Approximate Tolerance Intervals, Based on Maximum Likelihood Estimates

Abstract

Let X ₁, …, X_n be independent random variables with distributions depending on a possibly multidimensional θ. Let Y be an unobserved continuously distributed random variable whose distribution depends on θ. A tolerance interval for Y is desired, satisfying P[Y ε I(X ₁, …, X_n )] = β. A naive interval would estimate θ from the X's and construct the interval assuming that the estimate is exactly correct. This article assumes standard regularity conditions and uses Taylor approximations to construct correction terms of order 1/n. The resulting interval is longer than the naive interval because it takes into account the uncertainty in the estimate of θ. Two examples, one simple and one complex, illustrate the method.

Key Words:

• Prediction interval
• Asymptotic bias of MLE
• Negative binomial distribution