Testing for goodness rather than lack of fit of continuous probability distributions.

Stefan Wellek,Mauro Gasparini

doi:10.1371/journal.pone.0256499

Abstract

The vast majority of testing procedures presented in the literature as goodness-of-fit tests fail to accomplish what the term is promising. Actually, a significant result of such a test indicates that the true distribution underlying the data differs substantially from the assumed model, whereas the true objective is usually to establish that the model fits the data sufficiently well. Meeting that objective requires to carry out a testing procedure for a problem in which the statement that the deviations between model and true distribution are small, plays the role of the alternative hypothesis. Testing procedures of this kind, for which the term tests for equivalence has been coined in statistical usage, are available for establishing goodness-of-fit of discrete distributions. We show how this methodology can be extended to settings where interest is in establishing goodness-of-fit of distributions of the continuous type.

Highlights

Goodness-of-fit tests belong to the oldest and most frequently used methods of statistical inference
Many of the most widely used methods of statistical analysis rely on the assumption that the data follow a specific distributional law, and it is widespread practice to make sure of the adequacy of this assumption in a preliminary test
Since 1.0793 is an inner point of this interval, the conclusion is that the goodness-of-fit test for standard normality at level α = 0.05 leads under the chosen specifications to rejecting the null hypothesis of lack-of-fit

Summary

Introduction

Goodness-of-fit tests belong to the oldest and most frequently used methods of statistical inference. All inferential procedures presented in the existing literature as tests of goodness-of-fit, share one crucial feature: The statement that the model to be fitted coincides with the true distribution from which the data are taken, plays the role of the null hypothesis, implying that a significant result indicates lack rather than goodness of fit of the model. This is clearly at variance with the fact that in the vast majority of applications, interest will be in proving rather than falsifying the model so that such a test typically fails to serve the purpose of its user.

Assumptions and hypotheses formulation

Numerical results on the UMP test for goodness-of-fit

Xn f00 xi À

Illustrating example

Discussion