Estimation of a k‐monotone density: characterizations, consistency and minimax lower bounds

Fadoua Balabdaoui and Jon A. Wellner

Statistica Neerlandica, 2010, vol. 64, issue 1, 45-70

Abstract: The classes of monotone or convex (and necessarily monotone) densities on can be viewed as special cases of the classes of k‐monotone densities on . These classes bridge the gap between the classes of monotone (1‐monotone) and convex decreasing (2‐monotone) densities for which asymptotic results are known, and the class of completely monotone (∞‐monotone) densities on . In this paper we consider non‐parametric maximum likelihood and least squares estimators of a k‐monotone density g0. We prove existence of the estimators and give characterizations. We also establish consistency properties, and show that the estimators are splines of degree k−1 with simple knots. We further provide asymptotic minimax risk lower bounds for estimating the derivatives , at a fixed point x0 under the assumption that .

Date: 2010
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https://doi.org/10.1111/j.1467-9574.2009.00438.x

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