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基于周期图的分形性质估计量AI 解读

We suggest an estimator, based on the periodogram, of the fractal index and fractal dimension of a continuous, stationary Gaussian process. We argue that the cosine part of the periodogram is more appropriate than the full periodogram for this application. The term "semiperiodogram" is used to describe the cosine component, and our estimator is based on simple linear regression of the logarithm of the semiperiodogram on the algorithm of frequency. Theoretical properties of…

摘要暂缺

混沌 Logistic 映射的不变分布与 Lyapunov 指数性质AI 解读

SUMMARY Statistical scientists have recently focused sharp attention on properties of iterated chaotic maps, with a view to employing such processes to model naturally occurring phenomena. In the present paper we treat the logistic map, which has earlier been studied in the context of modelling biological systems. We derive theory describing properties of ‘invariant’ or ‘stationary’ distributions under logistic maps and apply those results in conjunction with numerical work…

非线性小波密度估计的平均积分平方误差公式AI 解读

We provide an asymptotic formula for the mean integrated squared error (MISE) of nonlinear wavelet-based density estimators. We show that, unlike the analogous situation for kernel density estimators, this MISE formula is relatively unaffected by assumptions of continuity. In particular, it is available for densities which are smooth in only a piecewise sense. Another difference is that in the wavelet case the classical MISE formula is valid only for sufficiently small…

置信区间的一致覆盖误差界与 Edgeworth 展开的 Berry–Esseen 定理AI 解读

We derive upper bounds for the coverage error of confidence intervals for a population mean uniformly over large classes of populations and different types of confidence intervals. It is shown that the order of these bounds is achieved by the normal approximation method for constructing confidence intervals, uniformly over distributions with finite third moment, and, by an empirical Edgeworth correction of this approach, uniformly over smooth distributions with finite fourth…

改进的可变窗宽核密度估计AI 解读

Variable window width kernel density estimators, with the width varying proportionally to the square root of the density, have been thought to have superior asymptotic properties. The rate of convergence has been claimed to be as good as those typical for higher-order kernels, which makes the variable width estimators more attractive because no adjustment is needed to handle the negativity usually entailed by the latter. However, in a recent paper, Terrell and Scott show…

曲线估计中的部分局部平滑规则AI 解读

We compare the performances of local and global rules for smoothing parameter choice, in terms of asymptotic mean squared errors of the resulting estimators. In some instances there is surprisingly little to choose between local and global approaches; our analysis identifies contexts where the differences are small or large. This work motivates development of smoothing rules that form a ‘half-way house’ between local and global smoothing. There, interpolation provides a…

依赖数据自助法的分块选择规则AI 解读

SUMMARY: We address the issue of optimal block choice in applications of the block bootstrap to dependent data. It is shown that optimal block size depends significantly on context, being equal to n1/3, n1/4 and n1/5 in the cases of variance or bias estimation, estimation of a onesided distribution function, and estimation of a two-sided distribution function, respectively. A clear intuitive explanation of this phenomenon is given, together with outlines of theoretical…

一般重抽样算法及其在分布估计中的表现AI 解读

Recent work of several authors has focussed on first-order properties (e.g., consistency) of general bootstrap algorithms, where the numbers of times that data values are resampled form an exchangeable sequence. In the present paper we develop second-order properties of such algorithms, in a very general setting. Performance is discussed in the context of distribution estimation, and formulae for higher-order moments and cumulants are developed. Arguing thus, necessary and…

协方差函数的非参数估计AI 解读

We describe kernel methods for estimating the covariance function of a stationary stochastic process, and show how to ensure that the estimator has the positive semidefiniteness property. From a practical viewpoint, our method is significant because it does not demand a parametric model for covariance. From a technical angle, our results exhibit a striking departure from those in more familiar cases of kernel estimation. For example, in the context of covariance estimation,…

有限总体的 Bootstrap 方法AI 解读

We show that the familiar bootstrap plug-in rule of Efron has a natural analog in finite population settings. In our method a characteristic of the population is estimated by the average value of the characteristic over a class of empirical populations constructed from the sample. Our method extends that of Gross to situations in which the stratum sizes are not integer multiples of their respective sample sizes. Moreover, we show that our method can be used to generate…

通过最小化平均绝对偏差估计曲线及其含义AI 解读

The local median regression method has long been known as a robustified alternative to methods such as local mean regression. Yet, its optimal statistical properties are largely unknown. In this paper, we show via decision-theoretic arguments that a local weighted median estimator is the best least absolute deviation estimator in an asymptotic minimax sense, under $L_1$-loss. We also study asymptotic efficiency of the local median estimator in the class of all possible…

