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

• 成果类型: Article
• 署名作者: Balabdaoui, Fadoua; Wellner, Jon A.
• 署名单位: University of Gottingen; University of Washington; University of Washington Seattle
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053607000000262
• 发表日期: 2007
• 页码: 2536-2564
• 关键词: maximum-likelihood-estimation; population pharmacokinetics; regression

摘要

We study the asymptotic behavior of the Maximum Likelihood and Least Squares Estimators of a k-monotone density go at a fixed point x(0) when k > 2. We find that the jib 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-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 - I with simple knots. Establishing the order of the random gap tau(+)(n) - tau(-)(n) where tau(+/-)(n) 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.