Invariant Interval Estimation of a Location Parameter

Abstract

In the general problem of statistical estimation, it is possible to obtain from a random sample a single estimate, known as a point estimate, of a population parameter. But this estimate is not very meaningful unless associated with a measure of its reliability. One approach to this problem consists of giving a point estimate with its standard error, but this has a major drawback. In using this approach, one typically fails to make an assertion regarding the error involved in estimating the standard error. In this paper, the problem of interval estimation is considered within the framework of decision theory, in which the cost to the statistician depends on the true value of the parameter and the interval chosen. For example, if $\theta$ is the true value of the parameter and $(a, b)$ is the interval chosen, a typical loss function is \begin{align*}L(\theta, (a, b)) = h(a - \theta, b - \theta) \text{if} a < \theta < b \\ = h(a - \theta, b - \theta) + 1 \text{if} \theta < a \text{or} > b\end{align*} where $h(a, b)$ is defined on $\{(a, b); a \leqq b\}$. Thus, the statistician would like to choose the interval $(a, b)$ small to make the payment $h(a, b)$ small, and yet he wants it large enough to have a good chance of containing $\theta$ so that he will not have to pay the "extra" unit. We consider two methods for finding optimal decision rules for the problem of interval estimation. The first one uses the Bayes principal and the second uses the invariance principle. The invariance principle is available only in decision problems which are invariant under certain transformations. Here we find a form of a best invariant interval estimate for the location parameter, and give certain conditions under which the best invariant interval estimate is minimax. In analogy of Blackwell and Girshick's suggestions, we present a loss function for which a best invariant interval estimate exists but is not minimax. (See the example in Section 4) Finally, in Section 5, we show that a best invariant interval estimate for a scale parameter of a distribution can be found by transforming the scale parameter problem to the location parameter problem.