Asymptotics of conditional moments of the summand in Poisson - PowerPoint PPT Presentation

Literatura Asymptotics of conditional moments of the summand in Poisson compound Tomasz Rolski (joint work with Agata Tomanek) Conference in Honour of Søren Asmussen Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound

Literatura Introduction N is a Z + -valued r.v. Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound

Literatura Introduction N is a Z + -valued r.v. X , X 1 , X 2 , . . . a sequence of i.i.d. Z + r.v.s independent of N . Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound

Literatura Introduction N is a Z + -valued r.v. X , X 1 , X 2 , . . . a sequence of i.i.d. Z + r.v.s independent of N . We are interested in   � N � �  N  . N k = d X j = k � j = 1 In particular we want to know the conditional mean E N k or the conditional variance V ar N k and their asymptotics for k → ∞ . In this talk N is Poisson with mean a . Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound

Literatura Introduction Suppose X is Poisson with mean b . We will call this case as (Poi ( a ) ,Poi ( b ) ). Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound

Literatura Introduction Suppose X is Poisson with mean b . We will call this case as (Poi ( a ) ,Poi ( b ) ). Compute � ∞ m ! e − a ( mb ) k m = 0 m a m k ! e − bm = B c ( k + 1 ) E N k = , � ∞ m ! e − a ( mb ) k B c ( k ) a m k ! e − bm m = 0 where ∞ � m k c m B c ( k ) = m ! e − c m = 1 is the k -th moment of the Poisson distribution and c = ae − b . Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound

Literatura Introduction More generally, E ( N k ) l = B c ( k + l ) , B c ( k ) � B c ( k + 2 ) � V ar N k = B c ( k + 1 ) B c ( k + 1 ) − B c ( k + 1 ) . B c ( k ) B c ( k ) Therefore of particular interest is ratio J c ( k ) = B c ( k + 1 ) / B c ( k ) . Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound

Literatura Introduction Interest in asymptotics formulas can be helpful. Jessen et al (2010) show J c ( k ) ∼ k / log k , as k → ∞ . Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound

Literatura Introduction Interest in asymptotics formulas can be helpful. Jessen et al (2010) show J c ( k ) ∼ k / log k , as k → ∞ . For c = 1, the asymptotics of J 1 ( k ) = J ( k ) was earlier written in Harper (66), however with redundant e in the denominator. Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound

Literatura Introduction Interest in asymptotics formulas can be helpful. Jessen et al (2010) show J c ( k ) ∼ k / log k , as k → ∞ . For c = 1, the asymptotics of J 1 ( k ) = J ( k ) was earlier written in Harper (66), however with redundant e in the denominator. Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound

Literatura Introduction Unfortunately, this asymptotics is extremely slow: Tomasz Rolski (joint work with Agata Tomanek) Rysunek: Ratio of J ( k ) / ( k / log k ) . Asymptotics of conditional moments of the summand in Poisson compound

Literatura Historical comments Studies of B ( k ) = B 1 ( k ) has a long history. Bell numbers: the k -th number: the number of partitions of a set of size k . Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound

Literatura Historical comments Studies of B ( k ) = B 1 ( k ) has a long history. Bell numbers: the k -th number: the number of partitions of a set of size k . Dobinski (1877): B ( k ) is equal to the k -th Bell number. Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound

Literatura Historical comments Studies of B ( k ) = B 1 ( k ) has a long history. Bell numbers: the k -th number: the number of partitions of a set of size k . Dobinski (1877): B ( k ) is equal to the k -th Bell number. De Bruijn (1981) gave � log log n � log B ( n ) = log n − log log n − 1 + o . n log n Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound

Literatura Historical comments Lov´ asz (93)(who quotes Moser and Wyman) B ( k ) ∼ k − 1 / 2 [Λ( k )] k + 1 / 2 e Λ( k ) − k − 1 , where Λ( x ) is the function defined by Λ( x ) log Λ( x ) = x . Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound

Literatura Historical comments Lov´ asz (93)(who quotes Moser and Wyman) B ( k ) ∼ k − 1 / 2 [Λ( k )] k + 1 / 2 e Λ( k ) − k − 1 , where Λ( x ) is the function defined by Λ( x ) log Λ( x ) = x . The function Λ is related to the Lambert W-function by W ( x ) = x / Λ( x ) . Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound

Literatura Historical comments From de Bruijn (1981) � log log x � W ( x ) = log x − log log x + O , log x and hence � � � log log x � 2 x 1 + log log x Λ( x ) ∼ + O ( ) . log x log x log x Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound

Literatura Historical comments From de Bruijn (1981) � log log x � W ( x ) = log x − log log x + O , log x and hence � � � log log x � 2 x 1 + log log x Λ( x ) ∼ + O ( ) . log x log x log x We also refer to Pitman (97) for interesting connections between Bell numbers and Poisson distributions. Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound

Literatura Historical comments Jessen et al (2010) � i m k e − c c m B c ( k ) = ( 1 + o ( 1 )) m ! , h k ( 1 − ǫ ) log k , k ( 1 + ǫ ) m ∈ log k Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound

Literatura Historical comments Jessen et al (2010) � i m k e − c c m B c ( k ) = ( 1 + o ( 1 )) m ! , h k ( 1 − ǫ ) log k , k ( 1 + ǫ ) m ∈ log k from which they concluded J c ( k ) = B c ( k + 1 ) / B c ( k ) ∼ k / log k . We will use their ideas of proof for other cases. Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound

Literatura Motivations: reserves in nonlife insurance Supose D k is the number of claims in a portfolio appearing in year 0 and paid in the year k k ∈ { 0 , 1 , . . . } . Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound

Literatura Motivations: reserves in nonlife insurance Supose D k is the number of claims in a portfolio appearing in year 0 and paid in the year k k ∈ { 0 , 1 , . . . } . Suppose we know ( D k ) , k = 0 , . . . , j , for some j ≥ 0. The aim is to estimate the reserves for years j + 1 , j + 2 , . . . . Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound

Literatura Motivations: reserves in nonlife insurance Supose D k is the number of claims in a portfolio appearing in year 0 and paid in the year k k ∈ { 0 , 1 , . . . } . Suppose we know ( D k ) , k = 0 , . . . , j , for some j ≥ 0. The aim is to estimate the reserves for years j + 1 , j + 2 , . . . . Natural estimator seems to be expected value conditioned on N 0 , . . . , N j : � � � � ˆ D j + l = E � D 0 , . . . , D j l = 0 , 1 , . . . . D j + l dla Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound

Literatura Motivations: reserves in nonlife insurance Supose D k is the number of claims in a portfolio appearing in year 0 and paid in the year k k ∈ { 0 , 1 , . . . } . Suppose we know ( D k ) , k = 0 , . . . , j , for some j ≥ 0. The aim is to estimate the reserves for years j + 1 , j + 2 , . . . . Natural estimator seems to be expected value conditioned on N 0 , . . . , N j : � � � � ˆ D j + l = E � D 0 , . . . , D j l = 0 , 1 , . . . . D j + l dla See e.g. Mack (1993, 1994) Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound

Literatura Motivations: reserves in nonlife insurance Jessen et al (2010) M –number of claims in year 0 q m = P ( M = m ) , m = 0 , 1 , . . . ; Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound

Literatura Motivations: reserves in nonlife insurance Jessen et al (2010) M –number of claims in year 0 q m = P ( M = m ) , m = 0 , 1 , . . . ; the m -th claim causes the stream K m of payments, where ( K m ) iid Poisson( µ ); Tomasz Rolski (joint work with Agata Tomanek) Asymptotics of conditional moments of the summand in Poisson compound