A Note on the Decomposition of a Random Sample Size


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1 A Note on the Decomposton of a Random Sample Sze Klaus Th. Hess Insttut für Mathematsche Stochastk Technsche Unverstät Dresden Abstract Ths note addresses some results of Hess 2000) on the decomposton of a random sample sze by usng the concept of the multvarate probablty generatng functon. 1 Introducton Usng the concept of the multvarate probablty functon we prove a nce formula for the vector of thnned random sample szes. Ths leads to new proofs of some of the results presented n Hess 2000). We use the followng vector notaton: Let d N and denote by e the th unt vector n R d. Defne 1 : d e. For x x 1... x d ) R d, n n 1... n d ) N d 0 and n N 0 such that n 1 n we defne usng the defnton 0 0 : 1) x n : n! : ) n n : x n n! n! n! n 1 n)! Throughout ths paper, let Ω, F, P) be a probablty space. 2 Probablty Generatng Functon In ths secton we consder the multvarate extenson of the probablty generatng functon; see also Zocher 2005). 1
2 Let N : Ω N d 0 be a random vector. The functon m N : [0, 1] d R wth m N t) E [ t N ] s called the probablty generatng functon of N. In the case d 1 ths defnton concdes wth that n the unvarate case. 2.1 Lemma. The probablty generatng functon m N has the followng propertes: a) m N s ncreasng wth respect to the coordnatewse order relaton and 0 m N t) m N 1) 1 holds for all t [0, 1] d. b) m N s contnuous. c) m N s nfntely often dfferentable on [0, 1) d. d) The dentty P [N n] 1 n! n m N t n t n 0) d holds for all n N d 0 and n : 1 n. In partcular, the dstrbuton of N s unquely defned by ts probablty generatng functon. The prevous lemma was proved by Zocher 2005) usng the representaton of the probablty generatng functon as a power seres n d varables. 2.2 Lemma. For all {1,..., d} the probablty generatng functon of the coordnate N fulflls m N t) m N 1 e + te ) The asserton follows drectly from the defnton. 2.3 Lemma. The random vector N has ndependent coordnates f and only f d holds for all t [0, 1] d. m N t) m N t ) Proof. If the coordnates are ndependent, then measurable functons of the coordnates are ndependent as well and the product formula follows. Conversely, f the product formula holds, then Lemma 2.1 yelds for all n N d P [N n] 1 n m N n! t n t n 0) d d 1 d n m N 0) n! dt n 2 P [N n ]
3 and hence the coordnates of N are ndependent. 2.4 Examples. a) Multnomal dstrbuton: The random vector N has the multnomal dstrbuton Multn, η) wth n N and η 0, 1) d such that 1 η 1 f P [N n] ) n η n 1 1 η) n 1 n n holds for all n N d 0 such that 1 n n. In ths case the probablty generatng functon of N satsfes Snce m N t) t η η)) n 1 1 η + t η) n m N t) m N 1 1 t)e ) 1 η + t η ) n the one dmensonal margnal dstrbutons of Multn, η) are the bnomal dstrbutons Bnn, η ). b) Multvarate Posson dstrbuton: The random vector N has the multvarate Posson dstrbuton MPoα) wth α 0, ) d f P [N n] αn n! e 1 α holds for all n N d 0. Ths means that N has ndependent coordnates whch are Posson dstrbuted. The probablty generatng functon satsfes m N t) m N t ) e α 1 t ) e α 1 t) c) Negatvemultnomal dstrbuton: The random vector N has the negatvemultnomal dstrbuton NMultβ, η) wth β 0, ) and η 0, 1) d such that 1 η < 1 f P [N n] Γβ + 1 n) 1 1 η) β η n Γβ) n! 3
