{'fraction_non_alphanumeric': 0.015625, 'fraction_numerical': 0.0020380434782608695, 'mean_word_length': 5.432314410480349, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Tolerance interval for the exponential distribution with a specified ratio of the tail probabilities and generalizations

Abstract Tolerance intervals are widely used in various industrial settings and business activities, such as product design, reliability analysis, and quality control. A -content tolerance interval (TI) is a statistical interval where at least 100 of the distribution of a random variable fall with a given confidence level In this paper, we consider an exact two-sided -content tolerance interval for exponential random variables with a specified ratio of the tail probabilities, called a tolerance interval. The proposed tolerance interval gives the practitioner more flexibility on how to distribute the probabilities in the tails. A comparison of the proposed TI with existing -content tolerance intervals for exponential random variables shows that the proposed TI is better on the basis of the expected coverage, standard deviation of coverage, and the sample size required to achieve the same level of accuracy. A study is conducted to examine the robustness of the proposed TI for some mild departures from the exponential distribution. Extensions to other distributions are presented, including the Weibull distribution with an unknown scale parameter. Illustrations are provided for construction and implementation of the proposed TI using some complete, Type-II censored and record values data from real situations found in the literature.