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球面数据的核密度估计AI 解读

We study two natural classes of kernel density estimators for use with spherical data. Members of both classes have already been used in practice. The classes have an element in common, but for the most part they are disjoint. However, all members of the first class are asymptotically equivalent to one another, and to a single element of the second class. In this sense the second class ‘contains’ the first. It includes some estimators which out-perform all those in the first…

Bootstrap 与基于似然的置信区域AI 解读

We describe a method for constructing likelihood-based confidence regions for a vector parameter, using the bootstrap and nonparametric density estimation. The technique is illustrated by application to a numerical example, and its theoretical properties are elucidated. It is argued that likelihood-based regions should not be approximated by ellipses, if we are to have any hope of capturing first-order departures from normality. Bootstrap algorithms for constructing…

构造置信区间需要多少次 Bootstrap 模拟AI 解读

We make two points about the number, $B$ of bootstrap simulations needed to construct a percentile-$t$ confidence interval based on an $n$ sample from a continuous distribution: (i) The bootstrap's reduction of error of coverage probability, from $O(n^{-1/2})$ to $O(n^{-1})$, is available uniformly in $B$, provided nominal coverage probability is a multiple of $(B + 1)^{-1}$. In fact, this improvement is available even if the number of simulations is held fixed as $n$ increas…

Bootstrap 与置信区间AI 解读

We derive an explicit formula for the first term in an unconditional Edgeworth-type expansion of coverage probability for the nonparametric bootstrap technique applied to a very broad class of "Studentized" statistics. The class includes sample mean, $k$-sample mean, sample correlation coefficient, maximum likelihood estimators expressible as functions of vector means, etc. We suggest that the bootstrap is really an empiric one-term Edgeworth inversion, with the bootstrap…

正交级数密度估计量的收敛速率AI 解读

SUMMARY We derive a lower bound to the rate of convergence of nonparametric density estimators on compact sets, and show that it is achieved by estimators based on orthogonal series. Unlike earlier work on this type of problem, the bound depends on the size of Fourier coefficients rather than explicitly on smoothness assumptions.

图像恢复中的若干平滑技术AI 解读

SUMMARY The problem is considered of restoring a blurred and/or noisy image using various regularization prescriptions. Preliminary work concerns the invertibility of point spread functions and the construction of a stochastic model for images. A simple linear regularization procedure is introduced, as well as the special cases of constrained deconvolution, maximum entropy restoration and least-squares filtering. Optimal choices for the degree of smoothing are obtained for…

正交级数密度估计中的项选择AI 解读

We show that Kronmal and Tarter's well-known rule for selecting the terms in an orthogonal series density estimator can lead to poor performance and even inconsistency in certain cases. These difficulties arise when the underlying density has a nonmonotone sequence of Fourier coefficients, as is likely to be the case with sharply peaked or multimodal distributions. We suggest a way of overcoming these shortcomings.

正交级数密度估计中的项选择AI 解读

We show that Kronmal and Tarter's well-known rule for selecting the terms in an orthogonal series density estimator can lead to poor performance and even inconsistency in certain cases. These difficulties arise when the underlying density has a nonmonotone sequence of Fourier coefficients, as is likely to be the case with sharply peaked or multimodal distributions. We suggest a way of overcoming these shortcomings.

利用未分类数据改进混合分布成分密度的非参数估计AI 解读

SUMMARY We present a very simple, utilitarian technique for combining categorized and uncategorized data to construct a nonparametric estimator of a component mixture density. The method can be used for a wide class of estimators, and provides a reduction in the variance compared with the estimator based solely on categorized data.

针对特定备择设计的拟合优度检验AI 解读

SUMMARY We examine a large class of distributional tests equivalent to tests based on degenerate U-statistics. The factors which influence the performance of these tests are isolated. An analysis of the factors leads us to suggest tests with improved power properties against certain types of alternatives. One such test is based on a chi squared statistic which employs overlapping cells, rather than disjoint cells. It is shown that this test has power superior to that of the…

估计强度时对线段计数边界效应的校正AI 解读

A formula is suggested for correcting segment or fibre counts for edge effects. The correction results in an unbiased estimate of segment intensity, provided only that the observation field is convex. No assumptions are made about the distribution of fibre length. A method is proposed for constructing confidence intervals for segment intensity, using the corrected segment count.

