Smoothed nonparametric tests and their properties

TL;DR

This paper introduces new smoothed versions of sign and Wilcoxon's signed rank tests using kernel estimators, analyzing their properties, asymptotic behavior, and efficiency, showing they are nearly equivalent to traditional tests.

Contribution

The paper develops and analyzes smoothed nonparametric tests based on kernel estimators, providing asymptotic properties and efficiency comparisons with classical tests.

Findings

Smoothed tests are asymptotically equivalent to classical tests in efficiency.

Asymptotic expectations and variances are distribution-independent under the null hypothesis.

Edgeworth expansions for the tests are derived with residuals independent of the underlying distribution.

Abstract

In this paper we propose new smoothed sign and Wilcoxon's signed rank tests, which are based on a kernel estimator of the underlying distribution function of data. We discuss approximations of $p$-values and asymptotic properties of these tests. The new smoothed tests are equivalent to the ordinary sign and Wilcoxon's tests in the sense of the Pitman's asymptotic relative efficiency, and the differences of the ordinary and the new tests converge to zero in probability. Under the null hypothesis, the main terms of the asymptotic expectations and variances of the tests do not depend on the underlying distribution. Though the smoothed tests are not distribution-free, we can obtain Edgeworth expansions with residual term $o(n^{-1})$, which do not depend on the underlying distribution.