Conditional Inference in Small Sample Scenarios Using a Resampling Approach

Clemens Draxler,Andreas Kurz

doi:10.3390/stats4040049

Abstract

This paper discusses a non-parametric resampling technique in the context of multidimensional or multiparameter hypothesis testing of assumptions of the Rasch model. It is based on conditional distributions and it is suggested in small sample size scenarios as an alternative to the application of asymptotic or large sample theory. The exact sampling distribution of various well-known chi-square test statistics like Wald, likelihood ratio, score, and gradient tests as well as others can be arbitrarily well approximated in this way. A procedure to compute the power function of the tests is also presented. A number of examples of scenarios are discussed in which the power function of the test does not converge to 1 with an increasing deviation of the true values of the parameters of interest from the values specified in the hypothesis to be tested. Finally, an attempt to modify the critical region of the tests is made aiming at improving the power and an R package is provided.

Highlights

IntroductionVarious resampling approaches have been suggested in the statistical literature, notably permutation tests, cross-validation, the jackknife, and the bootstrap
Sample Scenarios Using a ResamplingVarious resampling approaches have been suggested in the statistical literature, notably permutation tests, cross-validation, the jackknife, and the bootstrap
A taxonomy of resampling techniques is, for example, provided by Rodgers [1]. These approaches are applicable to a wide range of statistical problems. They are utilized in cases where the probability distribution of an estimator or a test statistic is unknown

Summary

Introduction

Various resampling approaches have been suggested in the statistical literature, notably permutation tests, cross-validation, the jackknife, and the bootstrap. A taxonomy of resampling techniques is, for example, provided by Rodgers [1]. These approaches are applicable to a wide range of statistical problems. In principle, they are utilized in cases where the probability distribution of an estimator or a test statistic is unknown. In a number of practically relevant scenarios, the distribution can only be derived asymptotically, i.e., for large samples. When the sample size is small or in the case of sparse data resampling techniques can be used instead

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