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经验似然可作 Bartlett 校正AI 解读

It is shown that, in a very general setting, the empirical likelihood method for constructing confidence intervals is Bartlett-correctable. This means that a simple adjustment for the expected value of log-likelihood ratio reduces coverage error to an extremely low $O(n^{-2})$, where $n$ denotes sample size. That fact makes empirical likelihood competitive with methods such as the bootstrap which are not Bartlett-correctable and which usually have coverage error of size $n^{-1}$…

核密度估计中最优的数据驱动带宽选择AI 解读

A bandwidth selection method is proposed for kernel density estimation. This is based on the straightforward idea of plugging estimates into the usual asymptotic representation for the optimal bandwidth, but with two important modifications. The result is a bandwidth selector with the, by nonparametric standards, extremely fast asymptotic rate of convergence of n−½ where n ↑ ∞ denotes sample size. Comparison is given to other bandwidth selection methods, and small sample…

Bootstrap 的重要性重抽样AI 解读

We introduce an empirical method of importance resampling, which does not require analytical calculation of the resampling probabilities. Our method can easily be used as part of a general algorithm for Monte Carlo calculation of bootstrap confidence intervals and hypothesis tests. It produces consistent, efficient and unbiased Monte Carlo approximations. We also present a very general but elementary account of importance resampling, which shows that even optimal importance…

核密度估计中最优的数据驱动带宽选择AI 解读

A bandwidth selection method is proposed for kernel density estimation. This is based on the straightforward idea of plugging estimates into the usual asymptotic representation for the optimal bandwidth, but with two important modifications. The result is a bandwidth selector with the, by nonparametric standards, extremely fast asymptotic rate of convergence of n−½ where n ↑ ∞ denotes sample size. Comparison is given to other bandwidth selection methods, and small sample…

自助法中的重要性重抽样AI 解读

We introduce an empirical method of importance resampling, which does not require analytical calculation of the resampling probabilities. Our method can easily be used as part of a general algorithm for Monte Carlo calculation of bootstrap confidence intervals and hypothesis tests. It produces consistent, efficient and unbiased Monte Carlo approximations. We also present a very general but elementary account of importance resampling, which shows that even optimal importance…

非参数回归中的自适应 M 估计AI 解读

A method for robust nonparametric regression is discussed. We consider kernel $M$-estimates of the regression function using Huber's $\psi$-function and extend results of Hardle and Gasser to the case of random designs. A practical adaptive procedure is proposed consisting of simultaneously minimising a cross-validatory criterion with respect to both the smoothing parameter and a robustness parameter occurring in the $\psi$-function. This method is shown to possess a…

非参数回归中均值函数差异的 Bootstrap 检验AI 解读

A bootstrap test is proposed for detecting a difference between two mean functions in the setting of nonparametric regression. Error distributions in the regression model are permitted to be arbitrary and unequal. The test enjoys power properties akin to those in a parametric setting, in the sense that it can distinguish between regression functions distant only n −1/2 apart, where n is the sample size. It has exceptional level accuracy, with level error of only n −2, and…

非参数回归中渐近最优的差分型方差估计AI 解读

Abstract We define and compute asymptotically optimal difference sequences for estimating error variance in homoscedastic nonparametric regression. Our optimal difference sequences do not depend on unknowns, such as the mean function, and provide substantial improvements over the suboptimal sequences commonly used in practice. For example, in the case of normal data the usual variance estimator based on symmetric second-order differences is only 64% efficient relative to the…

摘要暂缺

可变带宽曲线估计量的偏差AI 解读

A major difficulty in understanding the properties of variable bandwidth methods (Breiman, Meisel & Purcell, 1977; Abramson, 1982) is that extremely lengthy and complex algebra is needed to assess the influence of bias. Indeed, the complexity is so great that it has forced investigators to use computer algebraic manipulation to determine formulae for bias. In this note we completely eliminate these algebraic obstacles by presenting a simple easy-to-use formula which gives…

摘要暂缺

删失线性回归半参数估计中的带宽选择AI 解读

Quantile and semiparametric M estimation are methods for estimating a censored linear regression model without assuming that the distribution of the random component of the model belongs to a known parametric family. Both methods require estimating derivatives of the unknown cumulative distribution function of the random component. The derivatives can be estimated consistently using kernel estimators in the case of quantile estimation and finite difference quotients in the…

