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分位数的平滑经验似然置信区间AI 解读

Standard empirical likelihood confidence intervals for quantiles are identical to sign-test intervals. They have relatively large coverage error, of size $n^{-1/2}$, even though they are two-sided intervals. We show that smoothed empirical likelihood confidence intervals for quantiles have coverage error of order $n^{-1}$, and may be Bartlett-corrected to produce intervals with an error of order only $n^{-2}$. Necessary and sufficient conditions on the smoothing parameter,…

插值构造的非参数预测区间与置信区间AI 解读

SUMMARY In several important statistical problems, prediction intervals and confidence intervals can be constructed with coverage levels which are known precisely but cannot be rendered equal to predetermined levels such as 0.95. One solution to this difficulty is to interpolate between such intervals. We show that simple linear interpolation reduces the order of coverage error, but that higher orders of interpolation produce no further improvement. The error is reduced by a…

高维数据低维投影的近似线性性质AI 解读

This paper studies the shapes of low dimensional projections from high dimensional data. After standardization, let $\mathbf{x}$ be a $p$-dimensional random variable with mean zero and identity covariance. For a projection $\beta'\mathbf{x}, \|\beta\| = 1$, find another direction $b$ so that the regression curve of $b'\mathbf{x}$ against $\beta'\mathbf{x}$ is as nonlinear as possible. We show that when the dimension of $\mathbf{x}$ is large, for most directions $\beta$ even…

密度估计局部平滑的插件规则AI 解读

Optimal local smoothing of a curve estimator requires knowledge of various derivatives of the curve in the neighbourhood of the point at which estimation is being conducted. One empirical approach to selecting the amount of smoothing is to employ pilot estimators to approximate those derivatives, and substitute the approximate values into an analytical formula for the desired local bandwidth. In the present paper we study how bandwidth choice for the pilot estimators affects…

用于自助法的平衡重要性重抽样AI 解读

We show that the method of importance resampling, introduced by Vernon Johns and Anthony Davison, may be enhanced by balancing the resamples. It is demonstrated that "balanced importance resampling" improves on both "balanced uniform resampling" and "random importance resampling", from the viewpoint of statistical efficiency. Moreover, the range of applications for which efficient resampling methods may be applied is extended to include statistics which are smooth functions…

单指标模型中的最优平滑AI 解读

Single-index models generalize linear regression. They have applications to a variety of fields, such as discrete choice analysis in econometrics and dose response models in biometrics, where high-dimensional regression models are often employed. Single-index models are similar to the first step of projection pursuit regression, a dimension-reduction method. In both cases the orientation vector can be estimated root-n consistently, even if the unknown univariate function (or…

密度估计非参数反卷积的傅里叶方法AI 解读

SUMMARY We consider the problem of constructing a nonparametric estimate of a probability density function h from independent random samples of observations from densities a and f, when a represents the convolution of h and f. Our approach is based on truncated Fourier inversion, in which the truncation point plays the role of a smoothing parameter. We derive the asymptotic mean integrated squared error of the estimate and use this formula to suggest a simple practical…

分形维数盒计数估计量的表现AI 解读

Box-counting estimators are popular for estimating fractal dimension. However, very little is known of their stochastic properties, despite increasing statistical interest in their application. We show that, if the irregular curve to which the estimators are applied is modelled by a Gaussian process, concise formulae may be developed for asymptotic bias and variance of box-counting estimators. These formulae point to critical differences between a native form of the…

随机系数回归中的系数分布估计AI 解读

Random coefficient regression models are important in representing linear models with heteroscedastic errors and in unifying the study of classical fixed effects and random effects linear models. For prediction intervals and for bootstrapping in random coefficient regressions, it is necessary to estimate the distributions of the random coefficients consistently. We show that this is often possible and provide practical representative estimators of these distributions.

条件分布的Bootstrap估计AI 解读

Techniques are developed for bootstrap estimation of conditional distributions, with application to confidence intervals and hypothesis tests for one parameter, conditional on the value of an estimator of another. Both Monte Carlo and saddlepoint methods for approximating bootstrap distributions are considered, and empirical methods are suggested for implementing these techniques. For example, in the case of Monte Carlo methods, we suggest empirical techniques for selecting…

渐近展开启发的插值规则AI 解读

We suggest a simple interpolation rule, based on Edgeworth and Cornish-Fisher expansions, which permits highly accurate numerical approximation in problems involving calculation of distributions or quantiles without requiring explicit calculation of terms in the expansions.

有限总体分布函数估计量的性质AI 解读

Two leading estimators of the finite population distribution function are studied. Their large-sample mean squared errors are compared, under a given model. Neither estimator is universally better; circumstances are identified where the model-based estimator would do poorly, despite employing the correct model.

