Estimation of a k-Monotone Density: Limit Distribution Theory and the Spline Connection, with Complete Proofs

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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. In Balabdaoui and Wellner (2004a), it was proved that both estimators exist and are splines of degree k−1 with simple knots. These knots, which are also the jump points of the (k−1)- st derivative of the estimators, cluster around a point x0 > 0 under the assumption that g0 has a continuous k-th derivative in a neighborhood of x0 and (−1)kg(k) 0 (x0) > 0. If τ − n and τ + n are two successive knots, we prove that the random “gap” τ + n − τ − n is Op(n−1/(2k+1)) for any k > 2 if a conjecture about the upper bound on the error in a particular Hermite interpolation via odd-degree splines holds. Based on the order of the gap, the asymptotic distribution of the Maximum Likelihood and Least Squares estimators can be established. We find that the j-th 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). AMS 2000 subject classifications: Primary 62G05, 60G99; secondary 60G15, 62E20.

Keywords and phrases: asymptotic distribution, completely monotone, convex, Hermite interpolation, inversion, k−fold integral of Brownian motion, least squares, maximimum likelihood, minimax risk, mixture models, multiply monotone, nonparametric estimation, rates of convergence, shape constraints, splines.


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