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

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 H_k 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 τ_n⁺−τ_n⁻, where τ_n^± 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.