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观测误差下的变点、边界与前沿估计AI 解读

A range of problems in economics and statistics involve calculation of the boundary, or frontier, of the support of a distribution. Several practical and attractive solutions exist if the sampled distribution has a sharp discontinuity at the frontier, but accuracy can be greatly diminished if the data are observed with error. Indeed, if the error is additive and has variance σ2 then inaccuracies are usually of order σ, for small σ. In this article we suggest an elementary…

依赖性对密度估计随机精度度量的影响AI 解读

In kernel density estimation, those data values that make a nondegenerate contribution to the estimator (computed at a given point) tend to be spaced well apart. This property has the effect of suppressing many of the conventional consequences of long-range dependence, for example, slower rates of convergence, which might otherwise be revealed by a traditional loss-or risk-based assessment of performance. From that viewpoint, dependence has to be very long-range indeed…

密度估计中的高阶数据锐化AI 解读

Summary It is shown that data sharpening can be used to produce density estimators that enjoy arbitrarily high orders of bias reduction. Practical advantages of this approach, relative to competing methods, are demonstrated. They include the sheer simplicity of the estimators, which makes code for computing them particularly easy to write, very good mean-squared error performance, reduced ‘wiggliness' of estimates and greater robustness against undersmoothing.

提高非参数预测区间的覆盖准确性AI 解读

Summary Methods are suggested for improving the coverage accuracy of intervals for predicting future values of a random variable drawn from a sampled distribution. It is shown that properties of solutions to such problems may be quite unexpected. For example, the bootstrap and the jackknife perform very poorly when used to calibrate coverage, although the jackknife estimator of the true coverage is virtually unbiased. A version of the smoothed bootstrap can be employed for…

We examine the way in which empirical bandwidth choice affects distributional properties of nonparametric density estimators. Two bandwidth selection methods are considered in detail: local and global plug-in rules. Particular attention is focussed on whether the accuracy of distributional bootstrap approximations is appreciably influenced by using the resample version $\hat{h}*$,rather than the sample version $\hat{h}$, of an empirical bandwidth. It is shown theoretically…

利用局部似然追踪断裂线与边界AI 解读

Summary We suggest locally parametric methods for estimating curves, such as boundaries of density supports or fault lines in response surfaces, in a variety of spatial problems. The methods are based on spatial approximations to the local likelihood that the curve passes through a given point in the plane, as a function of that point. The local likelihood might be a regular likelihood computed locally, with kernel weights (e.g. in the case of support boundary estimation) or…

极值分析中选择阈值的一种诊断方法AI 解读

Summary A new approach is suggested for choosing the threshold when fitting the Hill estimator of a tail exponent to extreme value data. Our method is based on an easily computed diagnostic, which in turn is founded directly on the Hill estimator itself, ‘symmetrized’ to remove the effect of the tail exponent but designed to emphasize biases in estimates of that exponent. The attractions of the method are its accuracy, its simplicity and the generality with which it applies.…

施加单调性约束的非参数核回归AI 解读

We suggest a method for monotonizing general kernel-type estimators, for example local linear estimators and Nadaraya .Watson estimators. Attributes of our approach include the fact that it produces smooth estimates, indeed with the same smoothness as the unconstrained estimate. The method is applicable to a particularly wide range of estimator types, it can be trivially modified to render an estimator strictly monotone and it can be employed after the smoothing step has…

非参数回归中降低偏差的数据锐化方法AI 解读

We consider methods for kernel regression when the explanatory and/or response variables are adjusted prior to substitution into a conven- tional estimator.This “data-sharpening” procedure is designed to preserve the advantages of relatively simple, low-order techniques, for example, their robustness against design sparsity problems, yet attain the sorts of bias reductions that are commonly associated only with high-order methods.We consider Nadaraya–Watson and local-linear…

构造双重自助法置信区间时插值的重要性AI 解读

Summary We show that, in the context of double-bootstrap confidence intervals, linear interpolation at the second level of the double bootstrap can reduce the simulation error component of coverage error by an order of magnitude. Intervals that are indistinguishable in terms of coverage error with theoretical, infinite simulation, double-bootstrap confidence intervals may be obtained at substantially less computational expense than by using the standard Monte Carlo…

基于任意阶自回归计算的 bootstrap 置信域AI 解读

Summary Given a linear time series, e.g. an autoregression of infinite order, we may construct a finite order approximation and use that as the basis for confidence regions. The sieve or autoregressive bootstrap, as this method is often called, is generally seen as a competitor with the better-understood block bootstrap approach. However, in the present paper we argue that, for linear time series, the sieve bootstrap has significantly better performance than blocking methods…

厚切片 Wicksell 问题中的密度估计方法AI 解读

A new, implicit method is suggested for density estimation in inverse problems, where data are drawn not from the target distribution, but rather from its image under a transformation. The approach that we propose produces density estimators that are themselves densities, without the negativity problems known to plague more explicit inversion techniques. We also suggest a general empirical approach to selecting the smoothing parameter so as to optimize performance in the…

