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论投影寻踪回归AI 解读

We construct a tractable mathematical model for kernel-based projection pursuit regression approximation. The model permits computation of explicit formulae for bias and variance of estimators. It is shown that the bias of an orientation estimate dominates error about the mean--indeed, the latter is asymptotically negligible in comparison with bias. However, bias and error about the mean are of the same order in the case of projection pursuit curve estimates. Implications of…

Bootstrap的对偶重抽样AI 解读

An antithetic variates method for the bootstrap is proposed and discussed. It is applicable quite generally, to bias estimation, distribution function estimation and quantile estimation, for example. It is based on an ‘antithetic permutation’ of the sample, which amounts to ranking values of a certain function of the data. Once this has been done B uniform resampling operations may be immediately converted into 2B ‘effective’ resampling operations, yielding greater…

高效 bootstrap 模拟AI 解读

It is shown that three new, efficient bootstrap algorithms are asymptotically equivalent. This is done in two ways. First, asymptotic formulae for variances and mean squared errors are derived, and shown to be identical. Secondly, it is demonstrated that two of the methods may be viewed as approximations to the third. The three algorithms considered are the balanced bootstrap and the linear approximation method proposed by Davison, Hinkley & Schechtman (1986), and a centring…

参数似然与经验似然函数的比较AI 解读

In this paper, parametric and empirical likelihood functions or surfaces are compared. In particular, first- and second-order expansions for log likelihood functions are developed in nonparametric and parametric situations, where the functional of interest is a smooth function of vector means. The two approaches are compared and the precise order of difference between the nonparametric and parametric log likelihood functions is obtained. In general, the surfaces need not…

密度反卷积的最优收敛速度AI 解读

Suppose that the sum of two independent random variables X and Z is observed, where Z denotes measurement error and has a known distribution, and where the unknown density f of X is to be estimated. One application is the estimation of a prior density for a sequence of location parameters. A second application arises in the errors-in-variables problem for nonlinear and generalized linear models, when one attempts to model the distribution of the true but unobservable…

随机复杂度与非参数密度估计AI 解读

We use the concepts of stochastic complexity, description length, and model selection to develop data-based methods for choosing smoothing parameters in nonparametric density estimation. In the case of histogram estimators, we derive a simple, exact formula for stochastic complexity when the prior distribution of cell probabilities is uniform over the class of all possible choices. The formula depends only on the data and the smoothing parameter, which is readily chosen…

对称 Bootstrap 置信区间AI 解读

SUMMARY It is shown that in quite general circumstances, symmetric percentile-t bootstrap confidence intervals have coverage error O(n –2), where n denotes sample size. This compares with an error of roughly constant n –1 for equal-tailed percentile-t confidence intervals. It is also proved that symmetric intervals are not necessarily any longer than equal-tailed intervals. Arguments based on Edgeworth and Cornish–Fisher expansions are used to predict circumstances in which…

We develop a unified framework within which many commonly used bootstrap critical points and confidence intervals may be discussed and compared. In all, seven different bootstrap methods are examined, each being usable in both parametric and nonparametric contexts. Emphasis is on the way in which the methods cope with first- and second-order departures from normality. Percentile-$t$ and accelerated bias-correction emerge as the most promising of existing techniques. Certain…

利用密度差进行非参数判别AI 解读

We propose a technique for nonparametric discrimination in which smoothing parameters are chosen jointly, according to a criterion based on the difference between two densities. The approach is suitable for categorical, continuous and mixed data, and uses information from both populations to determine the smoothing parameter for any one population. In the case of categorical data, optimal performance is sometimes achieved using negative smoothing parameters, a property which…

学生化分位数的分布AI 解读

SUMMARY The asymptotic variance of a sample quantile depends on the value of the population density at the population quantile. Therefore Studentizing a sample quantile involves density estimation, either explicitly or implicitly. One popular way of Studentizing is to use the Siddiqui–Bloch–Gastwirth estimator, whose construction depends crucially on the choice of a smoothing parameter m. We examine the effect which the selection of m has on the level error of tests or…

Bartlett 校正后似然比统计量的检验水平误差AI 解读

It is well that Bartlett adjustment reduces level-error of the likelihood ratio statistic from order n−1 to order n−3/2. In the present note we show that level-error of the adjusted statistic is actually order n−2.

