JOURNAL OF THE IRANIAN STATISTICAL SOCIETY (JIRSS)
Year:2005 | Volume:4 | Issue:1|Papers Count:5

The paper proposes empirical Bayes (EB) estimators for simultaneous estimation of means in the natural exponential family (NEF) with quadratic variance functions (QVF) models. Morris (1982, 1983a) characterized the NEF-QVF distributions which include among others the binomial, Poisson and normal distributions. In addition to the EB estimators, we provide approximations to the MSE’s of these estimators. Our approach generalizes the findings of Prasad and Rao (1990) for the random effects model where only area specific direct estimators and covariates are available. The EB estimators are derived using the theory of optimal estimating functions as proposed by Godambe and Thompson (1989). This is in contrast to the approach of Morris (1988) who found some approximate EB estimators for this problem. Also, unlike Morris (1988), we allow unequal number of observations in different clusters in the derivation of the EB estimators. In finding approximations to the MSE’s, we apply a bias-correction technique as proposed in Cox and Snell (1968). We illustrate our methodology by reanalyzing the toxoplasmosis data of Efron (1978, 1986).

We consider a sequence of independent and identically distributed (iid) random variables with absolutely continuous distribution function F(x) and probability density function (pdf) f(x). Let Rnl be the largest observation after observing nth record and R(ns) be the smallest observation after observing the nth record. Then we say Wnr=Rnl-R(ns), n>1, as the nth record range. We will consider some distributional properties of Wnr when f(x) = 1, 0£ x £1.

In this paper we specify the conditions on the parameters of pairs of gOS’s under which the corresponding generalized order statistics are ordered according to usual stochastic ordering, hazard rate ordering, likelihood ratio ordering and dispersive ordering. We consider this problem in one-sample as well as two-sample problems. We show that some of the results obtained by Franco et al. [Probab.Engrg. Inform. Sci. 2002, 16, 471-484] and Belzunce et al. [Probab.Engrg. Inform. Sci. (2004), to appear] for stochastic orderings of gOS’s are contained in our new results.

Given a sequence of letters generated independently from a finite alphabet, we consider the case when more than one, but not all, letters are generated with the highest probability. The length of the longest run of any of these letters is shown to be one greater than the length of the longest run in a particular state of an associated Markov chain. Using results of Foulser and Karlin (1987), a conjecture of a previous paper (Smythe, 2003) concerning the expectation of this length is verified.

We study the limiting behavior of weighted sums for negatively associated (NA) random variables. We extend results in Wu (1999) and a theorem in Chow and Lai (1973) for NA random variables.