Γ-minimax estimation with delayed observations from the multinomial distribution

Abstract

The problem of finding optimal stopping times and the corresponding sequential estimators for the unknown parameter vector p = (p 1 , . . . , p m ) of m independent multinomial distributions M(1, p i ), i = 1, . . . , m, is considered in the special case when the data arrive at random times. In the problem of finding optimal sequential estimation procedures, the intermediate approach between the Bayes and the minimax principle is applied in which it is assumed that a vague prior information on the distribution of the unknown parameter is available. It is supposed that the set of all prior distributions of the parameter p is restricted to a set consisting of the priors for which certain conditions on the moments are imposed. Using the weighted squared error loss functions defined by Eqs. (4) and (13) and the observation cost determined by a convex function of the moment of stopping, several classes of -minimax sequential procedures for estimating p are established, respectively, to different conditions imposed on the set of prior distributions.

Distributions of stopping times in some sequential estimation procedures

“…The function L h (t) can be interpreted as the total loss incurred if the process is stopped at time t. For example, the conditional expected losses of Bayes or minimax estimators in the statistical models considered in the papers of Starr et al (1976), Magiera (1982), Jokiel-Rokita and Magiera (1999Magiera ( , 2010 and Jokiel-Rokita (2006) are of the form given by (3). By the optimal stopping time τ 0 with respect to F t , t ≥ 0, we mean the random variable which minimizes…”

“…In the papers of Starr et al (1976) and Jokiel-Rokita and Magiera (1999) adaptive strategies are also proposed which require knowledge of neither n nor F. These strategies perform nearly as well as it is possible when n is large for large class of F.…”

“…when the distribution of the random variables which determine the moments of recording the observations is exactly known (papers of Starr et al 1976;Magiera 1982;Jokiel-Rokita andMagiera 1999, 2010;Jokiel-Rokita 2006); 2. when the distribution of the random variables which determine the moments of obtaining the observations is exponential with an unknown parameter (papers of Starr et al 1976;Jokiel-Rokita and Magiera 1999;Jokiel-Rokita 2006) or it belongs to a time transformed exponential family with an unknown parameter (paper Jokiel-Rokita and Magiera 2010).…”