On Two-Sided Tolerance Intervals for a Normal Distribution

Abstract

In [6], Wald and Wolfowitz present a method for obtaining approximate two-sided tolerance intervals for a normal distribution. Some readers of [6] have been left with the impression that Wald and Wolfowitz proved that the confidence level attained by their approximation converges to the nominal confidence level with increasing sample size $N$, and that the difference is $O(1/N^2)$ (see, e.g., [3]). However, Wald and Wolfowitz did not consider that problem in their paper, nor does the problem seem to have been considered elsewhere in the literature. The principal result of Wald and Wolfowitz is given at the end of Section 1. In Section 2 it is shown that the confidence level attained by the Wald-Wolfowitz approximation does converge to the nominal confidence level, that the difference is $O(1/N)$, and that $1/N$ is the exact order of the rate of convergence except for the confidence level .5. This is a corollary of a more general result obtained by considering the case in which $s^2 \sim \sigma^2\chi^2_n/n$, and independently $\bar{x} \sim \text{normal} (\mu, \sigma^2/N)$, where $n$ is not necessarily equal to $N - 1$. It is found, perhaps surprisingly, that as $N \rightarrow \infty$, the confidence level attained by the obvious generalization of the Wald-Wolfowitz approximation converges to the nominal confidence level fastest when $n$ is fixed, the difference being $O(1/N^2)$ in this case. Furthermore, if $n$ increases "too rapidly" as $N \rightarrow \infty$, then the confidence level attained does not converge to the nominal confidence level. These results are consequences of the two theorems proved in this paper. Theorem 1 in Section 2 states that if $n/N^2 \rightarrow 0$, then the confidence level attained by the generalization of the Wald-Wolfowitz approximation converges to the nominal confidence level, that the difference is $O(n/N^2)$, and that, with certain unimportant exceptions, $n/N^2$ is the exact order of the rate of convergence. In Theorem 2, Section 5, it is shown that if $n/N^2 \rightarrow \infty$, then the confidence level attained by the generalization of the Wald-Wolfowitz approximation converges to a limit which does not depend on the nominal confidence level. A modification of the generalized Wald-Wolfowitz approximation which has the desired convergence property in this case is presented. Certain facts used in Section 2 are verified in Sections 3 and 4. In Section 6, an heuristic explanation of the observed asymptotic behavior is given, and the possibility of improving the Wald-Wolfowitz approximation is discussed briefly. The basic notation and method of determining error bounds in this paper are essentially the same as those employed by Wald and Wolfowitz in [6]. Sections 2, 3 and 4 of this paper are counterparts, respectively, of Sections 4, 8 and 9 of the Wald-Wolfowitz paper. Equation (19) in the present paper plays the role that the basic Equation (4.1) did in the Wald and Wolfowitz paper.