4 holds for all n N d 0. In ths case the probablty generatng functon satsfes m N t) 1 1 ) β η 1 t η Snce m N t) m N 1 1 t)e ) 1 1 ) β η 1 1 η + η tη 1 η /1 1 η + η ) 1 t η /1 1 η + η ) the one dmensonal margnal dstrbutons of NMultn, η) are the negatvebnomal dstrbutons NBnβ, η /1 1 η + η )). ) β Snce we wll consder condtonal dstrbutons t s necessary to defne condtonal probablty generatng functons. Let Θ : Ω R be a random varable and let K : PN d 0) Ω [0, 1] be a Θ Markov kernel. For each ω Ω, K., ω) s a probablty measure on PN d 0) havng a probablty generatng functon m K., ω). Ths defnes a functon m K : [0, 1] d Ω R whch s called the Markov kernel generatng functon of K. For the random vector N : Ω N d 0 there exsts a Θ condtonal dstrbuton P N Θ of N. The Markov kernel generatng functon of P N Θ s called a Θ condtonal probablty generatng functon of N and wll be denoted by m N Θ. As usual for condtonal expectatons we wll drop the argument ω Ω. 2.5 Lemma. The condtonal probablty generatng functon m N Θ satsfes m N Θ t) E t N Θ ) and m N t) E [ m N Θ t) ] 2.6 Example. Let N : Ω N d 0 be a random vector and let N : Ω N be random varable such that P N N MultN, η) for some η 0, 1) d wth 1 η 1. Then we have m N N t) 1 1 η + t η) N and m N t) E [ 1 1 η + t η) N ] 4
5 3 Random Samples and ther Decomposton In ths secton we prove a formula on the probablty generatng functon of the vector of thnned random sample szes. Ths formula leads to new proofs of results of Hess 2000). Let M, M) be a measurable space. Gven a random varable N : Ω N 0 and a sequence {Y j } j N of random varables Y j : Ω M, the par N, {Yj } j N s called a random sample f the sequence {Y j } j N s..d. and ndependent of N. In ths case N, s called the random sample sze and Y j s called a sample varable. Let {C 1,..., C d } be a measurable partton of M such that P [Y 1 C ] > 0 holds for all {1,..., d}. Furthermore defne η : P [Y 1 C ] for all {1,..., d} and η : η 1... η d ). For each {1,..., d}, the thnned random sample sze of group s defned as N : N χ {Yj C } j1 and N : N 1,..., N d ) s called vector of the thnned random sample szes. The next theorem gves a nce formula for the probablty generatng functon of the vector of the thnned random sample szes: 3.1 Theorem. It holds m N N t) t η) N m N t) m N t η) m N t) m N 1 η + tη ) for all t [0, 1] d, t [0, 1] and {1,..., d}. In partcular, the condtonal dstrbuton of N gven N s the multnomal dstrbuton MN, η). Proof. we get Snce the random sample sze and the sample varables are ndependent, d m N N t) E E ) t N N d N ) j1 t χ {Y j C } N 5
6 [ d E n0 n E n0 j1 n E n0 j1 [ n d E n0 j1 n j1 t χ {Y j C } χ {Nn} t χ {Y j C } t χ {Y j C } t χ {Y j C } χ {Nn} χ {Nn} ] χ {Nn} ] P [N n] ) 1 χ{nn} Because of t χ {Y j C }ω) { t f Y j ω) C 1 f Y j ω) / C and snce {C 1,..., C d } s a partton of M, the prevous dentty yelds [ n d ] m N N t) E t χ {Yj C } χ {Nn} n0 j1 n d t P [Y j C ] χ {Nn} n0 j1 n d t η χ {Nn} n0 j1 d ) n t η χ {Nn} n0 h1 t η) N Ths proves the frst equaton. Usng ths equaton we get from Lemma 2.5 m N t) E [ m N N t) ] Lemma 2.2 yelds E [ t η) N ] m N t η) m N t) m N 1 e + te ) m N 1 e + te ) η ) m N 1 η + tη ) Snce 1 η 1, the fnal asserton follows from Example 2.6. Theorem 3.1 can be used to prove the followng corollary va probablty generatng functons, see Example 2.4: 6
