Showing papers on "U-statistic published in 2004"
TL;DR: In this article, the authors established the asymptotic normality of Un when the sample observations come from a nonlinear time series and linear processes and showed that Un is robust to linear processes.
Abstract: A weighted U-statistic based on a random sample X1,…,Xn has the form Un=∑1≤i,j≤nwi−jK(Xi,Xj), where K is a fixed symmetric measurable function and the wi are symmetric weights A large class of statistics can be expressed as weighted U-statistics or variations thereof This paper establishes the asymptotic normality of Un when the sample observations come from a nonlinear time series and linear processes
75 citations
TL;DR: In this article, the authors extend the DeLong's approach and use the theory of the jackknife methodology applied to correlated one sample generalized U-statistics to derive the distribution of correlated estimators of C indexes as well as a consistent estimate of the asymptotic variance.
Abstract: The area under the receiver operating characteristic curve (overall C) is a widely used measure of a prognostic model discrimination. In this paper, we develop a nonparametric test for the comparison of two correlated C indexes of two different models when applied to the same population. We extend the DeLong's approach and use the theory of the jackknife methodology applied to correlated one sample generalized U-statistics. We derive the distribution of correlated estimators of C indexes as well as a consistent estimate of the asymptotic variance, leading to an asymptotically normal test.
75 citations
TL;DR: In this paper, a case-cohort design for failure time data from the Atherosclerosis Risk in Communities (ARCC) study is presented, in which covariates are assembled only for a subco-hort randomly selected from the entire cohort, and any additional cases outside the sub-co hort.
Abstract: SUMMARY In a case-cohort design introduced by Prentice (1986), covariates are assembled only for a subcohort randomly selected from the entire cohort, and any additional cases outside the subcohort. Semiparametric transformation models are considered here for failure time data from the case-cohort design. Weighted estimating equations are proposed for esti mation of the regression parameters. The estimation procedure of survival probability at given covariate levels is also provided. Asymptotic properties are derived for the estimators using finite population sampling theory, U-statistics theory and martingale convergence results. The finite-sample properties of the proposed estimators, as well as the efficiency relative to the full cohort estimators, are assessed via simulation studies. A case-cohort dataset from the Atherosclerosis Risk in Communities study is used to illustrate the estimating procedure.
62 citations
TL;DR: In this article, a new estimator of a survival function in the random censorship model when the censoring indicator is missing at random for some study subjects is proposed and analyzed, whose asymptotic variance reduces to that of the Kaplan-Meier estimator.
Abstract: We propose and analyze a new estimator of a survival function in the random censorship model when the censoring indicator is missing at random for some study subjects. The proposed approach appeals to a known representation for the survival function, expressible as a smooth functional of a certain conditional probability and the cumulative hazard function of the observed minimum. Well-known estimators are substituted into this representation leading to a simple estimator of the survival function. The new estimator, whose asymptotic variance reduces to that of the Kaplan-Meier estimator when all the censoring indicators are observed, is shown to achieve the efficiency bound derived by van der Laan and McKeague. This research was supported by the National Institute of Health grant CA103845.
27 citations
TL;DR: In this paper, Zhao and Chen showed that the Hoeffding-ANOVA decomposition of a symmetric and square integrable statistic T(Xn(α,c)) is explicitly computed in terms of linear combinations of well chosen conditional expectations of T.
Abstract: Consider a (possibly infinite) exchangeable sequence X = {Xn:1 ≤ n 0, Xn(α,c) consists of the first n instants of a generalized Polya urn sequence. For every choice of α(⋅) and c, the Hoeffding-ANOVA decomposition of a symmetric and square integrable statistic T(Xn(α,c)) is explicitly computed in terms of linear combinations of well chosen conditional expectations of T. Our formulae generalize and unify the classic results of Hoeffding [Ann. Math. Statist. 19 (1948) 293–325] for i.i.d. variables, Zhao and Chen [Acta Math. Appl. Sinica 6 (1990) 263–272] and Bloznelis and Gotze [Ann. Statist. 29 (2001) 353–365 and Ann. Probab. 30 (2002) 1238–1265] for finite population statistics. Applications are given to construct infinite “weak urn sequences” and to characterize the covariance of symmetric statistics of generalized urn sequences.
