Pairs of successes in Bernoulli trials and a new n-estimator for the binomial distribution

Abstract

Abstract. The problem of estimating the number, n, of trials, given a sequence of k independent success counts obtained by replicating the n-trial experiment is reconsidered in this paper. In contrast to existing methods it is assumed here that more information than usual is available: not only the numbers of successes are given but also the number of pairs of consecutive successes. This assumption is realistic in a class of problems of spatial statistics. There typically k = 1, in which case the classical estimators cannot be used. The quality of the new estimator is analysed and, for k > 1, compared with that of a classical n-estimator. The theoretical basis for this is the distribution of the number of success pairs in Bernoulli trials, which can be determined by an elementary Markov chain argument.1. Introduction. The standard statistical problem associated with the binomial distribution is that of estimating its probability, p, of success.A much less well studied and considerably harder problem is that of estimating the number, n, of trials. The papers by Olkin, Petkau and Zidek [6] and Carroll and Lombard [3] study this problem for the case where k independent success counts s 1 , . . . , s k are given. Their methods cannot be applied if only one count is considered, k = 1. But just this case appears in some problems of spatial statistics.An important application consists in estimating the fraction of chips on a silicon wafer which are faulty because of technological reasons. In a general setting the spatial problem is as follows. A rectangle is divided into 1991 Mathematics Subject Classification: 62E25, 62F10, 62M30.

Bayes and Empirical Bayes Estimation of the Parameternin a Binomial Distribution

“…Classic literature on this problem includes Fisher (1942), Haldane (1942), Feldman and Fox (1968), Olkein et al (1981), Casella (1986), Hall (1994), and Kuhne et al (1994), and others. Classical procedures are mostly proposed by using the method of moments estimation (MME) and the maximum likelihood estimation (MLE).…”