Estimators of shift based on statistics of the Kolmogorov-Smirnov type

This paper is concerned with the estimation of a shift parameter δ_o, based on some nonnegative functional Hg₁ of the pair (D^δ_N(x), f?^δ_N(x)), where D^δ_N(x) = K_N/b {F_2,n(x)—F_1,m (x + δ)}, +^δ_N(x) = {mF_1,m (x + δ) + nF_2,n(x)}/N, where F_1,m and F_2,n are the empirical distribution functions of two independent random samples (N = m + n), and where K²_N = mn/N. First an estimator δ_N, is defined as a value of δ minimizing a functional H of the type of H₁. A second estimator δ¹_N is also defined which is a linearized version of the first. Finite and asymptotic properties of these estimators are considered. It is also shown that most well-known test statistics of the Kolmogorov-Smirnov type are particular cases of such functionals H₁. The asymptotic distribution and the asymptotic efficiency of some estimators are given.     

On the asymptotics of randomness statistics

Kiefer (1959) studied the asymptotics of q-sample Cramér-von Mises nonparametric statistics when q is fixed and the sample sizes tend to infinity. Here we prove the asymptotic normality of such statistics when the sample sizes stay fixed or small while the number of samples, q, becomes large.     

Cramér-von Mises tests of fit for the Poisson distribution

Goodness-of-fit tests based on the Cramér-von Mises statistics are given for the Poisson distribution. Power comparisons show that these statistics, particularly A², give good overall tests of fit. The statistic A² will be particularly useful for detecting distributions where the variance is close to the mean, but which are not Poisson.     

Robust weighted Cramér-von Mises Estimators of location,with minimax variance in ϵ-contamination neighbourhoods

We construct weighted Cramér-von Mises location estimators which are asymptotically normally distributed throughout an ?e-contamination neighbourhood of a given, strongly unimodal distribution function, and which minimize the maximum asymptotic variance in such neighbourhoods. Applications to the estimation of a normal or logistic mean are given.     

On a cramér-von mises type statistic for testing bivariate independence

A Cramér-von Mises type statistic for testing bivariate independence, proposed by Hoeffding (1948) and by Blum, Kiefer, and Rosenblatt (1961), is examined in greater detail. The statistic is decomposed into components in the manner of Durbin and Knott (1972), and the components are shown to be related to linear rank statistics. Asymptotic power properties of the Hoeffding statistic and its components in testing for independence with bivariate normal random observations are described; a Monte Carlo study comparing these statistics with other nonparametric statistics for bivariate independence is also reported.     

Simultaneous robust estimation of location and scale parameters: A minimum-distance approach

Simultaneous robust estimates of location and scale parameters are derived from minimizing a minimum-distance criterion function. The criterion function measures the squared distance between the pth power (p > 0) of the empirical distribution function and the pth power of the imperfectly determined model distribution function over the real line. We show that the estimator is uniquely defined, is asymptotically bivariate normal and for p > 0.3 has positive breakdown. If the scale parameter is known, when p = 0.9 the asymptotic variance (1.0436) of the location estimator for the normal model is smaller than the asymptotic variance of the Hodges-Lehmann (HL)estimator (1.0472). Efficiencies with respect to HL and maximum-likelihood estimators (MLE) are 1.0034 and 0.9582, respectively. Similarly, if the location parameter is known, when p = 0.97 the asymptotic variance (0.6158) of the scale estimator is minimum. The efficiency with respect to the MLE is 0.8119. We show that the estimator can tolerate more corrupted observations at oo than at – for p < 1, and vice versa for p > 1.     

Minimum Risk Invariant Estimators of a Continuous Cumulative Distribution Function

The problem of nonparametric minimum risk invariant estimation has engaged a good deal of attention in the literature and minimum risk invariant estimators (MRIE's) have been constructed for some special statistical models. We present a new and simple method of obtaining the MRIE's of a continuous cumulative distribution function (cdf) under a general invariant loss function. All the MRIE's, which are known from the literature, can be constructed by the method presented in the article, in particular, under the weighted quadratic, LINEX and entropy loss functions. This method enables also to construct the MRIE's in nonparametric statistical models which have not been considered until now. In particular, considering a family of nonparametric precautionary loss functions, a new class of MRIE's of the cdf has been found. We also give some general remarks on obtaining the MRIE's and a review concerning minimaxity and admissibility of MRIE's.     