正则变动参数的自适应估计AI 解读

The problem of estimating shape and scale parameters for a distribution with regularly varying tails is related to that of nonparametrically estimating a density at a fixed point, in that optimal construction of the estimators depends substantially upon unknown features of the distribution. We show how to overcome this problem by using adaptive methods. Our main results hold very generally, for a large class of adaptive estimators. Later we consider specific versions of…

估计线段强度时对边缘效应造成的计数偏差进行校正AI 解读

A formula is suggested for correcting segment or fibre counts for edge effects. The correction results in an unbiased estimate of segment intensity, provided only that the observation field is convex. No assumptions are made about the distribution of fibre length. A method is proposed for constructing confidence intervals for segment intensity, using the corrected segment count.

概率密度核估计量的一项最优性质AI 解读

SUMMARY Let f̂(x) be an asymptotically unbiased, nonparametric estimator of the density f(x) based on observations falling within the interval (x – ∈, x + ∈), where ∈ → 0 as the sample size diverges. We derive a lower bound to the variance of f(x), and prove that the bound is attained by a kernel estimator. This result is extended to estimates of f (r)(x), and to estimates of derivatives whose bias is improved by taking out the effects of unwanted terms.

正则变尾参数估计的最优可达收敛速率AI 解读

We derive lower bounds to rates of convergence for estimators of shape and scale parameters in distributions with regularly varying tails. We exhibit simple estimators which attain these rates.

We examine large-sample properties of cross-validation for estimating cell probabilities, starting from a completely general measure of loss. Necessary and sufficient conditions on the loss function are derived for the resulting estimator to be consistent, or to minimize expected loss. These results reveal that cross-validation is extremely sensitive to the shape of the loss function. Nevertheless, when the loss function is chosen correctly, cross-validation can be relied on…

混合比例的有效非参数估计AI 解读

SUMMARY By constructing a sequence of multinomial approximations and related maximum likelihood estimators, we derive a Cramér-Rao lower bound for nonparametric estimators of the mixture proportions and thereby characterize asymptotically optimal estimators. For the case of the sampling model M2 of Hosmer (1973) it is shown that the sequence of maximum likelihood estimators, which can be obtained explicitly, is asymptotically optimal in this sense. The results hold true even…

回归函数核估计量的积分平方误差性质AI 解读

Weak laws of large numbers and central limit theorems are proved for integrated square error of kernel estimators of regression functions. The regressor is assumed to take values in $\mathbb{R}^p$, and the regressand, $X$, to be real valued. It is shown that in many cases, integrated square error is asymptotically normally distributed and independent of the $X$-sample. As an application, a test for the regression function (such as that proposed by Konakov) is seen to be…

概率与概率密度非参数估计中的交叉验证AI 解读

We examine large-sample properties of cross-validation for estimating cell probabilities, starting from a completely general measure of loss. Necessary and sufficient conditions on the loss function are derived for the resulting estimator to be consistent, or to minimize expected loss. These results reveal that cross-validation is extremely sensitive to the shape of the loss function. Nevertheless, when the loss function is chosen correctly, cross-validation can be relied on…

密度估计中最小二乘交叉验证的大样本最优性AI 解读

We prove that the method of cross-validation suggested by A. W. Bowman and M. Rudemo achieves its goal of minimising integrated square error, in an asymptotic sense. The tail conditions we impose are only slightly more severe than the hypothesis of finite variance, and so least squares cross-validation does not exhibit the pathological behaviour which has been observed for Kullback-Leibler cross-validation. This is apparently the first time that a cross-validatory procedure…

摘要暂缺

分布函数的正交级数估计及其应用AI 解读

SUMMARY Motivated by applications to the non-parametric estimation of mixture proportions, we develop techniques for multivariate orthogonal series estimation of distribution functions. We approximate the empiric distribution function, rather than the true distribution function. This enables us to use trigonometric series methods to estimate distribution functions with unbounded support.

适用于定性与定量数据的正交级数方法AI 解读

We introduce and describe orthogonal series methods for estimating the density of qualitative, quantitative or mixed data. The techniques are completely nonparametric in character, and so may be used in situations where parametric models are difficult to construct. Just this situation arises in the context of mixed--both qualitative and quantitative--data, where there are few parametric models.

基于经验特征函数的正态性检验AI 解读

We present a simple test for normality against the alternative that the underlying distribution is long tailed. The test is based on the behaviour of the empirical characteristic function in the neighbourhood of the origin, and is very competitive with several well-known tests for normality.