二项分布试验次数N估计量的不稳定行为AI 解读

The problem of estimating the number of trials in a Binomial distribution is known to be particularly difficult in certain circumstances. Both the maximum likelihood and method-of-moments estimators can produce very erratic results, being extremely sensitive to minor fluctuations in the data. But existing asymptotic theory for these estimators does not provide any insight into this peculiar behavior, as the theory typically produces asymptotic Normal distributions with…

像素平均后错位图像的位移估计AI 解读

In problems where the recorded signal comprises two or more bands (i.e., wavelengths), it is often the case that the bands are not perfectly aligned in the focal plane. Such misregistration may be caused by atmospheric or oceanographic effects, which refract different bands to differing degrees, or it may result from features of the design of the recording equipment. This article develops two methods for estimating the lag, or the amount by which the bands are out of…

错位且经像素平均的图像中的位移估计AI 解读

In problems where the recorded signal comprises two or more bands (i.e., wavelengths), it is often the case that the bands are not perfectly aligned in the focal plane. Such misregistration may be caused by atmospheric or oceanographic effects, which refract different bands to differing degrees, or it may result from features of the design of the recording equipment. This article develops two methods for estimating the lag, or the amount by which the bands are out of…

摘要暂缺

通过有效分形维数估计刻画表面平滑度AI 解读

SUMMARY The fractal dimension D of stationary Gaussian surfaces may be expressed very simply in terms of behaviour of the covariance function near the origin. Indeed, only the covariance of line transect samples is required, and that fact makes practical estimation of D relatively straightforward. The case of non-Gaussian surfaces is more poorly understood, but we might define ‘effective fractal dimension’ in terms of the covariance function, as though the surface were…

Monte Carlo 近似与迭代自助法AI 解读

We discuss optimal choice of the numbers of resamples in the two stages of the iterated bootstrap, when that technique is used to calibrate bootstrap confidence intervals. If the numbers of resamples in the first and second stages are denoted by B and C respectively, we show that it is optimal to take C to be approximately a constant multiple of B½. The value of the constant is derived, and shown to depend only on the nominal coverage level of a confidence interval. However,…

测量误差模型中线性函数关系的 Bootstrap 置信域AI 解读

We suggest bootstrap methods for constructing confidence bands (and intervals) for an unknown linear functional relationship in an errors-invariables model. It is assumed that the ratio of error variances is known to lie within an interval $\Lambda = \lbrack\lambda_1, \lambda_2\rbrack$. A confidence band is constructed for the range of possible linear relationships when $\lambda \in \Lambda$. Meaningful results are obtained even in the extreme case $\Lambda = \lbrack 0, \infty\rbrack$…

L1回归估计量分布的解析与自助法近似AI 解读

Edgeworth and bootstrap approximations to estimator distributions in L1 regression are described. Analytic approximations based on Edgeworth expansions that mix lattice and nonlattice components and allow for an intercept term in the regression are developed under mild conditions, which do not even require a density for the error distribution. Under stronger assumptions on the error distribution, the Edgeworth expansion assumes a simpler form. Bootstrap approximations are…

摘要暂缺

非参数曲线估计中的Edgeworth展开与Bootstrap置信带AI 解读

SUMMARY The notion of pivoting and techniques of Edgeworth expansion have become basic tools for using and understanding bootstrap methods in finite parameter problems. This paper shows that these ideas have important contributions to make to infinite parameter problems, e.g. in the construction of simultaneous confidence bands for nonparametric density estimators. We propose a bootstrap method for confidence band construction based on a pivotal function estimator and…

Jackknife 分布函数估计量的一项改进AI 解读

In a recent paper, C. F. J. Wu showed that the jackknife estimator of a distribution function has optimal convergence rate $O(n^{-1/2})$, where $n$ denotes the sample size. This rate is achieved by retaining $O(n)$ data values from the original sample during the jackknife algorithm. Wu's result is particularly important since it permits a direct comparison of jackknife and bootstrap methods for distribution estimation. In the present paper we show that a very simple,…

利用非参数回归估计有限总体分布函数AI 解读

This paper considers estimators of the distribution function of a variable over a finite population, when a sample of units is available and values of a related auxiliary variable are known for the whole population. Theory is offered for several estimators which rely on nonparametric regression, and for calibration estimators which require a parametric model.