7 3.2 Corollary. a) If N has the bnomal dstrbuton Bnn, ϑ) wth n N and ϑ 0, 1), then N has the multnomal dstrbuton Multn, ϑη) and each N has the bnomal dstrbuton Bnn, ϑη ). b) If N has the Posson dstrbuton Poα) wth α 0, ), then N has the multvarate Posson dstrbuton MPoαη) and each N has the Posson dstrbuton Poαη ). In ths case, N has ndependent coordnates. c) If N has the negatvebnomal dstrbuton NBnβ, ϑ) wth β 0, ) and ϑ 0, 1), then N has the negatvemultnomal dstrbuton NMultβ, ϑη) and each N has the negatvebnomal dstrbuton NBnβ, ϑη /1 ϑ + ϑη )). Next we want to show that the Posson case s the only case n whch the coordnates are ndependent. A proof of ths result was gven by Hess and Schmdt 2002), but here we want to gve an analytc one. 3.3 Theorem. The thnned random sample szes are ndependent f and only f the orgnal sample sze has a Posson dstrbuton. Proof. In Corollary 3.2 we have establshed that the Posson dstrbuton of the orgnal sample sze mples ndependence of the thnned random sample szes. Now let the thnned random sample szes be ndependent. Usng 1 η 1, Theorem 3.1, and Lemma 2.3, we get ) d m N 1 η t η ) m N t η) m N t) m N t ) m N 1 η + t η ) m N 1 η t η )) for all t t 1,..., t d ) [0, 1] d. Lemma 2.1 yelds that m N s contnuous, ncreasng and fulflls m N 1) 1. Therefore m N s an ncreasng soluton of the functonal equaton ) d f 1 u f1 u ) for all u 1,..., u d [0, 1] fulfllng d u 1 n the class of all contnuous functons f : [0, 1] R fulfllng the ntal condton f1) 1. Defne ϕ : [0, 1] [0, 1] wth 7
8 ϕu) 1 u. Then ϕ s a bjecton. Ths leads to the functonal equaton d ) f ϕ) u f ϕ)u ) for all u 1,..., u d [0, 1] fulfllng d u 1 n the class of all contnuous functons f : [0, 1] R fulfllng the ntal condton f1) 1, and hence f ϕ)0) 1. Snce ths characterzed the exponental functon we get f ϕ)t) e αt for some α R and therefore ft) e α1 t). As mentoned before, m N s ncreasng. Therefore wet get α 0, ). Ths s the probablty generatng functon of the Posson dstrbuton Poα). We complete ths note wth a result on mxed Posson dstrbutons. 3.4 Theorem. If N has the mxed Posson dstrbuton wth parameter Θ, then N has the mxed multvarate Posson dstrbuton wth parameter Θη. Proof. Let n N d 0. Usng Theorem 3.1 we get ) [ ] d d P [N n] P N n N n P N n d ) n! d η n e Θ Θ d n d ) dp n! Ω n! Ω e Θη Θη ) n n! dp as was to be shown. For related results on the decomposton of sample varables see Hess 2000) who generalzed results by Franke and Macht 1995) and Hess, Schmdt and Macht 1995). For applcatons to countng processes see Schmdt 1996) and Hess 2003). Acknowledgement The author would lke to thank Klaus D. Schmdt Dresden) and Mathas Zocher Zürch) for a dscusson on the present note. References Franke, T.; Macht, W. 1995): Decomposton of Rsk Processes. Dresdner Schrften zur Verscherungsmathematk 2/1995. Hess, K. Th. 2000): Random parttons of samples. Dresdner Schrften zur Verscherungsmathematk 1/
9 Hess, K. Th. 2003): On the Decomposton of Mxed Posson Processes. Dresdner Schrften zur Verscherungsmathematk 3/2003. Hess, K. Th.; Macht, W.; Schmdt, K. D. 1995): Thnnng of Rsk Processes. Dresdner Schrften zur Verscherungsmathematk 1/1995. Hess, K. Th.; Schmdt, K. D. 2002): A comparson of models for the chan ladder method. Insurance: Math. Econom. 31, Schmdt, K. D. 1996): Lectures on Rsk Theory. Stuttgart: B. G. Teubner. Zocher, M. 2005): Multvarate Mxed Posson Processes. PhD Thess, TU Dresden, Klaus Th. Hess Insttut für Mathematsche Stochastk Technsche Unverstät Dresden D Dresden Germany e mal: nternet: September 1,
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