质量工程中最优性能准则的非参数估计AI 解读

Box and Leon, Shoemaker and Kackar have discussed the problem of closeness to target in quality engineering. If the mean response $f(x, z)$ depends on $(x, z)$, the variance function is a PERMIA if it is $g(z)$, i.e., depends only on $z$. The goal is to find $(x_0, z_0)$ which minimizes variance while achieving a target mean value. We pose and answer the question: For given smoothness assumptions about $f$ and $g$, how accurately can we estimate $x_0$ and $z_0$? As part of…

经验似然的伪似然理论AI 解读

It is proved that, except for a location term, empirical likelihood does draw contours which are second-order correct for those of a pseudo-likelihood. However, except in the case of one dimension, this pseudo-likelihood is not that which would commonly be employed when constructing a likelihood-based confidence region. It is shown that empirical likelihood regions may be adjusted for location so as to render them second-order correct. Furthermore, it is proved that…

可变带宽曲线估计量的偏差AI 解读

A major difficulty in understanding the properties of variable bandwidth methods (Breiman, Meisel & Purcell, 1977; Abramson, 1982) is that extremely lengthy and complex algebra is needed to assess the influence of bias. Indeed, the complexity is so great that it has forced investigators to use computer algebraic manipulation to determine formulae for bias. In this note we completely eliminate these algebraic obstacles by presenting a simple easy-to-use formula which gives…

非参数回归中的方差估计AI 解读

A simple explicit estimator of variance in nonparametric regression is suggested, based on the mean square of a sequence of residuals. The estimator is valid for general error distributions, has elementary bias and variance properties, and is optimal to both first and second orders.

方向数据的自助法置信区域AI 解读

Methods are proposed for constructing bootstrap confidence regions for the mean direction of a random p-dimensional unit vector X with an arbitrary unimodal distribution on the p sphere. The approach of this article differs from that of other authors in that it is based on pivotal statistics. A general pivotal method is introduced that produces a wide variety of confidence regions on general p-dimensional spheres; included are confidence cones and likelihood-based regions.…

方向数据的Bootstrap置信区域AI 解读

Methods are proposed for constructing bootstrap confidence regions for the mean direction of a random p-dimensional unit vector X with an arbitrary unimodal distribution on the p sphere. The approach of this article differs from that of other authors in that it is based on pivotal statistics. A general pivotal method is introduced that produces a wide variety of confidence regions on general p-dimensional spheres; included are confidence cones and likelihood-based regions.…

摘要暂缺

回归中的方差函数估计：估计均值的影响AI 解读

SUMMARY We consider estimation of a variance function g in regression problems. Such estimation requires simultaneous estimation of the mean function f. We obtain clear results on the extent to which the smoothness of f influences best rates of convergence for estimating g. For example, in nonparametric regression with two derivatives on g, ‘classical' rates of convergence are possible if and only if the unknown f satisfies a Lipschitz condition of order 13 or more. If a…

高效自助法模拟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…

模拟阶数对 Monte Carlo 检验水平精度与功效的影响AI 解读

SUMMARY We discuss the effect of simulation order on level accuracy and power of Monte Carlo tests, in a very general setting. Both parametric problems, with or without nuisance parameters, and nonparametric problems are treated by a single unifying argument. It is shown that if the level of a Monte Carlo test is known only nominally, not precisely, then the level error of a Monte Carlo test is an order of magnitude less than that of the corresponding asymptotic test. This…

平滑与自助法AI 解读

Recent attention has focussed on possible improvements in performance of estimators which might flow from using the smoothed bootstrap. We point out that in a great many problems, such as those involving functions of vector means, any such improvements will be only second-order effects. However, we argue that substantial and significant improvements can occur in problems where local properties of underlying distributions play a decisive role. This situation often occurs in…

探索性投影寻踪中的多项式投影指标AI 解读

We develop asymptotic theory for two polynomial-based methods of estimating orientation in projection pursuit density approximation. One of the techniques uses Legendre polynomials and has been proposed and implemented by Friedman [1]. The other employs Hermite functions. Issues of smoothing parameter choice and robustness are addressed. It is shown that each method can be used to construct $\sqrt n$-consistent estimates of the projection which maximizes distance from…