Uniformly Most Accurate Upper Tolerance Limits for Monotone Likelihood Ratio Families of Discrete Distributions.

There is not much in the literature on tolerance limits for discrete distributions, since generally one will find it difficult to derive explicit expressions for these limits. Distribution-free tolerance limits for discrete cases were discussed. If the form of the distribution function is known, one should try to determine the formula of the tolerance limits for the particular family of distributions to which it belongs. Otherwise, an important information is not utilized. In applications for sampling inspection reliability, inventory control, the observed random variables are very often assumed to have a binomial, negative-binomial or Poisson distribution. In these cases one can obtain explicit expressions for tolerance limits, which have desirable optimum properties. In the present article we derive uniformly most accurate upper tolerance limits for families of distributions with the monotone likelihood ratio property. For these families there exist uniformly most accurate upper confidence limits. Furthermore, the distribution functions which belong to monotone likelihood ratio families are stochastically ordered. Thus, monotone relationship exists in these cases between tolerance limits and confidence limits. This relationship is utilized to derive the formulae of uniformly most accurate tolerance limits. The general theory is presented in Section 2. Section 3 is devoted to uniformly most accurate upper confidence limits.