A least squares method for computing statistical tolerance limits

Abstract

Statistical analysis often requires the fitting of a probability density function to sample data in order to estimate either the population parameters or a particular percentile point of the population. In either case the exact determination of the population values cannot be expected because a sample generally does not completely describe the population. The theories of statistical confidence regions and tolerance limits allow inferences to be drawn regarding relationships between the estimates and the corresponding unknown population values. A least squares procedure has been devised which allows the construction of confidence regions and tolerance limits for arbitrary distribution functions. The class errors, ei, are defined as the difference between the sample histogram and the expected frequency based upon the density function. The confidence region is defined by the relation Rα = {θ : [R1(e)/R2(e)] < [m/(N − m)]F(α, m, N − m)}, where θ represents the m‐dimensional parameter vector of the distribution function, R1(e) and R2(e) represent decompositions of of ranks m and N − m, respectively, and F is Snedecor's F statistic. The η tolerance limit of the γ percentile point is found from the relation .