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Estimation of a k-monotone density: limit distribution theory and the Spline connection

Balabdaoui, Fadoua; Wellner, Jon (2007), Estimation of a k-monotone density: limit distribution theory and the Spline connection, Annals of Statistics, 35, 6, p. 2536-2564. http://dx.doi.org/10.1214/009053607000000262

Type
Article accepté pour publication ou publié
External document link
http://hal.archives-ouvertes.fr/hal-00363240/en/
Date
2007
Journal name
Annals of Statistics
Volume
35
Number
6
Publisher
Institute of Mathematical Statistics
Pages
2536-2564
Publication identifier
http://dx.doi.org/10.1214/009053607000000262
Metadata
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Author(s)
Balabdaoui, Fadoua
Wellner, Jon
Abstract (EN)
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 \mycite{balabwell:04a}, 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 $j=0, \ldots, k-1$. The limiting distribution depends on an almost surely uniquely defined stochastic process $H_k$ that stays above (below) the $k$-fold integral of Brownian motion plus a deterministic drift, when $k$ is even (odd).
Subjects / Keywords
asymptotic distribution; LSE; MLE; completely monotone; k-monotone; splines

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