Goodness-of-Fit Criteria Based on Observations Quantized by Hypothetical and Empirical Percentiles

Authored by: I. Vajda , E. C. van der Meulen

Handbook of Fitting Statistical Distributions with R

Print publication date:  October  2010
Online publication date:  April  2016

Print ISBN: 9781584887119
eBook ISBN: 9781584887126
Adobe ISBN:

10.1201/b10159-30

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Abstract

Goodness-of-fit disparity statistics are defined as appropriately scaled ?-disparities or ?-divergences of quantized hypothetical and empirical distributions. It is shown that the classical Pearson-type statistics are obtained if we quantize by means of hypothetical percentiles, and that new spacings-based disparity statistics are obtained if we quantize by means of empirical percentiles. The main attention is paid to the asymptotic properties of the new disparity statistics and their comparisons with the spacings-based statistics known from the literature. First the asymptotic equivalence between them is proved, and then for the new statistics a general law of large numbers is proved, as well as an asymptotic normality theorem both under local and fixed alternatives. Special attention is devoted to the limit laws for the power divergence statistics of orders ? ∈ IR. Parameters of these laws are evaluated for ? ∈ (−1, ∞) in a closed-form and their continuity in ? on the subinterval (−1/2, ∞) is proved. These closed form expressions are used to compare local asymptotic powers of the tests based on these statistics, which allows us to extend previous asymptotic optimality results to the class of power divergence statistics. Tables of values of the asymptotic parameters are presented for selected representative orders of ? > −1/2. Programs are provided for the evaluation of three families of spacings-based statistics, including two families of true divergence statistics. These programs are applied to compare the three families for a specific hypothetical distribution and two examples of datasets.

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