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Estimation of a k-monotone density, part 4: limit distribution theory and the spline connection | | Fadoua Balabdaoui
; Jon A. Wellner
; | | Date: |
4 Sep 2005 | | Subject: | Statistics MSC-class: 62G05, 60G99 (primary), 60G15, 62E20 (secondary) | math.ST | | Affiliation: | University of Goettingen), Jon A. Wellner (University of Washington | | Abstract: | We study the asymptotic behavior of the Maximum Likelihood and Least Squares estimators of a k-monotone density g_0 at a fixed point x_0 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 x_0 > 0 under the assumption that g_0 has a continuous k-th derivative in a neighborhood of x_0 and (-1)^k g^(k)_0(x_0) > 0. If tau^{-}_n and tau^{+}_n are two successive knots, we prove that the random ``gap’’ tau^{+}_n - tau^{-}_n is O_p (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 x_0 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-1-fold integral of Brownian motion plus a deterministic drift, when k is even (odd). The family of processes H_k is studied separately in the companion manuscript Balabdaoui and Wellner (2004c). | | Source: | arXiv, math.ST/0509081 | | Services: | Forum | Review | PDF | Favorites | |
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