The Annals of Statistics

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

Fadoua Balabdaoui and Jon A. Wellner

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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. 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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Ann. Statist., Volume 35, Number 6 (2007), 2536-2564.

First available in Project Euclid: 22 January 2008

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Zentralblatt MATH identifier

Primary: 62G05: Estimation 60G99: None of the above, but in this section
Secondary: 60G15: Gaussian processes 62E20: Asymptotic distribution theory

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


Balabdaoui, Fadoua; Wellner, Jon A. Estimation of a k -monotone density: Limit distribution theory and the spline connection. Ann. Statist. 35 (2007), no. 6, 2536--2564. doi:10.1214/009053607000000262.

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