Showing papers on "U-statistic published in 1997"
TL;DR: A new method is developed for the state estimation of linear discrete-time stochastic systems in the presence of an unknown disturbance that is optimal in the unbiased minimum variance sense.
259 citations
TL;DR: In this article, the authors consider the ranked-set best linear unbiased estimator and show that it is more efficient than the simple random sample mean, even for normal data, but particularly for skew data such as from an exponential distribution.
Abstract: Summary Ranked-set sampling is a widely used sampling procedure when sample observations are expensive or difficult to obtain. It departs from simple random sampling by seeking to spread the observations in the sample widely over the distribution or population. This is achieved by ranking methods which may need to employ concomitant information. The ranked-set sample mean is known to be more efficient than the corresponding simple random sample mean. Instead of the ranked-set sample mean, this paper considers the corresponding optimal estimator: the ranked-set best linear unbiased estimator. This is shown to be more efficient, even for normal data, but particularly for skew data, such as from an exponential distribution. The corresponding forms of the estimators are quite distinct from the ranked-set sample mean. Improvement holds where the ordering is perfect or imperfect, with this prospect of improper ordering being explored through the use of concomitants. In addition, the corresponding optimal linear...
80 citations
TL;DR: For the Gauss-Markov model, the authors showed that the best linear unbiased estimators of the model parameters remain unchanged if the predicted values of the dependent variable are used as observed values in estimating the parameters.
Abstract: This article shows, for the Gauss–Markov model, that the best linear unbiased estimators of the model parameters remain unchanged if the predicted values of the dependent variable (based on best linear unbiased predictors) are used as observed values in estimating the parameters. This result not only provides a useful insight into the interpretation of best linear unbiased predictors, but it also simplifies calculation of predictions in some cases. We also use this result to construct large-sample approximate predictors for scale and location-scale parameter distributions. Examples from life-testing are given.
34 citations
TL;DR: In this paper, a law of the iterated logarithm and an invariance principle for the statistic Fˆn(Un) for a class of strongly mixing sequences of random variables {Xi,i≥1}.
Abstract: We prove the almost sure representation, a law of the iterated logarithm and an invariance principle for the statistic Fˆn(Un) for a class of strongly mixing sequences of random variables {Xi,i≥1}. Stationarity is not assumed. Here Fˆn is the perturbed empirical distribution function and Un is a U-statistic based on X1,…,Xn.
30 citations
TL;DR: In this article, an unbiased unreliability estimator with reduced variance is obtained, which leads to a special stochastic optimization problem that can be demonstrated to be convex, such that efficient solution techniques apply.
Abstract: Recently, some authors have suggested usage models of Markov type as a technique of specifying the estimated operational use distribution of a given program. A main purpose of such models is the derivation of random test cases allowing unbiased estimates on the (un)reliability of the program in its intended environment. In this article, we show that by a shift of the transition probabilities of the Markov chain corresponding to such a model, prior information on the errorjproneness of single-program operations can be taken into account. An unbiased unreliability estimator with reduced variance is obtained. Furthermore, it is shown that minimization of the variance leads to a special stochastic optimization problem that can be demonstrated to be convex, such that efficient solution techniques apply. Some related questions are also treated in a more general, non-Markovian framework.
28 citations
TL;DR: In this paper, the H-decomposition of a jackknife estimator of the variance of U-statistic was obtained and an Edgeworth expansion with remainder term o (n −1 2 ) was established.
24 citations
TL;DR: In this paper, the authors present a simple method of derivation of these results that they feel will assist students in learning this method of estimation better, and use this simple approach to show some interesting properties of best linear unbiased estimators in the case of exponential distributions.
Abstract: Best linear unbiased estimators of location and scale parameters based on order statistics (from either complete or Type-II censored samples) are usually illustrated with exponential and uniform distributions. But the derivations in these two cases involve the explicit inverse of a diagonal matrix of Type 2 and extensive algebraic manipulations. In this note we present a simple method of derivation of these results that we feel will assist students in learning this method of estimation better. Furthermore, we use this simple approach to show some interesting properties of best linear unbiased estimators in the case of exponential distributions.
