Likelihood Ratio Test for Monotonicity Density

Abstract

We study the likelihood ratio test statistic for a lasting, but unconventional, hypothesis testing problem on whether a random sample follows a distribution with a nonincreasing density function. The obtained test statistic has a surprisingly simple asymptotic null distribution, which is Gaussian, instead of the well-known chi-square for generic likelihood ratio tests. Also, the limiting distribution does not depend on the underlying density but on some universal constants such as the Euler-Mascheroni constant. The underlying density only plays a role in the higher-order effect, whereby modifications to the test statistic are suggested.