The t-Static for Underlying Skew/Stretched-Tail Distributions

Arthur, S.P. (1980). The t-Static for Underlying Skew/Stretched-Tail Distributions. IIASA Professional Paper. IIASA, Laxenburg, Austria: PP-80-005

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Abstract

For underlying skew distributions, Student-t confidence intervals about the mean have unequal tail probabilities -- the interval does not cover the mean in a "balanced" way. This study uses Monte-Carlo methods to estimate, for a class of highly skew, stretched-tail distributions, the population characteristic covered by the t-interval with symmetric loss. Results indicate that this "balanced" population characteristic depends on the degree of skewness and stretch, the desired significance level, and the sample size.

Estimates of the balanced population characteristic can be used to modify Student-t confidence intervals about the mean to achieve symmetric loss. The resulting tail probabilities are estimated and are found to be reasonably close to desired levels for many cases. Most of the discrepancy between true tail probabilities and tabled Student-t values is corrected, for the distributions of this study, by this simple modification.

The results of this study are applicable to a family of underlying distributions that are more skew and stretched-tail than generally considered in robustness studies of the t-statistic. Furthermore, the approach does not require large samples -- results are given for small to moderate sample sizes.

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