Estimators Based on Data‐Driven Generalized Weighted Cramér‐von Mises Distances under Censoring – with Applications to Mixture Models

Abstract. Estimators based on data‐driven generalized weighted Cramér‐von Mises distances are defined for data that are subject to a possible right censorship. The function used to measure the distance between the data, summarized by the Kaplan–Meier estimator, and the target model is allowed to depend on the sample size and, for example, on the number of censored items. It is shown that the estimators are consistent and asymptotically multivariate normal for every p dimensional parametric family fulfiling some mild regularity conditions. The results are applied to finite mixtures. Simulation results for finite mixtures indicate that the estimators are useful for moderate sample sizes. Furthermore, the simulation results reveal the usefulness of sample size dependent and censoring sensitive distance functions for moderate sample sizes. Moreover, the estimators for the mixing proportion seem to be fairly robust against a ‘symmetric’ contamination model even when censoring is present.     

Linear estimation of the location and scale parameters based on selected order statistics

This paper gives a review of the best linear estimates of the location and/or scale parameters based on a few order statistics selected from a complete or censored sample. Small sample and large sample cases are considered and compared. Some examples of the practical applications of the estimates are outlined.     

Tests for symmetry about an unknown value based on the empirical distribution function

A class of Kolmogorov-Smirnov and Cramér-von Mises type statistics for testing symmetry about an unknown value is described. These statistics are not distribution-free, however, and, indeed, are not readily amenable to calculation. A linear rank statistic analog of the first component of the Cramér-von Mises type statistic is investigated. Asymptotic non-null properties of these procedures in the normal case are studied, and an efficiency comparison of the Cramér-vonMises statistic, the linear rank statistic analog, the modified Wil-coxon statistic, and the likelihood ratio test is reported.     

Relative Errors of Difference-Based Variance Estimators in Nonparametric Regression

Difference-based estimators for the error variance are popular since they do not require the estimation of the mean function. Unlike most existing difference-based estimators, new estimators proposed by Müller et al. (2003 Müller , U. , Schick , A. , Wefelmeyer , W. ( 2003 ). Estimating the error variance in nonparametric regression by a covariate-matched U-statistic . Statistics 37 : 179 – 188 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and Tong and Wang (2005 Tong , T. , Wang , Y. ( 2005 ). Estimating residual variance in nonparametric regression using least squares . Biometrika 92 : 821 – 830 .[Crossref], [Web of Science ®] , [Google Scholar]) achieved the asymptotic optimal rate as residual-based estimators. In this article, we study the relative errors of these difference-based estimators which lead to better understanding of the differences between them and residual-based estimators. To compute the relative error of the covariate-matched U-statistic estimator proposed by Müller et al. (2003 Müller , U. , Schick , A. , Wefelmeyer , W. ( 2003 ). Estimating the error variance in nonparametric regression by a covariate-matched U-statistic . Statistics 37 : 179 – 188 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), we develop a modified version by using simpler weights. We further investigate its asymptotic property for both equidistant and random designs and show that our modified estimator is asymptotically efficient.     

Estimation of common location parameter of several heterogeneous exponential populations based on generalized order statistics

In this article, several independent populations following exponential distribution with common location parameter and unknown and unequal scale parameters are considered. From these populations, several independent samples of generalized order statistics (gos) are drawn. Under the setup of gos, the problem of estimation of common location parameter is discussed and various estimators of common location parameter are derived. The authors obtained maximum likelihood estimator (MLE), modified MLE and uniformly minimum variance unbiased estimator of common location parameter. Furthermore, under scaled-squared error loss function, a general inadmissibility result of invariant estimator is proposed. The derived results are further reduced for upper record values which is a special case of gos. Finally, simulation study and real life example are reported to show the performances of various competing estimators in terms of percentage risk improvement.     

