Showing papers on "U-statistic published in 1994"
TL;DR: In this paper, a semiparametric regression estimator that maximizes a $U$-process of order 3 is shown to be consistent and asymptotically normally distributed.
Abstract: Maximal inequalities for degenerate $U$-processes of order $k, k \geq 1$, are established. The results rest on a moment inequality (due to Bonami) for $k$th-order forms and on extensions of chaining and symmetrization inequalities from the theory of empirical processes. Rates of uniform convergence are obtained. The maximal inequalities can be used to determine the limiting distribution of estimators that optimize criterion functions having $U$-process structure. As an application, a semiparametric regression estimator that maximizes a $U$-process of order 3 is shown to be $\sqrt n$-consistent and asymptotically normally distributed.
220 citations
TL;DR: In this paper, a class of estimators of the semiparametric censored regression model under the assumption that the error terms are i.i.d. and independent of the covariates is proposed.
171 citations
TL;DR: An expression is derived for a Cramer-Rao lower bound on the variance of unbiased estimators of complex parameters.
Abstract: An expression is derived for a Cramer-Rao lower bound on the variance of unbiased estimators of complex parameters. >
109 citations
TL;DR: In this paper, the optimal combination of two unbiased estimators whose variances and correlations are known is proposed to estimate a location parameter for the uniform distribution, and it is shown that the optimal mix between mean and median is (3/2) − (1/2)-median.
Abstract: This article illuminates some aspects of unbiased estimation. It addresses optimal combination of two dependent unbiased estimators whose variances and correlations are known. When applied to the problem of estimating a location parameter for the uniform distribution, it was found that the optimal mix between mean and median is (3/2)(mean) — (1/2)(median). In cases in which uniformly minimum variance unbiased (UMVU) estimators exist, the approach can be used to give simple expressions for the correlation between the UMVU estimator and any other unbiased estimator. This correlation is always positive. The theory and examples given are well suited for classroom use at the senior undergraduate or beginning graduate level.
70 citations
Book•
12 Aug 1994
TL;DR: In this article, the authors review the impact of Hoeffding's work on sequential analysis and propose a non-parametric test of independence, which is based on the central limit theorem for dependent random variables.
Abstract: Three Articles Reviewing Hoeffding's Work.- Wassily Hoeffding' s Work in the Sixties.- The Impact of Wassily Hoeffding's Work on Sequential Analysis.- The Impact of Wassily Hoeffding's Research on Nonparametrics.- Publications.- Scale-Invariant Correlation Theory (English Translation).- Scale-Invariant Correlations for Discontinuous Distributions (English Translation).- Stochastic Dependence and Functional Relationships (English Translation).- On the Distribution of the Rank Correlation Coefficient r when the Variates are not Independent.- A Class of Statistics with Asymptotically Normal Distribution.- The Central Limit Theorem for Dependent Random Variables (with Herbert Robbins).- A Non-Parametric Test of Independence.- 'Optimum' Non-Parametric Tests.- A Combinatorial Central Limit Theorem.- The Large-Sample Power of Tests Based on Permutations of Observations.- On the Distribution of the Expected Values of the Order Statistics.- A Lower Bound for the Average Sample Number of a Sequential Test.- Bounds for the Distribution of a Sum of Independent, Identically Distributed Random Variables (with S.S. Shrikhande).- The Efficiency of Tests (with J.R. Rosenblatt).- The Extrema of the Expected Value of a Function of Independent Random Variables.- On the Distribution of the Number of Successes in Independent Trials.- The role of Assumptions in Statistical Decisions.- Distinguishability of Sets of Distributions. (The case of Independent and Identically Distributed Random Variables) (with J Wolfowitz).- Lower Bounds for the Expected Sample Size and the Average Risk of a Sequential Procedure.- An Upper Bound for the Variance of Kendall's 'Tau' and of Related Statistics.- Lower Bounds for the Expected Sample Size of a Sequential Test.- On Sequences of Sums of Independent Random Vectors.- Probability Inequalities for Sums of Bounded Random Variables.- On a Theorem of V.M. Zolotarev.- Asymptotically Optimal Tests for Multinomial Distributions (with Discussion).- On Probabilities of Large Deviations.- Some Recent Developments in Nonparametric Statistics.- Unbiased Coin Tossing with a Biased Coin (with Gordon Simons).- Discussion of J. Hajek, Miscellaneous Problems of Rank Test Theory.- Discussion of H. Witting, On the Theory of Nonparametric Tests.- The L1 norm of the Approximation Error for Bernstein-Type Polynomials.- On the Centering of a Simple Linear Rank Statistic.- The L1 norm of the Approximation Error for Splines with Equidistant Knots.- Harold Hotelling, 1895-1973.- H.Hotelling.- Some Incomplete and Boundedly Complete Families of Distributions.- More on Incomplete and Boundedly Complete Families of Distributions.- A Statistician's Progress from Berlin to Chapel Hill.- Unbiased Range-Preserving Estimators.- Range Preserving Unbiased Estimators in the Multinomial Case.- Entries in The Encyclopedia of Statistical Sciences edited by S. Kotz, N.L. Johnson and C.B. Read. New York: Wiley.- Asymptotic Normality.- Hajek's Projection Lemma.- Hoeffding's Independence Test.- Probability Inequalities for Sums of Bounded Random Variables.- Range-Preserving Estimators.- Book Reviews.- S.S. Wilks, Mathematical Statistics.- G. Udny Yule Sz M. G. Kendall, An Introduction to the Theory of Statistics, 14th Edition.- D.A.S. Fraser, Nonparametric Methods in Statistics.- B.V. Gnedenko & A.N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables.- J. Hajek & S. Sidak, Theory of Rank Tests.- E.L. Lehmann, Testing Statistical Hypotheses.