密度估计中的核阶数选择AI 解读

The selection of the order, i.e., number of vanishing moments, of the kernel in a kernel density estimator is considered from two points of view. First, theoretical properties are investigated by a mean integrated squared error analysis of the problem. Second, and perhaps more importantly, cross validation is proposed as a practical method of choice, and theoretical backing for this is provided through an asymptotic optimality result.

自动选择的回归平滑参数距最优值有多远？AI 解读

We address the problem of smoothing parameter selection for nonparametric curve estimators in the specific context of kernel regression estimation. Call the “optimal bandwidth” the minimizer of the average squared error. We consider several automatically selected bandwidths that approximate the optimum. How far are the automatically selected bandwidths from the optimum? The answer is studied theoretically and through simulations. The theoretical results include a central…

摘要暂缺

随机复杂度与非参数密度估计AI 解读

We use the concepts of stochastic complexity, description length, and model selection to develop data-based methods for choosing smoothing parameters in nonparametric density estimation. In the case of histogram estimators, we derive a simple, exact formula for stochastic complexity when the prior distribution of cell probabilities is uniform over the class of all possible choices. The formula depends only on the data and the smoothing parameter, which is readily chosen…

Bootstrap 重抽样与迭代AI 解读

We propose a single unifying approach to bootstrap resampling, applicable to a very wide range of statistical problems. It enables attention to be focused sharply on one or more characteristics which are of major importance in any particular problem, such as coverage error or length for confidence intervals, or bias for point estimation. Our approach leads easily and directly to a very general form of bootstrap iteration, unifying and generalizing present disparate accounts…

利用密度差进行非参数判别AI 解读

We propose a technique for nonparametric discrimination in which smoothing parameters are chosen jointly, according to a criterion based on the difference between two densities. The approach is suitable for categorical, continuous and mixed data, and uses information from both populations to determine the smoothing parameter for any one population. In the case of categorical data, optimal performance is sometimes achieved using negative smoothing parameters, a property which…

Bartlett 调整后似然比统计量的检验水平误差AI 解读

It is well that Bartlett adjustment reduces level-error of the likelihood ratio statistic from order n−1 to order n−3/2. In the present note we show that level-error of the adjusted statistic is actually order n−2.

Kullback—Leibler损失与密度估计AI 解读

"Discrimination information," or Kullback-Leibler loss, is an appropriate measure of distance in problems of discrimination. We examine it in the context of nonparametric kernel density estimation and show that its asymptotic properties are profoundly influenced by tail properties of the kernel and of the unknown density. We suggest ways of choosing the kernel so as to reduce loss, and describe the extent to which likelihood cross-validation asymptotically minimises loss.…

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 与连续性校正AI 解读

SUMMARY A continuity correction to the bootstrap is suggested, useful when the bootstrap is employed to approximate distributions of lattice-valued sample means. Versions of the correction are available quite generally, even in multivariate cases. Asymptotic theory reveals that the correction can reduce maximum rounding error by up to 50%. An example, where continuity correction is used in the construction of confidence intervals, illustrates the performance of the procedure…

自助法与基于似然的置信域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…

核密度估计带宽选择所固有的噪声AI 解读

In the setting of kernel density estimation, data-driven bandwidth, i.e., smoothing parameter, selectors are considered. It is seen that there is a well-defined, and surprisingly restrictive, bound on the rate of convergence of any automatic bandwidth selection method to the optimum. The method of least squares cross-validation achieves this bound.

回归问题中平滑参数选择技术的共同结构AI 解读

SUMMARY Two general methods have been used for choosing the degree of smoothing in both linear ridge-regression and nonparametric regression. One is based on a criterion of minimum risk and the other is based on a measure of fit to the data as summarised by a statistic based on the residuals. The comparative effects of these two approaches are investigated and one popular version of the second method is shown to oversmooth quite drastically. The work also generates…