24 citations
01 Sep 1997
TL;DR: In this paper, the authors consider the situation where a life-testing experiment yields a Type-II progressively censored sample and develop large-sample approximations to the best linear unbiased estimators for the scale-parameter as well as for the location-scale parameter families of distributions.
Abstract: In this paper, we consider the situation where a life-testing experiment yields a Type-II progressively censored sample. We then develop large-sample approximations to the best linear unbiased estimators for the scale-parameter as well as for the location-scale parameter families of distributions. Large-sample expressions are also derived for the variances and covariance of these estimators. These results are used further to develop large-sample approximations to the best linear unbiased predictors of future failures. Finally, we present two examples in order to illustrate the methods of inference developed in this paper.
24 citations
01 Jan 1997
TL;DR: In this paper, the authors presented an approach to generate equivariant conditionally unbiased estimators of the factor scores in a confirmatory factor analytic model, which is achieved by only the weighted (WLS) and generalized least squares (GLS) estimators.
Abstract: Some approaches to generating equivariant conditionally unbiased estimators of the factor scores in a confirmatory factor analytic model are defined. In order to select among these estimators, weighted (WLS) and generalized least squares (GLS) functions are optimized, optimal minimum variance estimators are obtained, and fifteen multivariate criterion functions are used to define optimality. Optimality is achieved by only the WLS and GLS estimators, which are found to be equivalent under standard conditions. It is shown that when the factor loading matrix consists of a simple cluster structure, and some conditions hold on the loadings and unique variances, the estimators simplify substantially and some simple estimators used in practice can be justified. The well-known Bartlett WLS estimator is not applicable when the unique variance matrix is singular, but the GLS estimator can be applied more generally. However, it also can break down when the observed variables are linearly dependent. A modified estimator remains valid.
17 citations
TL;DR: In this paper, the authors obtain an unbiased estimator of the rth central moment of a distribution, which is unbiased for all distributions for which the first r moments exist, by finding the kernel which allows the rTH central moment to be written as a regular statistical functional.
Abstract: We obtain an estimator of the rth central moment of a distribution, which is unbiased for all distributions for which the first r moments exist. We do this by finding the kernel which allows the rth central moment to be written as a regular statistical functional. The U-statistic associated with this kernel is the unique symmetric unbiased estimator of the rth central moment, and, for each distribution, it has minimum variance among all estimators which are unbiased for all these distributions.
16 citations
TL;DR: In this article, a new theory for nonparametric analyses of r x k contingency tables, where the columns correspond to k ordered categories, was developed for non-parametric analysis of k-ordered contingency tables and the exact moments of U are simple functions of k and the two sample sizes.
Abstract: A new theory is developed for nonparametric analyses of r x k contingency tables, where the columns correspond to k ordered categories. The work is motivated by concern that conventional nonparametric theory yields inferences concerning the given categories, but results are often interpreted as if for a continuum. Asymptotic distributions are derived under random cut-points models for the Mann-Whitney (-Wilcoxon) U statistic. A model is recommended under which the exact moments of U are simple functions of k and the two sample sizes. For two more general models, the asymptotic variance is given for k ≤ 12 and k = 20, 30. Tables of empirical critical values show that the recommended model yields P values very close to P values under the general models.
TL;DR: In this article, it was shown that for independent and identically distributed random vectors, for which the components are independent and exponentially distributed with a common shift, one can construct unbiased estimators of their density, derived from the Uniform Minimum Variance Unbiased Estimator (UMVUE) of their distribution function.
TL;DR: In this paper, a class of shrinkage estimators for the variance of a normal population and its properties were proposed and compared with the usual unbiased estimator, minimum mean squared error (MMSE) estimator and Pandey and Singh, South African Statistical Journal (1976) and J. Indian Statistical Assoc. 15, 141−150 (1977) estimators.
TL;DR: In this article, the positivity of the best linear unbiased estimator of the scale parameter of a location-scale family of distributions in terms of complete or selected set of order statistics has only been conjectured in the literature based solely on empirical evidence.