26 citations
01 Mar 2004
TL;DR: In this article, the authors consider unbiased linear estimators for a regression model with non-stochastic regressors and consider both the rate of convergence to the true value and the asymptotic distribution of the normalized error of the linear unbiased estimators.
Abstract: Under the symmetric α-stable distributional assumption for the disturbances, Blattberg et al (1971) consider unbiased linear estimators for a regression model with non-stochastic regressors. We consider both the rate of convergence to the true value and the asymptotic distribution of the normalized error of the linear unbiased estimators. By doing this, we allow the regressors to be stochastic and disturbances to be heavy-tailed with either finite or infinite variances, where the tail-thickness parameters of the regressors and disturbances may be different.
22 citations
TL;DR: In this article, the authors consider a random permutation superpopulation model with labels and positions in the permutation, and develop optimum estimators of the linear combinations of the unit parameters and optimum predictors of the random effects.
19 citations
TL;DR: In this article, collections of two-sample U-statistics are considered as a U-process indexed by a class of kernels and sufficient conditions for a functional central limit theorem in the non-degenerate case are given and a uniform law of large numbers is obtained.
17 citations
TL;DR: In this paper, the authors considered the class of U-statistics which are based on a collection of associated random variables and proved a strong law of large numbers for this class of statistics in the case where the "kernel" of the U-Statistic belongs to a large family of functions.
14 citations
TL;DR: In this paper, an estimator with U-statistic structure is constructed and its asymptotic normality is established for the tail index of heavy-tailed distributions and a large deviation principle is proved.
14 citations
TL;DR: In this article, a representation of the KMintegral in terms of the estimator of a censoring distribution is obtained, which may be useful not only to calculate the KMIntegral but also to characterize the KM-integral from a point view of the censoring distributions and the biasedness.
Abstract: sample size. A natural estimator of � ϕdF is a KMintegral, � ϕdFn. However, it is known that KMintegrals have serious biases for unbounded ϕ’s. A representation of the KMintegral in terms of the KMestimator of a censoring distribution is obtained. The representation may be useful not only to calculate the KMintegral but also to characterize the KMintegral from a point view of the censoring distribution and the biasedness. A class of unbiased estimators under the condition that the censoring distribution is known is considered, and the estimators are compared.
TL;DR: In this article, the weighted bootstrap for U-statistics and its properties were investigated and it was shown that it provides second-order accurate approximations to the distribution of U-Statistics.
TL;DR: In this article, a survey of U-statistics within the general framework of sequential and change-point literature is surveyed and some recent developments are discussed and extended, including new sequential testing strategies based on Wiener process approximations, and empirical studies explore the finite sample performance of these tests.
Abstract: Research on U-statistics within the general framework of sequential and change-point literature is surveyed. Some recent developments are discussed and extended. New sequential testing strategies based on Wiener process approximations are proposed, and empirical studies explore the finite sample performance of these tests. It allows users to choose one that is appropriate for their application.
TL;DR: In this article, it is shown how to solve singularities in measurement experiments under special restrictions on model parameters, and the estimability of model parameters is studied and unbiased estimators are given in explicit forms.
Abstract: In modelling a measurement experiment some singularities can occur even if the experiment is quite standard and simple. Such an experiment is described in the paper as a motivation example. It is presented in the papar how to solve these situations under special restrictions on model parameters. The estimability of model parameters is studied and unbiased estimators are given in explicit forms.
Journal Article•
TL;DR: In this article, moment inequalities for the supremum of empirical processes of U-Statistic structure were derived and applied to kernel type density estimation and estimation of the distribution function for functions of observations.
Abstract: We derive moment inequalities for the supremum of empirical processes of U-Statistic structure and give application to kernel type density estimation and estimation of the distribution function for functions of observations.
TL;DR: A general framework which unifies already existing asymptotic theory for projection matrices as well as matrices of all-iid entries for permanents of random matrices with independent columns of exchangeable components is provided.
