Donsker-type theorems for nonparametric maximum likelihood estimators
Donsker-type theorems for nonparametric maximum likelihood estimators
复制标题
非参数最大似然估计的 Donsker 型定理
DOI：
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
Richard Nickl
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文献类型：
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作者：
Richard Nickl
Let $${mathcal{P}}$$ be a nonparametric probability model consisting of smooth probability densities and let $${hat{p}_{n}}$$ be the corresponding maximum likelihood estimator based on n independent observations each distributed according to the law $${mathbb{P}}$$ . With $$hat{mathbb{P}}_{n}$$ denoting the measure induced by the density $${hat{p}_{n}}$$ , define the stochastic process $${hat{
u}}_{n}: flongmapsto sqrt{n} int fd({hat{mathbb{P}}}_{n} -mathbb{P})$$ where f ranges over some function class $${mathcal{F}}$$ . We give a general condition for Donsker classes $${mathcal{F}}$$ implying that the stochastic process $$hat{
u}_{n}$$ is asymptotically equivalent to the empirical process in the space $${ell ^{infty }(mathcal{F})}$$ of bounded functions on $${ mathcal{F}}$$ . This implies in particular that $$hat{
u}_{n}$$ converges in law in $${ell ^{infty }(mathcal{F})}$$ to a mean zero Gaussian process. We verify the general condition for a large family of Donsker classes $${mathcal{ F}}$$ . We give a number of applications: convergence of the probability measure $${hat{mathbb{P}}_{n}}$$ to $${mathbb{P}}$$ at rate $${sqrt{n}}$$ in certain metrics metrizing the topology of weak(-star) convergence; a unified treatment of convergence rates of the MLE in a continuous scale of Sobolev-norms; $${sqrt{n}}$$ -efficient estimation of nonlinear functionals defined on $${mathcal{P}}$$ ; limit theorems at rate $${sqrt{n}}$$ for the maximum likelihood estimator of the convolution product $${mathbb{Past P}}$$ .
Let $${mathcal{P}}$$ be a nonparametric probability model consisting of smooth probability densities and let $${hat{p}_{n}}$$ be the corresponding maximum likelihood estimator based on n independent observations each distributed according to the law $${mathbb{P}}$$ . With $$hat{mathbb{P}}_{n}$$ denoting the measure induced by the density $${hat{p}_{n}}$$ , define the stochastic process $${hat{
u}}_{n}: flongmapsto sqrt{n} int fd({hat{mathbb{P}}}_{n} -mathbb{P})$$ where f ranges over some function class $${mathcal{F}}$$ . We give a general condition for Donsker classes $${mathcal{F}}$$ implying that the stochastic process $$hat{
u}_{n}$$ is asymptotically equivalent to the empirical process in the space $${ell ^{infty }(mathcal{F})}$$ of bounded functions on $${ mathcal{F}}$$ . This implies in particular that $$hat{
u}_{n}$$ converges in law in $${ell ^{infty }(mathcal{F})}$$ to a mean zero Gaussian process. We verify the general condition for a large family of Donsker classes $${mathcal{ F}}$$ . We give a number of applications: convergence of the probability measure $${hat{mathbb{P}}_{n}}$$ to $${mathbb{P}}$$ at rate $${sqrt{n}}$$ in certain metrics metrizing the topology of weak(-star) convergence; a unified treatment of convergence rates of the MLE in a continuous scale of Sobolev-norms; $${sqrt{n}}$$ -efficient estimation of nonlinear functionals defined on $${mathcal{P}}$$ ; limit theorems at rate $${sqrt{n}}$$ for the maximum likelihood estimator of the convolution product $${mathbb{Past P}}$$ .