Showing papers on "U-statistic published in 2007"

On local U-statistic processes and the estimation of densities of functions of several sample variables

Abstract: A notion of local U-statistic process is introduced and central limit theorems in various norms are obtained for it. This involves the development of several inequalities for U-processes that may be useful in other contexts. This local U-statistic process is based on an estimator of the density of a function of several sample variables proposed by Frees [J. Amer. Statist. Assoc. 89 (1994) 517-525] and, as a consequence, uniform in bandwidth central limit theorems in the sup and in the Lp norms are obtained for these estimators.

On local $U$-statistic processes and the estimation of densities of functions of several sample variables

Abstract: A notion of local $U$-statistic process is introduced and central limit theorems in various norms are obtained for it. This involves the development of several inequalities for $U$-processes that may be useful in other contexts. This local $U$-statistic process is based on an estimator of the density of a function of several sample variables proposed by Frees [J. Amer. Statist. Assoc. 89 (1994) 517--525] and, as a consequence, uniform in bandwidth central limit theorems in the sup and in the $L_p$ norms are obtained for these estimators.

Smoothed Rank Regression With Censored Data

Abstract: A weighted rank estimating function is proposed to estimate the regression parameter vector in an accelerated failure time model with right censored data. In general, rank estimating functions are discontinuous in the regression parameter, creating difficulties in determining the asymptotic distribution of the estimator. A local distribution function is used to create a rank based estimating function that is continuous and monotone in the regression parameter vector. A weight is included in the estimating function to produce a bounded influence estimate. The asymptotic distribution of the regression estimator is developed and simulations are performed to examine its finite sample properties. A lung cancer dataset is used to illustrate the methodology.

Ratio estimators in adaptive cluster sampling

Abstract: In most surveys data are collected on many items rather than just the one variable of primary interest. Making the most use of the information collected is a issue of both practical and theoretical interest. Ratio estimates for the population mean or total are often more efficient. Unfortunately, ratio estimation is straightforward with simple random sampling, but this is often not the case when more complicated sampling designs are used, such as adaptive cluster sampling. A serious concern with ratio estimates introduced with many complicated designs is lack of independence, a necessary assumption. In this article, we propose two new ratio estimators under adaptive cluster sampling, one of which is unbiased for adaptive cluster sampling designs. The efficiencies of the new estimators to existing unbiased estimators, which do not utilize the auxiliary information, for adaptive cluster sampling and the conventional ratio estimation under simple random sampling without replacement are compared in this article. Related result shows the proposed estimators can be considered as a robust alternative of the conventional ratio estimator, especially when the correlation between the variable of interest and the auxiliary variable is not high enough for the conventional ratio estimator to have satisfactory performance. Copyright © 2007 John Wiley & Sons, Ltd.

Laws of the Iterated Logarithm for the Local U-Statistic Process

Abstract: Laws of the iterated logarithm are established for the local U-statistic process. This entails the development of probability inequalities and moment bounds for U-processes that should be of separate interest. The local U-statistic process is based upon an estimator of the density of a function of several i.i.d. variables proposed by Frees (J. Am. Stat. Assoc. 89, 517–525, 1994). As a consequence, our results are directly applicable to the derivation of exact rates of uniform in bandwidth consistency in the sup and in the L p norms for these estimators.

A decomposition for invariant tests of uniformity on the sphere

Abstract: We introduce a U-statistic on which can be based a test for uniformity on the sphere. It is a simple function of the geometric mean of distances between points of the sample and consistent against all alternatives. We show that this type of U-statistic, whose kernel is invariant by isometries, can be separated into a set of statistics whose limiting random variables are independent. This decomposition is obtained via the so-called canonical decomposition of a group representation. The distribution of the limiting random variables of the components under the null hypothesis is given. We propose an interpretation of Watson type identities between quadratic functionals of Gaussian processes in the light of this decomposition.

Efficient estimation of population mean using incomplete survey data on study and auxiliary characteristics

Abstract: This paper considers the problem of estimating the population mean using the ratio and product methods when some observations in the sample data are missing at random and the population mean of the auxiliary characteristic is not known. Besides an unbiased estimator arising from the total discard of incomplete pairs of observations, four generally biased estimators are presented. The first two estimators arise form the partial utilization of data while the remaining two are based on full utilization. A comparative study of the efficiency properties of estimators is reported and the choice of estimators is discussed.

