Abstract
Statistical tolerance intervals are another tool for making statistical inference on anunknown population. The tolerance interval is an interval estimator based on the resultsof a calibration experiment, which can be asserted with stated confidence level 1 ? ,for example 0.95, to contain at least a specified proportion 1 ? , for example 0.99, ofthe items in the population under consideration. Typically, the limits of the toleranceintervals functionally depend on the tolerance factors. In contrast to other statisticalintervals commonly used for statistical inference, the tolerance intervals are used relativelyrarely. One reason is that the theoretical concept and computational complexity of thetolerance intervals is significantly more difficult than that of the standard confidence andprediction intervals.In this paper we present a brief overview of the theoretical background and approachesfor computing the tolerance factors based on samples from one or several univariate normal(Gaussian) populations, as well as the tolerance factors for the non-simultaneousand simultaneous two-sided tolerance intervals for univariate linear regression. Such toleranceintervals are well motivated by their applicability in the multiple-use calibrationproblem and in construction of the calibration confidence intervals. For illustration, wepresent examples of computing selected tolerance factors by the implemented algorithmin MATLAB.
Highlights
Statistical tolerance intervals are interval estimators used for making statistical inference on population(s), which can be fully described by a probability distribution from a given family of distributions
We have developed a MATLAB algorithm for efficient and highly precise computation of the exact tolerance factors for the non-simultaneous as well as simultaneous two-sided tolerance intervals for several independent univariate normal populations
Based on (12), we have developed the MATLAB algorithm ToleranceFactorGK, that computes the tolerance factors κ for the two-sided tolerance intervals by using an adaptive GaussKronod quadrature
Summary
Statistical tolerance intervals are interval estimators used for making statistical inference on population(s), which can be fully described by a probability distribution from a given family of distributions (as e.g., the family of normal distributions). We have developed a MATLAB algorithm for efficient and highly precise computation of the exact tolerance factors for the non-simultaneous as well as simultaneous two-sided tolerance intervals for several independent univariate normal populations. The methods and algorithms can be further used in the multiple-use calibration problem for constructing the appropriate simultaneous interval estimators (calibration confidence intervals) for values of the variable of primary interest, say x, based on possibly unlimited sequence of future observations of the response variable, say y, and on the results of the given calibration experiment, which was modeled/fitted by a linear regression model Such calibration intervals can be obtained by inverting the simultaneous tolerance intervals constructed for the regression (calibration) function. E.g., Scheffe (1973), Mee, Eberhardt, and Reeve (1991), Mee and Eberhardt (1996), Mathew and Zha (1997), and Chvostekova (2013b)
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