Goodness of Fit via Non‐parametric Likelihood Ratios

Abstract

To test if a density "f" is equal to a specified "f" 0, one knows by the Neyman-Pearson lemma the form of the optimal test at a specified alternative "f" 1. Any non-parametric density estimation scheme allows an estimate of "f". This leads to estimated likelihood ratios. Properties are studied of tests which for the density estimation ingredient use log-linear expansions. Such expansions are either coupled with subset selectors like the Akaike information criterion and the Bayesian information criterion regimes, or use order growing with sample size. Our tests are generalized to testing the adequacy of general parametric models, and to work also in higher dimensions. The tests are related to, but are different from, the 'smooth tests' that go back to Neyman [Skandinavisk Aktuarietidsskrift 20(1937) 149] and that have been studied extensively in recent literature. Our tests are large-sample equivalent to such smooth tests under local alternative conditions, but different from the smooth tests and often better under non-local conditions. Copyright 2004 Board of the Foundation of the Scandinavian Journal of Statistics..

A data-driven smooth test of symmetry

“…Applying Lemma 2 of Claeskens and Hjort (2004) yields N S p → 2L S as n → ∞, where N S and L S are given by (2) and (3) respectively. It follows that Pr(S = 1) → 1 under the selection rule (14) based on N s .…”

“…Kallenberg and Ledwina (1995) further established its consistency against essentially all fixed alternatives, allowing K n to increase with sample size at a proper rate. Claeskens and Hjort (2004) studied this test under local alternatives. The distribution of N s under the null hypothesis is given by the following Lemma.…”

Order selection tests with multiply imputed data

“…Such test was first considered for goodness-of-fit testing by Ledwina (1994). Claeskens and Hjort (2004) discuss some alternative schemes based on both BIC and AIC that have better power properties. Such tests could be of interest to investigate in the missing data setting as well.…”

Discussion: Approximating data

“…A natural idea is now to insert a nonparametric density estimate f for the unknown alternative position, leading to a genuine (and density-based) test statistic W n ( f ). Various tests worked with in Claeskens and Hjort (2004) are of this sort, and are shown there to have good theoretical and practical performance.…”