Simulating Univariate and Multivariate Nonnormal Distributions through the Method of Percentiles

Abstract

This article derives a standard normal-based power method polynomial transformation for Monte Carlo simulation studies, approximating distributions, and fitting distributions to data based on the method of percentiles. The proposed method is used primarily when (1) conventional (or L) moment-based estimators such as skew (or L-skew) and kurtosis (or L -kurtosis) are unknown or (2) data are unavailable but percentiles are known (e.g., standardized test score reports). The proposed transformation also has the advantage that solutions to polynomial coefficients are available in simple closed form and thus obviates numerical equation solving. A procedure is also described for simulating power method distributions with specified medians, inter-decile ranges, left-right tail-weight ratios (skew function), tail-weight factors (kurtosis function), and Spearman correlations. The Monte Carlo results presented in this study indicate that the estimators based on the method of percentiles are substantially superior to their corresponding conventional product-moment estimators in terms of relative bias. It is also shown that the percentile power method can be modified for generating nonnormal distributions with specified Pearson correlations. An illustration shows the applicability of the percentile power method technique to publicly available statistics from the Idaho state educational assessment.

Notes

The applied researcher may find it easier to think in terms of a subsample of size n drawn from a larger sample of size N.