Showing papers on "U-statistic published in 2006"
TL;DR: Tail-area-based confidence interval methods are developed which can be applied to very small samples or extreme outcomes and are equivalent to the area under the receiver operating characteristic curve.
Abstract: For two random variables X and Y, θ=Pr[Y>X] + ½Pr[Y=X] is advocated as a general measure of effect size to characterize the degree of separation of their distributions. It is estimated by U/mn, a generalization of the Mann–Whitney U statistic, derived by dividing U by the product of the two sample sizes. It is equivalent to the area under the receiver operating characteristic curve. It is readily visualized in terms of two Gaussian distributions with appropriately separated peaks. The effect of discretization of a continuous variable is explored. Tail-area-based confidence interval methods are developed which can be applied to very small samples or extreme outcomes. Copyright © 2005 John Wiley & Sons, Ltd.
149 citations
TL;DR: In this paper, a model for a truncated sample of pairs (Xi, Y) satisfying 1 > Xi is proposed, where a possible dependency between the truncation time and the variable of interest is modelled with a parametric family of copulas.
Abstract: SUMMARY The product-limit estimator calculated from data subject to random left-truncation relies on the testable assumption of quasi-independence between the failure time and the truncation time. In this paper, we propose a model for a truncated sample of pairs (Xi, Y) satisfying 1 > Xi. A possible dependency between the truncation time and the variable of interest is modelled with a parametric family of copulas. The model also features a distribution function Fx(.) and a survival distribution Sy(.) associated with the marginal behaviours of X and Y in the observable region Y> X. Semiparametric estimators for these two functions are proposed; they do not make any parametric assumption about either Fx(.) or Sy(.). We derive an estimator for the copula parameter cX based on the conditional Kendall's tau. We generalise the copula-graphic estimators of Zheng & Klein (1995) to truncated variables. The asymptotic distributions of all these estimators are then investigated. The methods are illustrated with a real dataset on HIV infection by transfusion and by simulations.
64 citations
TL;DR: In this article, the random integral of a function with respect to the k-fold product of the normalized signed measure is considered, where μ n denotes the empirical measure defined by the random variables ξ 1,..., ξ n and the probabilities Open image in new window for all x>0.
Abstract: Let a sequence of iid. random variables ξ 1, . . . ,ξ n be given on a space Open image in new window with distribution μ together with a nice class Open image in new window of functions f(x 1, . . . ,x k ) of k variables on the product space Open image in new window For all f ∈ Open image in new window we consider the random integral J n,k (f) of the function f with respect to the k-fold product of the normalized signed measure Open image in new window where μ n denotes the empirical measure defined by the random variables ξ 1, . . . ,ξ n and investigate the probabilities Open image in new window for all x>0. We show that for nice classes of functions, for instance if Open image in new window is a Vapnik–Cervonenkis class, an almost as good bound can be given for these probabilities as in the case when only the random integral of one function is considered. A similar result holds for degenerate U-statistics, too.
32 citations
TL;DR: In this article, the best quadratic unbiased estimators of the integrated variance in the presence of independent market microstructure noise were considered and the asymptotic normality of a feasible best unbiased estimator under the assumption of constant market noise was established.
Abstract: We consider the best quadratic unbiased estimators of the integrated variance in the presence of independent market microstructure noise. We establish the asymptotic normality of a feasible best quadratic unbiased estimator under the assumption of constant
28 citations
Proceedings Article•
16 Jul 2006TL;DR: This paper describes the complete set of all unbiased estimators, that is, for any possible unknown dynamics the estimate's expectation is always the agent's expected utility, and identifies the unbiased estimator with minimum variance.
Abstract: Evaluating the performance of an agent or group of agents can be, by itself, a very challenging problem. The stochastic nature of the environment plus the stochastic nature of agents' decisions can result in estimates with intractably large variances This paper examines the problem of finding low variance estimates of agent performance. In particular, we assume that some agent-environment dynamics are known, such as the random outcome of drawing a card or rolling a die. Other dynamics are unknown, such as the reasoning of a human or other black-box agent. Using the known dynamics, we describe the complete set of all unbiased estimators, that is, for any possible unknown dynamics the estimate's expectation is always the agent's expected utility. Then, given a belief abcut the unknown dynamics, we identify the unbiased estimator with minimum variance. If the belief is correct our estimate is optimal, and if the belief is wrong it is at least unbiased. Finally, we apply our unbiased estimator to the game of poker, demonstrating dramatically reduced variance and faster evaluation.
