Bewertung von sheätzungen bei einem linearen modell auf der grundlage des fiduzialkonzepts

Usingf fiducial inference the quality of estimations is investigated for a linear model under the conditionj that information about the vector to be estimated is derived from a small sample. By application of that model to the deduction of atmospheric temperature profiles the obtained results are illustrated numerically.     

Data‐driven choice of the smoothing parametrization for kernel density estimators

There are several levels of sophistication when specifying the bandwidth matrix H to be used in a multivariate kernel density estimator, including H to be a positive multiple of the identity matrix, a diagonal matrix with positive elements or, in its most general form, a symmetric positive‐definite matrix. In this paper, the author proposes a data‐based method for choosing the smoothing parametrization to be used in the kernel density estimator. The procedure is fully illustrated by a simulation study and some real data examples. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

On the asymptotic normality of functions of uniform spacings

A new proof is given for the asymptotic normality of sum functions of spacings, providing an alternative to the method of Le Cam (1958). The result is obtained under an optimal moment condition. The proof is based on a simple decomposition into a leading term, which is asymptotically normal, and a remainder term, which is shown to be negligible.     

Asymptotic distribution of data-driven smoothers in density and regression estimation under dependence

We consider automatic data-driven density, regression and autoregression estimates, based on any random bandwidth selector h/_T. We show that in a first-order asymptotic approximation they behave as well as the related estimates obtained with the “optimal” bandwidth h_T as long as h_T/h_T → 1 in probability. The results are obtained for dependent observations; some of them are also new for independent observations.     

Central limit theorem for the variable bandwidth kernel density estimators

In this paper we study the ideal variable bandwidth kernel density estimator introduced by McKay (1993a, b) and Jones et al. (1994) and the plug-in practical version of the variable bandwidth kernel estimator with two sequences of bandwidths as in Giné and Sang (2013). Based on the bias and variance analysis of the ideal and plug-in variable bandwidth kernel density estimators, we study the central limit theorems for each of them. The simulation study confirms the central limit theorem and demonstrates the advantage of the plug-in variable bandwidth kernel method over the classical kernel method.     

Bounded-influence rank estimation in the linear model

We introduce and study a class of rank-based estimators for the linear model. The estimate may be roughly described as being calculated in the same manner as a generalized M-estimate, but with the residual being replaced by a function of its signed rank. The influence function can thus be bounded, both as a function of the residual and as a function of the carriers. Subject to such a bound, the efficiency at a particular model distribution can be optimized by appropriate choices of rank scores and carrier weights. Such choices are given, with respect to a variety of optimality criteria. We compare our estimates with several others, in a Monte Carlo study and on a real data set from the literature.     

Numerical estimation of a probability measure

An iterative solution to the problem of maximizing a concave functional ø defined on the set of all probability measures on a topological space is considered. Convergence of this procedure and a rapidly converging algorithm are studied. Computational aspects of this algorithm along with the ones developed earlier by Wynn, Fedorov, Atwood, Wu and others are provided. Examples discussed are taken from the area of mixture likehoods and optimal experimental design.     

Partially linear models and polynomial spline approximations for the analysis of unbalanced panel data

In this paper, we study the estimation of the unbalanced panel data partially linear models with a one-way error components structure. A weighted semiparametric least squares estimator (WSLSE) is developed using polynomial spline approximation and least squares. We show that the WSLSE is asymptotically more efficient than the corresponding unweighted estimator for both parametric and nonparametric components of the model. This is a significant improvement over previous results in the literature which showed that the simply weighting technique can only improve the estimation of the parametric component. The asymptotic normalities of the proposed WSLSE are also established.     

On multivariate associated kernels to estimate general density functions

Multivariate associated kernel estimators, which depend on both target point and bandwidth matrix, are appropriate for distributions with partially or totally bounded supports and generalize the classical ones such as the Gaussian. Previous studies on multivariate associated kernels have been restricted to products of univariate associated kernels, also considered having diagonal bandwidth matrices. However, it has been shown in classical cases that, for certain forms of target density such as multimodal ones, the use of full bandwidth matrices offers the potential for significantly improved density estimation. In this paper, general associated kernel estimators with correlation structure are introduced. Asymptotic properties of these estimators are presented; in particular, the boundary bias is investigated. Generalized bivariate beta kernels are handled in more details. The associated kernel with a correlation structure is built with a variant of the mode-dispersion method and two families of bandwidth matrices are discussed using the least squared cross validation method. Simulation studies are done. In the particular situation of bivariate beta kernels, a very good performance of associated kernel estimators with correlation structure is observed compared to the diagonal case. Finally, an illustration on a real dataset of paired rates in a framework of political elections is presented.     

On the asymptotlc normality for l2-error of wavelei density estimator with application

The l₂ error of linear wavelet estimator of a density from a random sample is shown to be asymptotically normal without imposing smoothing conditions on the density. As an application, the goodness-of-fit test and the approximation of its power for closeness alternatives are considered.     

Comparing relative importance of the same component in a series system in two environments

The relative performance of a component of a series system in two different environments is considered. The conditional probability of the failure of the system due to the failure of the specified component given that the system failed before time t is regarded as a measure of relative importance of the component to the system. A U-statistic test for checking the equality of the relative importance of the component to the system in two different environments against the alternative that the relative importance is smaller in one of the environments, is proposed. Some simulation results for estimating the power of the test are reported. The proposed test is applied to one real data set and it is seen that a different aspect of the data is brought out by this comparison than that by the comparisons of the absolute importance functions such as the subsurvival functions, considered in earlier studies.     