53 citations
TL;DR: In this paper, two distinct types of models are used for handling nonresponse in survey sampling theory: a parametric model and a response model, where the propensity of survey response is modeled as a random process, an additional phase of sample selection.
Abstract: Two distinct types of models are used for handling nonresponse in survey sampling theory. In a response (or quasi-randomization) model, the propensity of survey response is modeled as a random process, an additional phase of sample selection. In a parametric (or superpopulation) model, the survey data are themselves modeled. These two models can be used simultaneously in the estimation of a population mean so that one provides some protection against the potential for failure in the other. Two different estimators are discussed in this article. The first is a regression estimator that is both unbiased under the parametric model and nearly quasi-design unbiased under the response model. The second is a direct expansion estimator with imputed missing values. The imputed values are such that the estimator is both nearly quasi-design unbiased and unbiased under the combination of the parametric model and the original sampling design. The article includes a discussion of variance estimation with the g...
33 citations
TL;DR: In this article, the authors introduce a generalized bootstrap procedure for U -statistics, which reweights the terms of the original U-statistic by stochastic weights.
27 citations
TL;DR: In this paper, the authors prove decoupling inequalities for random polynomials in independent random variables with coefficients in vector space, using various means of comparison, including rearrangement invariant norms (e.g., Orlicz and Lorentz norms), tail distributions, tightness, hypercontractivity and so forth.
Abstract: We prove decoupling inequalities for random polynomials in independent random variables with coefficients in vector space. We use various means of comparison, including rearrangement invariant norms (e.g., Orlicz and Lorentz norms), tail distributions, tightness, hypercontractivity and so forth.
22 citations
TL;DR: In this paper, the best linear unbiased estimators when the samples are singly and doubly Type II censored are tabulated and the values of the variances and covariance of these estimators are also presented.
Abstract: For the half–logistic distribution, Balakrishnan (1985) derived several recurrence relations satisfied by the single and the product moments of order statistics and applied them in a simple recursive process to compute the means, variances and covariances of order statistics for sample sizes up to 15. Balakrishnan and Puthenpura (1986) made use of those results to tabulate the coefficients of the best linear unbiased estimators of location and scale parameters based on complete samples. In this paper, we extend their results and tabulate the. best linear unbiased estimators when the samples are singly and doubly Type II censored. We also present the values of the variances and covariance of these estimators. Finally, we present two examples in which tables are used.
18 citations
TL;DR: In this paper, the canonical form for the comparison of certain linear estimators using Pitman's Measure of Closeness is generalized to the class of all linear estimation methods under the assumption of normality, and the equivalence of Pitman-closest linear unbiased estimators and best linear equivalent estimators is shown.
TL;DR: In this article, a large sample test of the hypothesis that the copula of a bivariate distribution belongs to a family of copulas parametrized by Spearman's correlation such as the family of Morgenstern copulas is presented.
Abstract: This paper presents a large sample test of the hypothesis that the copula of a bivariate distribution belongs to a family of copulas parametrized by Spearman's correlation such as the family of Morgenstern copulas. The test statistic is based on the fact that if a copula belongs to the family then Kendall’s τ may be expressed as a function of Spearman’s ρ. Using this relationship along with U-statistic estimators of τ and ρ, we construct a test statistic that is asymptotically normal. To simplify calculations we suggest that the variance be estimated with a jackknife estimator.
01 Jan 1994
TL;DR: In this paper, the authors further explore the concept of RSS for the problem of estimation of a lognormal mean with a known coefficient of variation, and show that the use of RSS and its suitable modifications results in much improved estimators compared to the using of a simple random sample.
Abstract: SUMMARY. When the experimental or sampling units in a study can be more easily ranked than quantified, McIntyre (1952) observed that to estimate the population mean, the mean of n units based on a ranked set sample (RSS) provides an unbiased estimator with a smaller variance compared to a simple random sample of the same size n. In this paper we further explore the concept of RSS for the problem of estimation of a lognormal mean with a known coefficient of variation, and show that the use of RSS and its suitable modifications results in much improved estimators compared to the use of a simple random sample.
