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Estimation of a \(k\)-monotone density: limit distribution theory and the spline connection. (English) Zbl 1129.62019

Summary: 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 \(k>2\). We find that the \(j\) th derivative of the estimators at \(x_{0}\) converges at the rate \(n^{ - (k - j)/(2k+1)}\) for \(j=0, \cdots , k - 1\). 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 \(k - 1\) with simple knots. Establishing the order of the random gap \(\tau _n^{+} - \tau _n^{ - }\), where \(\tau _n^{\pm }\) 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.

MSC:

62G05 Nonparametric estimation
62E20 Asymptotic distribution theory in statistics
60J99 Markov processes
60G99 Stochastic processes
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