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非参数单调函数估计的贝叶斯方法AI 解读

Summary The paper proposes two Bayesian approaches to non-parametric monotone function estimation. The first approach uses a hierarchical Bayes framework and a characterization of smooth monotone functions given by Ramsay that allows unconstrained estimation. The second approach uses a Bayesian regression spline model of Smith and Kohn with a mixture distribution of constrained normal distributions as the prior for the regression coefficients to ensure the monotonicity of…

排序集样本的刻画及其在分布函数估计中的应用AI 解读

Ranked-set sampling has been shown to provide improved estimators of the mean and variance when actual measurement of the observations is difficult but ranking of the elements in a sample is relatively easy. This result holds even if the ranking is imperfect. In this article, we provide a characterization of a ranked-set sample that makes the source of this additional information intuitively clear. It is applied to show that the empirical distribution function of a…

空间模式的非参数最大似然估计AI 解读

Let $\mathbf{X}$ be an absolutely continuous random variable in $\mathbb{R}^k$ with distribution function $F(\mathbf{x})$ and density $f(\mathbf{x})$. Let $\mathbf{X}_1, \cdots, \mathbf{X}_n$ be independent random variables distributed according to $F$. Mapping the spatial distribution of $\mathbf{X}$ normally entails drawing a map of the isopleths, or level curves, of $f$. In this paper, it is shown how to map the isopleths of $f$ nonparametrically according to the…

保序众数回归的最大似然估计AI 解读

For each $t$ in an index set $T$, let $P_t$ be a probability measure with mode $M(t)$. In this paper we consider a maximum likelihood nonparametric estimator $\hat{M}_n(t)$ of $M(t)$ subject to the constraint that $\hat{M}_n(\cdot)$ be isotonic with respect to an order on $T$. The estimator is a solution to a minimization problem with zero-one loss. The estimator is not a max-min or min-max representation of "naive" estimators. Naive modal estimators are used but they are…

多元众数与等值轮廓的迭代估计方法AI 解读

An iterative procedure for estimating the mode and isopleths of a multivariate distribution is presented. The mode is estimated by selecting a point from the final set in a nested decreasing sequence of convex sets, each one of which is iteratively the smallest closed, convex subset containing a certain proportion of its predecessor's data points. An isopleth is estimated by the boundary of the convex subset corresponding to a fixed number of iterations (independent of n).…

多元分布众数的估计AI 解读

Consider a random sample from an absolutely continuous multivariate distribution. Let $\mathscr{J}$ be a class of sets which are not too long and thin. A point $\mathbf{\theta}_n$ chosen from a minimum volume set $S_n \in \mathscr{J}$ containing at least $r = r(n)$ of the data may be used as an estimate of the mode of the distribution. In this paper, it is shown that $\mathbf{\theta}_n$ converges almost surely to the true mode under very minor conditions on $\{r(n)\}$ and…

众数非参数估计的一致性AI 解读

Let $X$ be an absolutely continuous real-valued random variable with additional restrictions to be imposed later. Venter (1967) ("On estimation of the mode," Ann. Math. Statist. 37 1446-1455) estimated the mode of $X$ by a point from the shortest interval containing a specified number $r = r(n)$ of observations. Venter demonstrated that such an estimator is strongly consistent under appropriate conditions on the distribution of $X$ and on $r(n)$. It is the purpose of this…