A Hybrid Problem on the Exponential Family

Abstract

It is frequently of interest to an investigator to decide whether an unknown parameter lies in some interval and also to obtain a point estimate of the parameter. There are many instances in practice where one does this by first testing a hypothesis and then estimating. Sometimes the investigator will even allow the result of the hypothesis test to effect his method of estimation. Theoretical work on the procedure, where one follows a test of hypothesis by estimation, has been generally reviewed by Kitagawa [10]. Included in Kitagawa's review are references and discussions of other approaches to what we will call a hybrid problem. That is, we describe a hybrid problem as follows: Suppose we take $n$ observations on a random variable whose distribution is known to belong to a one-parameter family. Then, on the basis of these observations we decide whether the parameter (or some function of it) lies in a given interval of the parameter space (or the range space of the function of the parameter). Secondly, we estimate the parameter (or some function of it). In the next section we will explicitly state a decision theory formulation for a hybrid problem. Following the statement of the problem, we will give examples of situations where a hybrid problem is defined and for which the given formulation is appropriate. The formulation will be given for a hybrid problem when a single observation is made on a random variable whose distribution is of the exponential type, and when the decisions are made with regard to the expected value of the random variable. We decide whether the expected value, which is a function of the underlying parameter of the exponential distribution, lies in some given interval and we also estimate the expected value. For a large subclass of the exponential family, including the normal distribution with unknown mean, complete classes of procedures are found for the case when the given interval consists of a single point. These complete classes consist of procedures called interval-monotone procedures defined as follows: If the observation falls in a particular interval of the sample space, then decide that the expected value of the random variable lies in the given interval consisting of a single point and estimate it to be precisely that point. If the observation lies outside that particular interval of the sample space, decide that the expected value lies outside the given interval and estimate it by some monotone analytic function of the observation. Another result for the given formulation is concerned with the case when the distribution is normal with unknown mean and the given fixed interval is any symmetric interval about the origin. Here, a class of admissible procedures is found, where each procedure is a symmetric interval-monotone procedure, the symmetry pertaining both to the interval and the estimate. Furthermore, included in this class is the following intuitively appealing procedure: If the observation falls in some particular symmetric interval about the origin, then decide that the parameter lies in the given fixed interval of the parameter space and estimate the parameter by zero; while if the observation falls outside the particular interval of the sample space, then decide that the parameter lies outside the given interval and estimate it by the value of the observation. It is interesting to note that the estimate resulting from the above procedure was proved to be an inadmissible estimate for the squared error loss function. (See [5]). The analogue of the admissibility result for the normal case is next found for the cases where the probability distribution of the random variable is binomial, Poisson and gamma, and the left end point of the given interval of the parameter space is zero. In the next section then, we state the problem and give examples. In Section 3 we give complete class theorems. The admissibility results for the normal case and binomial, Poisson, and gamma cases follow in Sections 4 and 5, respectively.