Estimation of a k-monotone density: Limit distribution theory and the spline connection

The Annals of Statistics · 2007 · Volume 35 · Issue 6

Abstract

We study the asymptotic behavior of the Maximum Likelihood and Least Squares Estimators of a k-monotone density g0 at a fixed point x0 when k>2. We find that the jth derivative of the estimators at x0 converges at the rate n−(k−j)/(2k+1) for j=0, …, k−1. The limiting distribution depends on an almost surely uniquely defined stochastic process Hk that stays above (below) the k-fold integral of Brownian motion plus a deterministic drift when k is even (odd). Both the MLE and LSE are known to be splines of degree k−1 with simple knots. Establishing the order of the random gap τn+−τn−, where τn± denote two successive knots, is a key ingredient of the proof of the main results. We show that this “gap problem” can be solved if a conjecture about the upper bound on the error in a particular Hermite interpolation via odd-degree splines holds.

Keywords

60G15, 60G99, 62E20, 62G05, asymptotic distribution, completely monotone, convex, Hermite interpolation, inversion, k-fold integral of Brownian motion, least squares, maximum likelihood, minimax risk, Mixture models, multiply monotone, nonparametric estimation, rates of convergence, shape constraints, splines

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