Simultaneous estimation of Poisson means of the selected subset

Let _π1,_π2,…,_πp${\pi }_1,{\pi }_2,\dots ,{\pi }_p$ be p   independent Poisson populations with means _λ1,…,_λp${\lambda }_1,\dots ,{\lambda }_p$, respectively. Let {X₁,…,X_p} denote the set of observations, where X_i is from _πi${\pi }_i$. Suppose a subset of populations is selected using Gupta and Huang's (1975) selection rule which selects _πi${\pi }_i$ if and only if _Xi+1?_cX(1)$X_i+1?{\mathit{cX}}_{(1)}$, where X₍₁₎=max{X₁,…,X_p}, and 0<c<1$0<c<1$. In this paper, the simultaneous estimation of the Poisson means associated with the selected populations is considered for the k-normalized squared error loss function. It is shown that the natural estimator is positively biased. Also, a class of estimators that are better than the natural estimator is obtained by solving certain difference inequalities over the sample space. A class of estimators which dominate the UMVUE is also obtained. Monte carlo simulations are used to assess the percentage improvements and an application to a real-life example is also discussed.     

Nonparametric Bayes estimation in repair models

Consider a sequence of dependent random variables _X1,_X2,…,_Xn$X_1,X_2,\dots ,X_n$, where _X1$X_1$ has distribution F (or probability measure P  ), and the distribution of _Xi+1$X_{i+1}$ given _X1,…,_Xi$X_1,\dots ,X_i$ and other covariates and environmental factors depends on F   and the previous data, i=1,…,n-1$i=1,\dots ,n-1$. General repair models give rise to such random variables as the failure times of an item subject to repair. There exist nonparametric non-Bayes methods of estimating F in the literature, for instance, Whitaker and Samaniego [1989. Estimating the reliability of systems subject to imperfect repair. J. Amer. Statist. Assoc. 84, 301–309], Hollander et al. [1992. Nonparametric methods for imperfect repair models. Ann. Statist. 20, 879–896] and Dorado et al. [1997. Nonparametric estimation for a general repair model. Ann. Statist. 25, 1140–1160], etc. Typically these methods apply only to special repair models and also require repair data on N independent items until exactly only one item is left awaiting a “perfect repair”.     

An identity for the Fisher information and Mahalanobis distance

Consider a mixture problem consisting of k classes. Suppose we observe an s-dimensional random vector X   whose distribution is specified by the relations P(X∈A|Y=i)=_Pi(A)$P(X\in A|Y=i)=P_i(A)$, where Y   is an unobserved class identifier defined on {1,…,k}$\{1,\dots ,k\}$, having distribution P(Y=i)=_pi$P(Y=i)=p_i$. Assuming the distributions _Pi$P_i$ having a common covariance matrix, elegant identities are presented that connect the matrix of Fisher information in Y   on the parameters _p1,…,_pk$p_1,\dots ,p_k$, the matrix of linear information in X, and the Mahalanobis distances between the pairs of P  's. Since the parameters are not free, the information matrices are singular and the technique of generalized inverses is used. A matrix extension of the Mahalanobis distance and its invariant forms are introduced that are of interest in their own right. In terms of parameter estimation, the results provide an independent of the parameter upper bound for the loss of accuracy by esimating _p1,…,_pk$p_1,\dots ,p_k$ from a sample of X^′$X^{\prime }$s, as compared with the ideal estimator based on a random sample of Y^′$Y^{\prime }$s.