厚切片 Wicksell 问题的密度估计方法AI 解读

A new, implicit method is suggested for density estimation in inverse problems, where data are drawn not from the target distribution, but rather from its image under a transformation. The approach that we propose produces density estimators that are themselves densities, without the negativity problems known to plague more explicit inversion techniques. We also suggest a general empirical approach to selecting the smoothing parameter so as to optimize performance in the…

响应曲面中光滑断裂线的追踪估计AI 解读

We suggest a sequential, or “tracking,”algorithm for estimating a smooth fault line in a response surface. The method starts with an approximation to a point on the line, and from there the line is tracked as it meanders through the plane. The technique differs from recent approaches in that it does not require a large part of the plane to be searched for evidence of a fault line. This offers potential computational savings, and produces a method that is invariant under…

通过线性函数校准检验回归均值的单调性AI 解读

A new approach to testing for monotonicity of a regression mean, not requiring computation of a curve estimator or a bandwidth, is suggested. It is based on the notion of “running gradients ” over short intervals, although from some viewpoints it may be regarded as an analogue for monotonicity testing of the dip/excess mass approach for testing modality hypotheses about densities. Like the latter methods, the new technique does not suffer difficulties caused by almost-flat…

用于自助法迭代的加权自助法AI 解读

Summary The operation of resampling from a bootstrap resample, encountered in applications of the double bootstrap, maybe viewed as resampling directly from the sample but using probability weights that are proportional to the numbers of times that sample values appear in the resample. This suggests an approximate approach to double-bootstrap Monte Carlo simulation, where weighted bootstrap methods are used to circumvent much of the labour involved in compounded Monte Carlo…

减轻分布污染影响的有偏 Bootstrap 方法AI 解读

Summary Contamination of a sampled distribution, for example by a heavy-tailed distribution, can degrade the performance of a statistical estimator. We suggest a general approach to alleviating this problem, using a version of the weighted bootstrap. The idea is to ‘tilt’ away from the contaminated distribution by a given (but arbitrary) amount, in a direction that minimizes a measure of the new distribution's dispersion. This theoretical proposal has a simple empirical…

困难情形下的众数检验AI 解读

Usually, when testing the null hypothesis that a distribution has one mode against the alternative that it has two, the null hypothesis is interpreted as entailing that the density of the sampling distribution has a unique point of zero slope, which is a local maximum. In this paper we argue that a more appropriate null hypothesis is that the density has two points of zero slope, of which one is a local maximum and the other is a shoulder. We show that when a test for a…

条件分布函数的估计方法AI 解读

Motivated by the problem of setting prediction intervals in time series analysis, we suggest two new methods for conditional distribution estimation. The first method is based on locally fitting a logistic model and is in the spirit of recent work on locally parametric techniques in density estimation. It produces distribution estimators that may be of arbitrarily high order but nevertheless always lie between 0 and 1. The second method involves an adjusted form of the…

有意偏置的 Bootstrap 方法AI 解读

Summary A class of weighted bootstrap techniques, called biased bootstrap or b-bootstrap methods, is introduced. It is motivated by the need to adjust empirical methods, such as the ‘uniform’ bootstrap, in a surgical way to alter some of their features while leaving others unchanged. Depending on the nature of the adjustment, the b-bootstrap can be used to reduce bias, or to reduce variance or to render some characteristic equal to a predetermined quantity. Examples of the…

规则网格数据下光滑曲线近似的全局表现AI 解读

Approximating boundaries using data recorded on a regular grid induces discrete rounding errors in both vertical and horizontal directions. In cases where grid points exhibit at least some degree of randomness, an extensive theory has been developed for local-polynomial boundary estimators. It is inapplicable to regular grids, however. In this paper we impose strict regularity of the grid and describe the performance of local linear estimators in this context. Unlike the…

随机曲线分布众数的非参数估计AI 解读

Summary Motivated by the need to develop meaningful empirical approximations to a ‘typical’ data value, we introduce methods for density and mode estimation when data are in the form of random curves. Our approach is based on finite dimensional approximations via generalized Fourier expansions on an empirically chosen basis. The mode estimation problem is reduced to a problem of kernel-type multivariate estimation from vector data and is solved using a new recursive…

超额质量与 Dip 峰态检验的校准AI 解读

Summary Nonparametric tests of modality are a distribution-free way of assessing evidence about inhomogeneity in a population, provided that the potential sub populations are sufficiently well separated. They include the excess mass and dip tests, which are equivalent in univariate settings and are alternatives to the bandwidth test. Only very conservative forms of the excess mass and dip tests are available at presently, however, and for that reason they are generally not…

核与小波曲线估计中的块阈值规则AI 解读

Motivated by recently developed threshold rules for wavelet estimators, we suggest threshold methods for general kernel density estimators, including those of classical Rosenblatt–Parzen type. Thresholding makes kernel methods competitive in terms of their adaptivity to a wide variety of aberrations in complex signals. It is argued that term-by-term thresholding does not always produce optimal performance, since individual coefficients cannot be estimated sufficiently…