Goodness-of-Fit Tests Based on P-P Probability Plots

Abstract

Percentage-percentage (P-P) probability plots constructed from standardized observations possess some attractive features that are not shared by more commonly used quantile-quantile (Q-Q) plots. In particular, the identification of viable alternatives to a proposed probability model can be greatly facilitated by displaying curves on P-P plots to represent families of alternative models. A single curve can represent an entire family of alternatives indexed by both location and scale parameters. Two goodness-of-fit statistics, based on measures of linearity for standardized P-P plots, are proposed and simple approximations for percentage points of these statistics are presented for testing the fit of exponential, Gumbel (Weibull), and normal (lognormal) probability models with unknown parameters. Results of extensive Monte Carlo power comparisons with other goodness-of-fit tests are summarized. The proposed tests are shown to have superior power for detecting light-tailed and moderate-tailed alternatives to the exponential distribution, but statistics based on measures of linearity for Q-Q plots tend to be more powerful against relatively heavy-tailed alternatives to hypothesized distributions with more moderate tails, such as the normal and Gumbel distributions. Q-Q plots give greater emphasis to the tails of proposed distributions than P-P plots. The Anderson- Darling test is also shown to provide superior power against moderate-tailed and heavy-tailed alternatives to the normal and Gumbel distributions.

KEY WORDS:

• Chi-squared tests
• EDF statistics
• Exponential, Gumbel, and normal distributions
• Power comparisons
• Rectangle test
• Shapiro–Wilk statistic.