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December 2007 Estimation of a k-monotone density: Limit distribution theory and the spline connection
Fadoua Balabdaoui, Jon A. Wellner
Ann. Statist. 35(6): 2536-2564 (December 2007). DOI: 10.1214/009053607000000262


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−(kj)/(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.


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Fadoua Balabdaoui. Jon A. Wellner. "Estimation of a k-monotone density: Limit distribution theory and the spline connection." Ann. Statist. 35 (6) 2536 - 2564, December 2007.


Published: December 2007
First available in Project Euclid: 22 January 2008

zbMATH: 1129.62019
MathSciNet: MR2382657
Digital Object Identifier: 10.1214/009053607000000262

Primary: 60G99 , 62G05
Secondary: 60G15 , 62E20

Keywords: 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

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 6 • December 2007
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