TL;DR: In this article, a class of unbiased ratio-type estimators in two (double) phases sampling has been constructed and an empirical study is carried out to illustrate the performances of constructed estimator over various other estimators.
Abstract: A class of unbiased ratio-type estimators in two (double) phases sampling has been constructed. It has been shown that Sukhatme's (1962) estimator is a particular member of the proposed class. Variance expression of the proposed class has been derived and optimum estimator in the class is identified. An empirical study is carried out to illustrate the performances of constructed estimator over various other estimators. Further, the cost aspect has also been considered.
TL;DR: In this paper, the covariance structure of stochastic flows generated by linear time-dependent differential equations driven by certain harmonizable or ortho- gonal increment noise processes is considered.
01 Dec 1994
TL;DR: In this article, it was shown that the criteria proposed by ishii (1969) based on matrices and the one proposed by Bibby and Toutenburg (1977), based on quadratic loss in the class of vector unbiased predictors are equivalent.
Abstract: A characterization of optimal vector unbiased predictor is obtained. Some properties of optimal unbiased predictors are established. It is shown that simultaneous prediction of future random variables is equivalent to marginal prediction of these random variables. Following Kale and Chandrasekar (1983) and Chandrasekar (1984), it is shown that the criteria proposed by ishii (1969) based on matrices and the one proposed by Bibby and Toutenburg (1977) based on quadratic loss in the class of vector unbiased predictors are equivalent. The above approach is illustrated with some examples.
01 Jan 1994
TL;DR: In this article, the authors compare six almost unbiased ratio estimators with respect to bias and efficiency for finite populations and infinite populations in which the joint distribution of the characters under study is bivariate normal.
Abstract: In this paper, we compare six almost unbiased ratio estimators with respect to bias and efficiency for (i) finite populations, and (ii) infinite populations in which the joint distribution of the characters under study is bivariate normal
TL;DR: In this paper, the asymptotic consistency of the bootstrap approximation of the vector of the marginal generalized quantiles of U-statistic structure (multivariate U-quantiles for short) is established.
TL;DR: In this paper, the authors characterise linear unbiased strategies for estimating the population total, admitting uniformly nonnegative unbiased variance estimators, and derive necessary and sufficient conditions for the nonnegativity of unbiased estimators vis-a-vis nonnegative definite matrices.
TL;DR: In this paper, an unbiased estimator which combines both X1 and X2 is developed and its risk behavior is studied, assuming that σ21 is known, a motivation for the best shrinkage estimator in a class of estimators that shrink X 1 toward X 2 is given.
TL;DR: In this article, it was shown that the number of non-void arrangements of a random variable X with distribution defined by, which occur in connection with the rencontre problem is the uniquely determined unbiased estimator of.
Abstract: It is shown that the number of non-void arrangements of {1, … k}, where is some realization of a random variable X with distribution defined by ,which occur in connection with the rencontre problem (as the probability for m fixed points of permutations of {1, … ,n} selected with probability ), is the uniquely determined unbiased estimator of . Here stands for the one-point mass at zero. Furthermore, the class of all unbiased estimators of n with respect to the family , where the case n = 0 is excluded, is determined and all locally MVU estimators at every in the model are characterized. In particular, it turns out that there does not exist a uniformly MVU resp. a locally MVU estimator for n at every in the model .
TL;DR: In this article, the authors highlight the twofold role of stratification in that it not only improves the efficiency of the ratio method of estimation but also enhances the chances of getting uniformly nonnegative unbiased variance estimators.
Abstract: When the ratio method is appropriate for estimating the population total one is faced with the problem of nonavailability of uniformly nonnegative unbiased variance estimators (nnuve). Here we highlight the twofold role of stratification in that it not only improves the efficiency of the ratio method of estimation but it also enhances the chances of getting uniformly nonnegative unbiased variance estimators.
TL;DR: In this paper, the least squares estimators and minimum-variance linear unbiased estimators of the unknown mean for homogeneous and isotropic random fields observed on sets of a special kind (a system of concentric circles, a ring).
Abstract: Least-squares estimators and minimum-variance linear unbiased estimators of the unknown mean are studied for homogeneous and isotropic random fields observed on sets of a special kind (a system of concentric circles, a ring).
TL;DR: In this article, the Cramer-Rao bound is used as a measure to determine the efficiency of unbiased estimators, which can potentially lead to a vast underestimated inefficiency of the used estimator conditions.
TL;DR: In this article, it is shown that it is possible to adopt optimal procedures of statistical inference like uniformly minimum variance unbiased estimation, uniformly most powerful and uniformly powerful unbiased test procedures, and Wald-type sequential probability ratio tests for uniparametric as well as multiparametric random processes belonging to the exponential family by employing random stopping times.