( r ) ∗ , 1 � r � n , are called generalized order statistics based on F , if their joint density is � ¯ � n − 1 � � ¯ � m i f ( x i ) f X ( 1 ) ∗ ,..., X ( n ) � γ n − 1 f ( x n ) , ∗ ( x 1 , . . . , x n ) = c n − 1 � F ( x i ) F ( x n ) i = 1 for −∞ < x 1 � . . . � x n < + ∞

Introduction and some known results Equivalent condition Special cases for ℓ = 2 Generalized order statistics Fix n � 1 and parameters γ 1 , . . . , γ n > 0 Define c n − 1 = � n i = 1 γ i and m i = γ i − γ i + 1 − 1 , 1 � i � n − 1 . Assume that F is absolutely continuous cdf with density f and the support ( α, β ) Let ¯ F ( x ) = 1 − F ( x ) denote the survival function of F The random variables X ( r ) ∗ , 1 � r � n , are called generalized order statistics based on F , if their joint density is � ¯ � n − 1 � � ¯ � m i f ( x i ) f X ( 1 ) ∗ ,..., X ( n ) � γ n − 1 f ( x n ) , ∗ ( x 1 , . . . , x n ) = c n − 1 � F ( x i ) F ( x n ) i = 1 for −∞ < x 1 � . . . � x n < + ∞

Introduction and some known results Equivalent condition Special cases for ℓ = 2 Special cases γ i = n − i + 1 , 1 � i � n — order statistics X 1 : n � . . . � X n : n γ i = 1 — record values γ i = k — k th record values progressively censored order statistics

Introduction and some known results Equivalent condition Special cases for ℓ = 2 Special cases γ i = n − i + 1 , 1 � i � n — order statistics X 1 : n � . . . � X n : n γ i = 1 — record values γ i = k — k th record values progressively censored order statistics

Introduction and some known results Equivalent condition Special cases for ℓ = 2 Special cases γ i = n − i + 1 , 1 � i � n — order statistics X 1 : n � . . . � X n : n γ i = 1 — record values γ i = k — k th record values progressively censored order statistics

Introduction and some known results Equivalent condition Special cases for ℓ = 2 Special cases γ i = n − i + 1 , 1 � i � n — order statistics X 1 : n � . . . � X n : n γ i = 1 — record values γ i = k — k th record values progressively censored order statistics

Introduction and some known results Equivalent condition Special cases for ℓ = 2 Statement of the problem Fix r , ℓ � 1 and parameters γ 1 , . . . , γ r + ℓ > 0 Assume that F is absolutely continuous cdf with the support ( α, β ) Consider generalized order statistics X ( 1 ) ∗ , . . . , X ( r ) ∗ , . . . , X ( r + ℓ ) with parameters ∗ γ 1 , . . . , γ r + ℓ based on F Assume that function h : ( α, β ) → R is strictly increasing and continuous � �� � X ( r + ℓ ) If E � h � < ∞ , then we define the regression function � � ∗ � � X ( r + ℓ ) � X ( r ) � � � ξ ( x ) = E = x x ∈ ( α, β ) h , ∗ ∗ Problem: Does ξ determine F uniquely? What is the expression for F in terms of ξ and h ?

Introduction and some known results Equivalent condition Special cases for ℓ = 2 Statement of the problem Fix r , ℓ � 1 and parameters γ 1 , . . . , γ r + ℓ > 0 Assume that F is absolutely continuous cdf with the support ( α, β ) Consider generalized order statistics X ( 1 ) ∗ , . . . , X ( r ) ∗ , . . . , X ( r + ℓ ) with parameters ∗ γ 1 , . . . , γ r + ℓ based on F Assume that function h : ( α, β ) → R is strictly increasing and continuous � �� � X ( r + ℓ ) If E � h � < ∞ , then we define the regression function � � ∗ � � X ( r + ℓ ) � X ( r ) � � � ξ ( x ) = E = x x ∈ ( α, β ) h , ∗ ∗ Problem: Does ξ determine F uniquely? What is the expression for F in terms of ξ and h ?

Introduction and some known results Equivalent condition Special cases for ℓ = 2 Statement of the problem Fix r , ℓ � 1 and parameters γ 1 , . . . , γ r + ℓ > 0 Assume that F is absolutely continuous cdf with the support ( α, β ) Consider generalized order statistics X ( 1 ) ∗ , . . . , X ( r ) ∗ , . . . , X ( r + ℓ ) with parameters ∗ γ 1 , . . . , γ r + ℓ based on F Assume that function h : ( α, β ) → R is strictly increasing and continuous � �� � X ( r + ℓ ) If E � h � < ∞ , then we define the regression function � � ∗ � � X ( r + ℓ ) � X ( r ) � � � ξ ( x ) = E = x x ∈ ( α, β ) h , ∗ ∗ Problem: Does ξ determine F uniquely? What is the expression for F in terms of ξ and h ?

Introduction and some known results Equivalent condition Special cases for ℓ = 2 Statement of the problem Fix r , ℓ � 1 and parameters γ 1 , . . . , γ r + ℓ > 0 Assume that F is absolutely continuous cdf with the support ( α, β ) Consider generalized order statistics X ( 1 ) ∗ , . . . , X ( r ) ∗ , . . . , X ( r + ℓ ) with parameters ∗ γ 1 , . . . , γ r + ℓ based on F Assume that function h : ( α, β ) → R is strictly increasing and continuous � �� � X ( r + ℓ ) If E � h � < ∞ , then we define the regression function � � ∗ � � X ( r + ℓ ) � X ( r ) � � � ξ ( x ) = E = x x ∈ ( α, β ) h , ∗ ∗ Problem: Does ξ determine F uniquely? What is the expression for F in terms of ξ and h ?

Introduction and some known results Equivalent condition Special cases for ℓ = 2 Statement of the problem Fix r , ℓ � 1 and parameters γ 1 , . . . , γ r + ℓ > 0 Assume that F is absolutely continuous cdf with the support ( α, β ) Consider generalized order statistics X ( 1 ) ∗ , . . . , X ( r ) ∗ , . . . , X ( r + ℓ ) with parameters ∗ γ 1 , . . . , γ r + ℓ based on F Assume that function h : ( α, β ) → R is strictly increasing and continuous � �� � X ( r + ℓ ) If E � h � < ∞ , then we define the regression function � � ∗ � � X ( r + ℓ ) � X ( r ) � � � ξ ( x ) = E = x x ∈ ( α, β ) h , ∗ ∗ Problem: Does ξ determine F uniquely? What is the expression for F in terms of ξ and h ?