Testing Multiple Inequality Hypotheses : A Smoothed Indicator Approach

TL;DR

This paper introduces a new smoothed indicator method for testing multiple inequalities that eliminates the need for simulation to determine critical values, ensuring correct size and unbiasedness under broad conditions.

Contribution

It develops a novel origin-smooth approximation for the sum-of-negative-part statistic, enabling fixed critical value testing without simulation, and establishes its optimality properties.

Findings

Test maintains correct asymptotic size under weak assumptions.

Method is unbiased for a wide class of local alternatives.

Compared favorably with existing tests in structure, theory, and simulations.

Abstract

This paper proposes a class of origin-smooth approximators of indicators underlying the sum-of-negative-part statistic for testing multiple inequalities. The need for simulation or bootstrap to obtain test critical values is thereby obviated. A simple procedure is enabled using fixed critical values. The test is shown to have correct asymptotic size in the uniform sense that supremum finite-sample rejection probability over null-restricted data distributions tends asymptotically to nominal significance level. This applies under weak assumptions allowing for estimator covariance singularity. The test is unbiased for a wide class of local alternatives. A new theorem establishes directions in which the test is locally most powerful. The proposed procedure is compared with predominant existing tests in structure, theory and simulation.