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THE Cramer-Rao inequality, which is usually associated with the case of fixed sample size, has been extended by Wolfowitz (1947) to the sequential case. In both cases, if this inequality, called sometimes the information inequality (cf. Degroot, 1959), becomes equality for a given parameter value, then the estimator is said to be efficient at that value. For a sampling plan, if there exists at least one estimator which is efficient at all parameter values, then the sampling plan is said to be efficient. Efficient estimators and sampling plans have been studied by Degroot (1959) for the binomial population. In this paper we discuss them for the multinomial population with parameter p = (Pli ... ,Pr+i), where we assume without loss of generality that pj >0, Ypj = 1. We also assume that p cannot be expressed in terms of fewer than r parameters. Let g(p) be a parametric function. The problem is to obtain an estimator of g(p) which is efficient at all parameter points p. Since the sampling plan will be efficient in that case, the problem is also equivalent to that of characterizing efficient sampling plans and parametric functions which are estimable efficiently. This problem is solved using the ideas and methods given in Girshick et al. (1946) and in Degroot (1959) for the case of the binomial population. We first prove that the information inequality becomes an equality for sampling plans whose boundary points are certain linear functions of the co-ordinates. Then we prove that such sampling plans, if they are to terminate with probability 1, should necessarily be either single sampling plans or inverse multinomial sampling plans (to be defined in Section 3). Since these plans can be proved to be efficient, these are the only efficient sampling plans. We also deduce the most general form of efficient estimators of the parametric functions for. a multinomial population. These conclusions are similar to those derived for the binomial population. |
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