Bootstrap Approximation to Distributions of Finite Population U-statistics

Abstract: We show that the without replacement bootstrap of Booth, Butler and Hall (J. Am. Stat. Assoc. 89, 1282–1289, 1994) provides second order correct approximation to the distribution function of a Studentized U-statistic based on simple random sample drawn without replacement. In order to achieve similar approximation accuracy for the bootstrap procedure due to Bickel and Freedman (Ann. Stat. 12, 470–482, 1984) and Chao and Lo (Sankhya Ser. A 47, 399–405, 1985) we introduce randomized adjustments to the resampling fraction.

Second-order and resampling approximation of finite population U-statistics based on stratified samples

Abstract: We construct one-term Edgeworth expansions to distributions of U statistics and Studentized U-statistics, based on stratified samples drawn without replacement. Replacing the cumulants defining the expansions by consistent jackknife estimators, we obtain empirical Edgeworth expansions. The expansions provide second-order approximations that improve upon the normal approximation. Theoretical results are illustrated by a simulation study where we compare various approximations to the distribution of the commonly used Gini's mean difference estimator.

Jackknifing a convex combination of one-sample U-statistics

Abstract: As an estimator of an estimable parameter, we consider a convex combination of U-statistics (Toda and Yamato, 2001) Yn which includes U-statistic, V-statistic, S-statistic, and limit of Bayes estimate. We introduce the bias-reduced jackknifed estimator based on this convex combination. This statistic is written as a linear combination of U-statistics. From this, we know that jackknifed version of the V-statistic is not equal to the U-statistic for the degree more than or equal to three. To see the effects of jackknifing the statistic Yn, we show the difference between Yn and by the second-order deficiency. By second-order deficiency, we can not see the difference between Un and and between different two 's. Therefore, we show the differences between them by the fourth-order deficiency. We also give some examples using the V-statistic, limit of Bayes estimate, and S-statistic.

Unbiased chi(2) estimator for linear function fit involving correlated data

Abstract: An unbiased factorized chi-square estimator is constructed to deal with the correlated data for linear function fit The difference between the biased and unbiased chi-square fitting is expounded In addition, the simplified R-value measurement is quoted to test the conclusion quantitatively

Tekrarlamalı Gauss-Seidel Yardımcı Değişkenler Algoritması İle Transfer Fonksiyonu Parametrelerinin Yansız Tahmini

Abstract: In this paper, a recursive algorithm based on the use of Gauss-Seidel iterations and instrumental variables together is introduced for unbiased estimation of transfer function parameters of linear time invariant discrete-time systems. Furthermore, a stochastic convergence analysis of the proposed algorithm is performed and it is shown that the proposed algorithm is an unbiased parameter estimator that gives optimum solution of normal equations even if the measurement noise is colored. The proposed algorithm is used for estimation of transfer function parameters of a sample second order system and compared with similar algorithms by a simulation study. According to the results obtained, it is shown that the proposed algorithm is a good alternative to the others by viewpoints of computational complexity and convergence rate.

More General Credibility Models

Abstract: This communication gives some extensions of the original BA¼hlmann model. The paper is devoted to semi-linear credibility, where one examines functions of the random variables representing claim amounts, rather than the claim amounts themselves.

Variance Estimation for Establishment Surveys: An Introductory Overview

Abstract: This overview begins by discussing the uses of models in establishment surveys. These range from providing some statistical structure when using estimation strategies with cutoff or convenience samples to improving the accuracy of more robust strategies based on probability sampling. In both situations, a statistician usually constructs an estimator unbiased (or nearly so) under a simple, yet plausible, linear model. A model-based measure of the estimator’s variance given a particular sample can be estimated. Alternatively, one can measure the estimator’s variance under the combination of the model and the random-sampling mechanism. Of particular interest are ratio estimators that are both unbiased under certain assumed models and nearly unbiased under the random-sampling mechanism. Robust model-based variance estimators for a ratio often possess better coverage properties than the conventional probability-sampling alternative. Results for the ratio extend to calibration estimators. Models can also be helpful in variance estimation when there is unequal-probability sampling within strata, additional phases of sample selection, reweighting for unit nonresponse, and imputation for item nonresponse.

A note on the U, V method of estimation ∗

Abstract: The U, V method of estimation provides unbiased estimators or predictors of random quantities. The method was introduced by Robbins (3) and subsequently studied in a series of papers by Robbins and Zhang. (See Zhang (5).) Practical applications of the method are featured in these papers. We demonstrate that for one U function (one for which there is an important application) the V estimator is inadmissible for a wide class of loss functions. For another important U function the V estimator is admissible for the squared error loss function.