TL;DR: The paper solves a longstanding problem in simulation: How to unbiasedly estimate analytic functions of expectations when the expectations must be simulated and applies these to Simulated Maximum Likelihood (SML) estimation.
Abstract: The paper solves a longstanding problem in simulation: How to unbiasedly estimate analytic functions of expectations when the expectations must be simulated. It then applies these to Simulated Maximum Likelihood (SML) estimation. The results include unbiased estimation of finite degree polynomials and other analytic functions, unbiased simulation of the score and likelihood, and the asymptotic properties of SML using these simulators. The motivating application is estimation in the mixed logit model. There are some older related results spread throughout the non-parametric and sequential estimation literatures, these seem unknown to both simulation researchers and practitioners, so they are collected here and presented, in context, with the new results.
TL;DR: In this article, a certain class of rectangular designs for incomplete U-statistics based on Latin squares were considered and shown to be optimal with respect to the minimal variance criterion. But they are not asymptotically efficient when compared with the corresponding complete statistics, as well as uniformly more efficient than random subset selection.
TL;DR: In this paper, the tube method is used to approximate the tail probability of the maximum of a Gaussian random field. But the results of the test statistic Srange = Smax - Smin are not considered.
Posted Content•
TL;DR: In this article, studentized versions of such estimating functions are defined and considered asymptotic approximations as well as an estimating function bootstrap (EFB) method based on resampling the estimated terms in the estimating functions.
Abstract: Suppose that inference about parameters of interest is to be based on an unbiased estimating function that is U-statistic of degree 1 or 2. We define suitable studentized versions of such estimating functions and consider asymptotic approximations as well as an estimating function bootstrap (EFB) method based on resampling the estimated terms in the estimating functions. These methods are justified asymptotically and lead to confidence intervals produced directly from the studentized estimating functions. Particular examples in this class of estimating functions arise in La estimation as well as Wilcoxon rank regression and other related estimation problems. The proposed methods are evaluated in examples and simulations and compared with a recent suggestion for inference in such problems which relies on resampling an underlying objective functions with U-statistic structure.
TL;DR: In this paper, a family of almost unbiased estimators for Y, the population mean of the study variable Y, is suggested and its properties analyzed under simple random sampling and without replacement (SRSWOR) scheme.
Abstract: Using Jackknife technique a family of almost unbiased estimators for Y, the population mean of the study variable Y, is suggested and its properties analysed under simple random sampling and without replacement (SRSWOR) scheme. An empirical investigation has been done to show the performance of the proposed unbiased strategies over the biased regression estimator.
TL;DR: In this article, a decision theoretic approach is taken with the quadratic loss function to derive the unbiased estimator of the essential part of the risk which is applicable for general estimators.
Abstract: We consider the problem of estimating the discriminant coefficients, = ~'~--1(0(1) -- 8 (2)) based on two independent normal samples from Np(O (*), ~) and Np(0 (2), ~E). We are concerned with the estimation of rl as the gradient of log-odds between two extreme situations. A decision theoretic approach is taken with the quadratic loss function. We derive the unbiased estimator of the essential part of the risk which is applicable for general estimators. We propose two types of new estimators and prove their dominance over the traditional estimator using this unbiased estimator.
01 Jan 2004
TL;DR: In this paper, two different methods of estimation were proposed to generate unbiased estimators of the population mean in the presence of two auxiliary variables for a two-phase sampling procedure, and the results showed that the two methods were more accurate than the single-parameter estimator.
Abstract: This paper considers two different methods of estimation to generate unbiased estimators of the population mean in the presence of two auxiliary variables for a two-phase sampling procedure.
07 Nov 2004
TL;DR: In this article, two extreme-based estimation of variance are considered, and their properties for uniform, Gaussian and Laplacian distributions are investigated for both uniform and Gaussian distributions.
Abstract: The nonlinear estimation, due to R. A. Fisher, of the mean of a uniformly-distributed random variable given by the average of the extreme values of N samples is unbiased. Furthermore, the variance of the estimation is very much less than that for the conventional linear estimation. Two extreme-based estimation of variance are considered, and their properties investigated for uniform, Gaussian and Laplacian distributions.