Statistica Sinica 14(2004), 217-229

A TOLERANCE INTERVAL FOR THE NORMAL

DISTRIBUTION WITH SEVERAL VARIANCE

COMPONENTS

C. T. Liao and H. K. Iyer

National Taiwan University and Colorado State University

Abstract: A tolerance interval procedure is derived from the concept of generalized pivotal quantities usually used to obtain confidence intervals in situations where standard procedures do not lead to useful solutions. We apply the generalized confidence intervals approach and propose a two-sided tolerance interval for the distribution $N(\theta,{\sum_{i=1}^q h_i \sigma_i^2)}$ based on mutually independent statistics $\hat{\theta}$, $S_1^2, \ldots, S_q^2$, where $\hat{\theta}$ is distributed as $N(\theta,{\sum_{i=1}^q c_i\sigma_i^2})$, $h_i$ and $c_i$ are known constants, and $n_iS_i^2/\sigma_i^2$ are independent chi-squared random variables with $n_i$ df, for $i=1,\ldots ,q$. Some practical examples are given to illustrate the applications of the proposed procedure. A simulation study is conducted to evaluate its frequentist coverage probability. The results indicate that the proposed method may be recommended for use in practical applications. The procedure provided in this paper can be applied to tolerance interval questions arising in arbitrary normal balanced mixed linear model situations.

Key words and phrases: Chi-squared approximation, generalized P-values, generalized confidence intervals, linear models, variance components.