变带宽密度估计量的全局性质AI 解读

It is argued that mean integrated squared error is not a useful measure of the performance of a variable bandwidth density estimator based on Abramson's square root law. The reason is that when the unknown density $f$ has even moderately light tails, properties of those tails drive the formula for optimal bandwidth, to the virtual exclusion of other properties of $f$. We suggest that weighted integrated squared error be employed as the performance criterion, using a weight…

非参数回归中的 Bootstrap 置信区间AI 解读

Several authors have developed bootstrap methods for constructing confidence intervals in nonparametric regression. On each occasion a nonpivotal approach has been employed. Nonpivotal methods are still the overwhelmingly popular choice when statisticians use the bootstrap to compute confidence intervals, but they are not necessarily the most appropriate. In this paper we point out some of the theoretical advantages of pivoting. They include a reduction in the error of the…

偏差估计对密度 Bootstrap 置信区间覆盖精度的影响AI 解读

The bootstrap is a poor estimator of bias in problems of curve estimation, and so bias must be corrected by other means when the bootstrap is used to construct confidence intervals for a probability density. Bias may either be estimated explicitly, or allowed for by undersmoothing the curve estimator. Which of these two approaches is to be preferred? In the present paper we address this question from the viewpoint of coverage accuracy, assuming a given number of derivatives…

判别问题中bootstrap与交叉验证误判率估计的偏差和变异AI 解读

Simulation studies have shown that bootstrap and cross-validation estimators of aggregate error rate in discrimination problems have different properties, the former having less variability but greater bias. We show by use of a local alternative model that the main differences in bias and variability emerge only when the populations are close. In this context the bootstrap method has less variability but an order of magnitude greater bias than cross-validation.

利用次序统计量伴随变量估计分布：自助法 Monte Carlo 模拟应用AI 解读

SUMMARY We show that a simulation method suggested by Efron for approximating bootstrap distributions is closely related to techniques based on concomitants of order statistics and develop its asymptotic properties from that viewpoint. We prove that the method produces Monte Carlo approximations with variance and mean-squared error decreasing like B− 1 n− 1/2, where B denotes the number of simulations and n equals the sample size. Therefore Efron's method can be a competitor…

经验泛函与高效平滑参数选择AI 解读

SUMMARY A striking feature of curve estimation is that the smoothing parameter ĥ 0, which minimizes the squared error of a kernel or smoothing spline estimator, is very difficult to estimate. This is manifest both in slow rates of convergence and in high variability of standard methods such as cross-validation. We quantify this difficulty by describing nonparametric information bounds and exhibit asymptotically efficient estimators of ĥ 0 that attain the bounds. The…

由渐近展开启发的插值规则AI 解读

We suggest a simple interpolation rule, based on Edgeworth and Cornish-Fisher expansions, which permits highly accurate numerical approximation in problems involving calculation of distributions or quantiles without requiring explicit calculation of terms in the expansions.

有限总体分布函数估计量的性质AI 解读

Two leading estimators of the finite population distribution function are studied. Their large-sample mean squared errors are compared, under a given model. Neither estimator is universally better; circumstances are identified where the model-based estimator would do poorly, despite employing the correct model.

判别问题中 bootstrap 与交叉验证误差率估计的偏差和变异性AI 解读

Simulation studies have shown that bootstrap and cross-validation estimators of aggregate error rate in discrimination problems have different properties, the former having less variability but greater bias. We show by use of a local alternative model that the main differences in bias and variability emerge only when the populations are close. In this context the bootstrap method has less variability but an order of magnitude greater bias than cross-validation.

交叉验证函数中的局部极小值AI 解读

SUMMARY The method of least squares cross-validation for choosing the bandwidth of a kernel density estimator has been the object of considerable research, through both theoretical analysis and simulation studies. The method involves the minimization of a certain function of the bandwidth. One of the less attractive features of this method, which has been observed in simulation studies but has not previously been understood theoretically, is that rather often the…

消除核型非参数回归边界效应的几何方法AI 解读

We introduce a simple geometric method for removing edge effects from kernel-type nonparametric regression estimators. It involves reflecting the data set in two estimated points, thereby generating a new data set with three times the range of the original data. The usual kernel-type estimator may be applied to the new, enlarged data set, without any danger of edge effects. This technique is applicable generally to both regularly spaced and randomly spaced designs and admits…

均匀重抽样与重要性重抽样的 Bahadur 表示及渐近相对效率AI 解读

We derive Bahadur-type representations for quantile estimates obtained from two different types of nonparametric bootstrap resampling--the commonly used uniform resampling method, where each sample value is drawn with the same probability, and importance resampling, where different sample values are assigned different resampling weights. These results are applied to obtain the relative efficiency of uniform resampling and importance resampling and to derive exact convergence…