Limiting distributions of Kolmogorov-Lévy-type statistics under the alternative

Let X_i, 1 ≤ i ≤ n, be independent identically distributed random variables with a common distribution function F, and let G be a smooth distribution function. We derive the limit distribution of □{ρ_α(F_n, G) - _α(F, G)}, where F_n is the empirical distribution function based on X₁,…,X_n and _α is a Kolmogorov-Lévy-type metric between distribution functions. For α ≤ 0 and two distribution functions F and G the metric p_α is given by p_α(F, G) = inf {? ≤ 0: G(x - α?) - ? F(x) ≤ G(x + α?) + ? for all x ?}.     

基于对数变换的小域的稳健估计量

小域估计是抽样调查的热点问题之一,其主流发展方向是基于模型的小域估计方法。但是这种方法依赖于模型的假定,若假定的模型错误,则估计效果很差。因此,利用对数变换和抽样设计权数得到小域的目标变量的稳健估计量,并通过模拟案例说明基于对数变换的方法是一种稳健有效的小域估计方法。     

Tests of fit for exponentiality based on a characterization via the mean residual life function

We study two new omnibus goodness of fit tests for exponentiality, each based on a characterization of the exponential distribution via the mean residual life function. The limiting null distributions of the tests statistics are the same as the limiting null distributions of the Kolmogorov-Smirnov and Cramér-von Mises statistics proposed when testing the simple hypothesis that the distribution of the sample variables is uniform on the interval [0, 1]. Work supported by the Deutsche Forschungsgemeinschaft     

On the Optimality of Prediction-based Selection Criteria and the Convergence Rates of Estimators

Several estimators of squared prediction error have been suggested for use in model and bandwidth selection problems. Among these are cross-validation, generalized cross-validation and a number of related techniques based on the residual sum of squares. For many situations with squared error loss, e.g. nonparametric smoothing, these estimators have been shown to be asymptotically optimal in the sense that in large samples the estimator minimizing the selection criterion also minimizes squared error loss. However, cross-validation is known not to be asymptotically optimal for some `easy' location problems. We consider selection criteria based on estimators of squared prediction risk for choosing between location estimators. We show that criteria based on adjusted residual sum of squares are not asymptotically optimal for choosing between asymptotically normal location estimators that converge at rate n ^1/2but are when the rate of convergence is slower. We also show that leave-one-out cross-validation is not asymptotically optimal for choosing between √ n -differentiable statistics but leave- d -out cross-validation is optimal when d ∞ at the appropriate rate.     

Goodness‐of‐fit tests for distributions estimated from complex survey data

We propose nonparametric procedures for comparing the empirical distribution function of data from a complex survey with a hypothesized parametric reference distribution. The hypothesized distribution may be fully specified, or it may be a family with the parameters to be estimated from the data. Of the procedures studied, a modification of the Cramér–von Mises test proposed by Lockhart, Spinelli & Stephens [Lockhart, Spinelli and Stephens, The Canadian Journal of Statistics 2007; 35, 125–133] is supported theoretically and performs well in two simulation studies. The methods are applied to examine the distribution of body mass index in the U.S. National Health and Nutrition Examination Survey. The Canadian Journal of Statistics 47: 409–425; 2019 © 2019 Statistical Society of Canada     

A Note on Asymptotic Behavior of the Nonparametric Density Estimators in Multivariate Mixtures

The asymptotic behavior of the nonparametric density estimator has been given for a multivariate mixture model. It has been observed that the estimator is asymptotically normally distributed with bias of size h ² and variance of size (nh)^?1.     

On tail index estimation based on multivariate data

This article is devoted to the study of tail index estimation based on i.i.d. multivariate observations, drawn from a standard heavy-tailed distribution, that is, of which Pareto-like marginals share the same tail index. A multivariate central limit theorem for a random vector, whose components correspond to (possibly dependent) Hill estimators of the common tail index α, is established under mild conditions. We introduce the concept of (standard) heavy-tailed random vector of tail index α and show how this limit result can be used in order to build an estimator of α with small asymptotic mean squared error, through a proper convex linear combination of the coordinates. Beyond asymptotic results, simulation experiments illustrating the relevance of the approach promoted are also presented.