TL;DR: In this paper, a least-squares approach to information dimension estimation of the invariant distribution of a dynamical system is proposed, which is computationally similar to the Grassberger-Procaccia algorithm.
TL;DR: Two classes of unbiased estimators of the density function of ergodic distribution for the diffusion process of observations are proposed in this article, and they are square-root consistent and asymptotically normal.
TL;DR: In this paper, the problem of estimating θ = p{X > Y} in the exponential model is reconsidered and the performance of the nonparametric unbiased estimator based on U-statistic is examined against the Uniformly Minimum Variance Unbiased Estimator.
Abstract: The problem of estimating θ = p{X > Y} in the exponential model is reconsidered. The performance of the nonparametric unbiased estimator based on U-statistic is examined against the Uniformly Minimum Variance Unbiased Estimator , proposed in the literature. Among other desirable properties, it is demonstrated that the estimator , its variance , and the U-statistic based unbiased estimator of the variance have simple expressions, easily amenable to recursive estimation. It is further shown that . Our numerical study indicates that generally we should be better off by using against possible outliers in the model.
01 Jan 1997
TL;DR: Sampling inspection as an instrument of intelligent statistical quality control should provide information about the process curve, i.
Abstract: Sampling inspection as an instrument of intelligent statistical quality control should provide information about the process curve, i. e. the long run distribution of product quality. Moreover, it should adapt the sampling strategy to this information.
TL;DR: In this paper, a multivariate approach was proposed to estimate the variance component of the covariance matrix of a one-way random effects model, including the error variance, using a delete-d jackknife procedure.
TL;DR: In this paper, an unbiased estimator of a finite population total and an unbiased variance estimator for it when samples are taken by usual procedures in the first two stages with varying probabilities but the third stage units are sampled for economy and convenience in a non-standard way from the pool of all sampled second stage units rather than independently from each of the latter separately containing the former.
Abstract: Presented are formulae for an unbiased estimator of a finite population total and an unbiased variance estimator for it when samples are taken by usual procedures in the first two stages with varying probabilities but the third stage units are sampled for economy and convenience in a non-standard way from the pool of all sampled second stage units rather than independently from each of the latter separately containing the former.
TL;DR: In this article, the nonexistence of unbiased estimators for the minimum and maximum parameters is discussed, and new estimators are proposed which work better than the existing estimators with respect to several criteria.
Abstract: Assuming independent random samples are drawn from two nonregular densities belonging to a general class of densities, an attempt has been made to unify and generalize the existing results from special models. The nonexistence of unbiased estimators for the minimum and maximum parameters is discussed. New estimators are proposed which work better than the existing estimators with respect to several criteria.
TL;DR: In this paper, a uniformly minimum variance unbiased estimator of population variance (of the sensitive character) has been obtained for eliciting sensitive information from a sample survey and a class of ordered sampling designs.
Abstract: Considering a class ofs randomized response trials for eliciting sensitive information from a sample survey and a class of ordered sampling designs, a uniformly minimum variance unbiased estimator of population variance (of the sensitive character) has been obtained. This note indicates that a theorem (theorem 3.9) of Cassel, Sarndal and Wretman (1977) and the results in the present note can be extended to estimation of any symmetric function of population values in the field of direct response surveys and randomized response surveys respectively.
TL;DR: In this paper, a new technique of optimal estimation of density functions for exponential shift families on a homogeneous space of a Lie group is proposed, which is essentially based on the algebraic properties of shift families.
Abstract: A new technique of optimal estimation of density functions for exponential shift families on a homogeneous space of a Lie group is proposed. In contrast to traditional methods, the approach considered is essentially based on the algebraic properties of shift families. Here we give a universal formula for consistent estimators of density functions covering different classes of estimators such as unbiased estimators with uniformly minimum variance and Bayesian estimators under two popular loss functions. The representations of some maximal invariant density functions are derived and simultaneously a close connection between the estimators and these density functions is established.