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平滑单调密度与单峰密度的极大似然估计AI 解读

We study the nonparametric estimation of univariate monotone and unimodal densities usingthe maximum smoothed likelihood approach. The monotone estimator is the derivative of the least concave majorant of the distribution correspondingto a kernel estimator.We prove that the mapping on distributions $\Phi$ with density $\varphi$, $$\varphi \mapsto \text{the derivative of the least concave majorant of $\Phi},$$ is a contraction in all $L^P$ norms $(1 \leq p \leq \infty)$, and…

Good 非参数最大似然密度估计量的最优收敛率AI 解读

We study maximum penalized likelihood density estimation using the first roughness penalty functional of Good. We prove a simple pointwise comparison result with a kernel estimator based on the two-sided exponential kernel. This leads to $L^1$ convergence results similar to those for kernel estimators. We also prove Hellinger distance bounds for the roughness penalized estimator.

非参数反卷积的非线性平滑EM密度估计与自动平滑选择AI 解读

We study a nonparametric deconvolution density estimation problem. The estimator is obtained by an EM algorithm for a smoothed maximum likelihood estimation problem, which has a unique continuous solution. We present an implementation of the procedure incorporating a data-driven discrepancy principle for selecting the smoothing parameter. Simulations illustrate the good properties of the resulting estimator when the unknown distribution is smooth and has regularly varying…

自动选择平滑参数的非线性平滑 EM 反卷积密度估计AI 解读

We study a nonparametric deconvolution density estimation problem. The estimator is obtained by an EM algorithm for a smoothed maximum likelihood estimation problem, which has a unique continuous solution. We present an implementation of the procedure incorporating a data-driven discrepancy principle for selecting the smoothing parameter. Simulations illustrate the good properties of the resulting estimator when the unknown distribution is smooth and has regularly varying…

逆问题中的最大平滑似然密度估计AI 解读

We consider the problem of estimating a pdf $f$ from samples $X_1, X_2, \ldots, X_n$ of a random variable with pdf $\mathscr{K}f$, where $\mathscr{K}$ is a compact integral operator. We employ a maximum smoothed likelihood formalism inspired by a nonlinearly smoothed version of the EMS algorithm of Silverman, Jones, Wilson and Nychka. We show that this nonlinearly smoothed algorithm is itself an EM algorithm, which helps explain the strong convergence properties of the…

拟合优度检验中 Cramér–von Mises 与非参数函数估计方法的渐近比较AI 解读

Two new statistics for testing goodness-of-fit are derived from the viewpoint of nonparametric density estimation. These statistics are closely related to the Neyman smooth and Cramer-von Mises statistics but are shown to have superior properties both through asymptotic and small sample analyses. Comparison of the proposed tests with the Cramer-von Mises statistic requires the development of a novel technique for comparing tests that are capable of detecting local…

平滑拟合优度检验：分位数函数方法AI 解读

A quantile function based analog to Neyman's smooth tests for goodness-of-fit hypotheses is developed. These procedures possess asymptotic and small-sample power properties similar to those for smooth tests. They are, however, computationally simple, applicable to both complete and Type II censored data, and easily focused in the direction of any specific set of alternatives. These results are used to develop a new test for exponentiality. Only location/scale null hypotheses…

用于多样本问题的 Pearson φ² 距离分量AI 解读

A general approach to the k-sample problem for continuous random variables, based on a generalization of Pearson's classical φ2 measure, is presented. Tests are developed by considering an analysis of variance type of orthogonal decomposition of a distance measure for a k-population goodness-to-fit hypothesis. Estimates of some of these components are shown to provide test statistics for the hypothesis that the k-populations have a common distribution. Several classical…

k 样本问题中 Pearson φ² 距离度量的分量AI 解读

Abstract A general approach to the k-sample problem for continuous random variables, based on a generalization of Pearson's classical φ2 measure, is presented. Tests are developed by considering an analysis of variance type of orthogonal decomposition of a distance measure for a k-population goodness-to-fit hypothesis. Estimates of some of these components are shown to provide test statistics for the hypothesis that the k-populations have a common distribution. Several…

由Pearson平方距离分量导出的检验统计量AI 解读

A general approach to the comparison of two absolutely continuous distributions is presented based on Pearson's θ2 measure. Although no simple estimates are available for θ2, it can be decomposed into components, all of which are easily estimated. Asymptotic distribution theory is derived for the component estimates, and the form of the components is considered for several standard problems. It is demonstrated that the components-of-θ2 approach provides a unified approach to…