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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References:

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