An Estimator of the Exponent of Regular Variation Based on K-Record Values

Let {X _n:n ≥ 1} be an i.i.d. sequence of random variables with a continuous distribution function F. Under the assumption that the upper tail of Fis regularly varying with exponent 1/α, α > 0, we study the asymptotic properties of an estimator of α based on k-record values.     

Nonparametric function estimation from inversely sampled record-breaking data

Often, in industrial stress testing, meteorological data analysis, and other similar situations, measurements may be made sequentially and only values smaller than all previous ones are recorded. When the number of records is fixed in advance, the data are referred to as inversely sampled record-breaking data. This paper is concerned with nonparametric estimation of the distribution and density functions from such data (successive minima). For a single record-breaking sample, consistent estimation is not possible except in the extreme left tail of the distribution. Hence, replication is required, and for m such independent record-breaking samples, the estimators are shown to be strongly consistent and asymptotically normal as m ∞ →. Computer simulations are used to investigate the effect of the bandwidth on the mean squared errors and biases of the smooth estimators, and are also used to provide a comparison of their performance with the analogous estimators obtained under random sampling for record values.     

Asymptotic normality and mean consistency for the weighted estimator in nonparametric regression models

In this paper, we mainly study the asymptotic properties of weighted estimator for the nonparametric regression model based on linearly negative quadrant dependent (LNQD, for short) errors. We obtain the rate of uniformly asymptotic normality of the weighted estimator which is nearly $O\left(n^{?1∕4}\right)$ when the moment condition is appropriate. The results generalize the corresponding ones of Yang (2003) from NA samples to LNQD samples and improve or extend the corresponding one of Li et al. (2012) for LNQD samples. Moreover, we obtain some results on mean consistency, uniformly mean consistency, and the rate of mean consistency for the weighted estimator. Finally we carry out some simulations to verify the validity of our results.     

Adaptive estimation of error density in nonparametric regression with small sample size

It has been established recently in Efromovich [2005. Estimation of the density of regression errors. Ann. Statist. 33, 2194–2227] that, under a mild assumption, the error density in a nonparametric regression can be asymptotically estimated with the accuracy of an oracle that knows underlying regression errors. The asymptotic nature of the result, and in particular the used methodology of splitting data for estimating nuisance functions and the error density, does not make an asymptotic estimator, suggested in that article, feasible for practically interesting cases of small sample sizes. This article continues the research and solves two important issues. First, it shows that the asymptotic holds without splitting the data. Second, a data-driven estimator, based on the new asymptotic, is suggested and then tested on real and simulated examples.     

A domain outlier robust design and smooth estimation approach

Outliers that commonly occur in business sample surveys can have large impacts on domain estimates. The authors consider an outlier‐robust design and smooth estimation approach, which can be related to the so‐called “Surprise stratum” technique [Kish, “Survey Sampling,” Wiley, New York (1965)]. The sampling design utilizes a threshold sample consisting of previously observed outliers that are selected with probability one, together with stratified simple random sampling from the rest of the population. The domain predictor is an extension of the Winsorization‐based estimator proposed by Rivest and Hidiroglou [Rivest and Hidiroglou, “Outlier Treatment for Disaggregated Estimates,” in “Proceedings of the Section on Survey Research Methods,” American Statistical Association (2004), pp. 4248–4256], and is similar to the estimator for skewed populations suggested by Fuller [Fuller, Statistica Sinica 1991;1:137–158]. It makes use of a domain Winsorized sample mean plus a domain‐specific adjustment of the estimated overall mean of the excess values on top of that. The methods are studied in theory from a design‐based perspective and by simulations based on the Norwegian Research and Development Survey data. Guidelines for choosing the threshold values are provided. The Canadian Journal of Statistics 39: 147–164; 2011 © 2010 Statistical Society of Canada     

Invalidity of average squared error criterion in density estimation

The average squared error has been suggested earlier as an appropriate estimate of the integrated squared error, but an example is given which shows their ratio can tend to infinity. The results of a Monte Carlo study are also presented which suggest the average squared error can seriously underestimate the errors inherent in even the simplest density estimations.     

Fully efficient estimation of coefficients of correlation in the presence of imputed survey data

Marginal imputation, that consists of imputing items separately, generally leads to biased estimators of bivariate parameters such as finite population coefficients of correlation. To overcome this problem, two main approaches have been considered in the literature: the first consists of using customary imputation methods such as random hot‐deck imputation and adjusting for the bias at the estimation stage. This approach was studied in Skinner & Rao 2002 . In this paper, we extend the results of Skinner & Rao 2002 to the case of arbitrary sampling designs and three variants of random hot‐deck imputation. The second approach consists of using an imputation method, which preserves the relationship between variables. Shao & Wang 2002 proposed a joint random regression imputation procedure that succeeds in preserving the relationships between two study variables. One drawback of the Shao–Wang procedure is that it suffers from an additional variability (called the imputation variance) due to the random selection of residuals, resulting in potentially inefficient estimators. Following Chauvet, Deville, & Haziza 2011 , we propose a fully efficient version of the Shao–Wang procedure that preserves the relationship between two study variables, while virtually eliminating the imputation variance. Results of a simulation study support our findings. An application using data from the Workplace and Employees Survey is also presented. The Canadian Journal of Statistics 40: 124–149; 2012 © 2011 Statistical Society of Canada