A likelihood ratio rest for shape-constrained density functions

Gary Chan

The celebrated Grenander (1956) estimator is the maximum likelihood estimator of a decreasing density function.  In contrast to alternative nonparametric density estimators, Grenander estimator does not require any smoothing parameters and is often viewed as a fully automatic procedure.  However, the monotonic density assumption might be questionable.  While testing qualitative constraints such as monotonicity are difficult in general, we show that a likelihood ratio test statistic Kₙ has an incredibly simple asymptotic null distribution:  ⸍²(Kₙ-γ), where γ is the Euler-Mascheroni constant, converges to a normal distribution with mean 0 and variance π²/6-1.  The results are shown based on a connection between the test statistic and uniform spacing distributions, and by establishing a leading Oₚ(n-¹⁶) remainder of the normalized test statistic.