| The Poisson distribution is a very important discrete type distribution. This article is divided four parts to carry on a research to it.Part I :Introduced the Poisson distribution origin and theapplication, especially in the application in the insurance,and it may the approximate calculation binomial distribution.Lemma1.3.1 In the n-Bernoulli trials , event A probability appearing in a tria is p n (be connected with n ),if at that time n→+∞, np n→λ(λ>0 constant) , then has ,and proved the Poisson distribution is admissible.PartⅡ:Has extracted the estimation for parameter of Poisson distribution: the maximum likelihood estimation, the moment law estimated, The maximum likelihood estimate of parameterλis The moment law estimated isPartⅢ:Has extracted the Bayes estimate for parameter of Poisson distribution .Lemma3.2.1 Under the square loss function L (λ,δ)=(λ-δ)2, with any priori distribution, the Bayes estimate is the latter examines averageδ( X )= E(λX).Under the square loss function, a priori distribution takes theΓdistribution, the Bayes estimate for parameter is: and it is admissible. And it is also proposed that the Bayes lowers confidence limit .PartⅣ:Under the square loss function, regarding current sample X and the partially missing data historical sample X1 , X2,Xn, constructe the empirical Bayesion estimation for parameter of Poisson distribution ,and show that it is admissible and asymptotically aptimal. Moreover gives the convergence rate of this estimate is O ( n-1).Main conclusion: Let X has a Poisson distribution of parameterθ, and probability density function isTheorem 4.3.1 the loss function is L (θ,d)= (θ?d)2, d∈( 0,+∞),a priori distribution takes theΓdistribution, the distribute density of parameterβis H (β)=B (p1, q)? (1+ββq+)1p+q, and the partial flaw data historical sample X 1 , X2,Xn, then has in whichis admissible and asymptotically aptimal. Let p > 0, q>4, when n1≥1, then has in which K >0 and nothing to n1 andβ.
|
PDF Full Text Request