28 citations
15 citations
01 Dec 2006
TL;DR: The design provides a characterization of estimator with delay, which eases the established necessary conditions for existence of unbiased estimators with zero-delay, and handles the noise affecting the system becomes correlated with the estimation error.
Abstract: We present a method for constructing linear minimum-variance unbiased state estimators for discrete-time linear stochastic systems with unknown inputs. Our design provides a characterization of estimators with delay, which eases the established necessary conditions for existence of unbiased estimators with zero-delay. A consequence of using delayed estimators is that the noise affecting the system becomes correlated with the estimation error. We handle this correlation by increasing the dimension of the estimator appropriately.
15 citations
TL;DR: In this article, the problem of unbiased estimation of the variance of an exponential distribution using a ranked set sample (RSS) was considered and the authors proposed some unbiased estimators each of which is better than the non-parametric minimum variance quadratic unbiased estimator based on a balanced ranked subset sample as well as the uniformly minimum variance unbiased estimulator based on simple random sample (SRS) of the same size.
14 citations
TL;DR: In this article, the authors considered the problem of testing the hypothesis about the sub-mean vector and showed that the asymptotic expansion of the null distribution of Rao's U-statistic under a general condition is obtained up to order of n^-^1.
13 citations
TL;DR: In this paper, the authors proposed a class of estimators of the variance of the systematic sample mean, which is unbiased under the assumption that the population follows a superpopulation model that satisfies some mild conditions.
12 citations
01 Nov 2006
TL;DR: In this article, the optimal unbiased minimum-variance estimation for systems with unknown inputs that affect both the system model and the measurements is considered. And the authors propose an optimal unbiased minimization filter based on matrix equation solution theory, which appears to have the most general form.
Abstract: This paper considers the optimal unbiased minimum-variance estimation for systems with unknown inputs that affect both the system model and the measurements. By making use of the well-known matrix equation solution theory, the optimal unbiased minimum-variance filter, which appears to have the most general form, is proposed. Specific forms of this new filter are also presented to illustrate their relationships with the existing literature results. A numerical example is included in order to illustrate the proposed results.
TL;DR: In this paper, the authors considered the problem of unbiased estimation of the distribution function of an exponential population using order statistics based on a random sample and presented a (unique) unbiased estimator based on single, say ith, order statistic and study some properties of the estimator for i = 2.
Abstract: In this paper we consider the problem of unbiased estimation of the distribution function of an exponential population using order statistics based on a random sample. We present a (unique) unbiased estimator based on a single, say ith, order statistic and study some properties of the estimator for i = 2. We also indicate how this estimator can be utilized to obtain unbiased estimators when a few selected order statistics are available as well as when the sample is selected following an alternative sampling procedure known as ranked set sampling. It is further proved that for a ranked set sample of size two, the proposed estimator is uniformly better than the conventional nonparametric unbiased estimator, further, for a general sample size, a modified ranked set sampling procedure provides an unbiased estimator uniformly better than the conventional nonparametric unbiased estimator based on the usual ranked set sampling procedure.
TL;DR: In this paper, the central limit theorem for U n when the U -statistic is degenerate or non-degenerate is established when the sample is a sequence of negatively associated random variables.
06 Feb 2006
TL;DR: In this paper, a method to approximate the eigenvalues of linear operators depending on an unknown distribution is introduced and applied to weighted sums of squared normally distributed random variables, which can be used to obtain an approximation of the asymptotic null distribution of certain degenerated U- and V-statistics.
Abstract: A method to approximate the eigenvalues of linear operators depending on an unknown distribution is introduced and applied to weighted sums of squared normally distributed random variables. This area of application includes the approximation of the asymptotic null distribution of certain degenerated U- and V-statistics.
TL;DR: In this paper, an Edgeworth expansion for the studentized product-limit estimator was proposed and shown to be asymptotically normal under left truncation and right censoring.
Abstract: The product-limit estimator under left truncation and right censoring was first proposed and shown to be asymptotically normal by Tsai et al. (1987). In this article, we shall develop an Edgeworth expansion for the studentized product-limit estimator. This is achieved by representing the studentized product-limit estimator as a U-statistic plus some negligible remainder terms. The bootstrap is employed to approximate the distribution of the studentized product-limit estimator and shown to be second-order accurate.
TL;DR: In this article, a general class of almost unbiased estimators is proposed for estimating population mean Y of the characteristic under study y when auxiliary information is available using systematic sampling plan, and explicit expressions for the variance of class of estimators are obtained to the first order of approximation.
Abstract: In this paper, a general class of almost unbiased estimators is proposed for estimating population mean ¯ Y of the characteristic under study y when auxiliary information is available using systematic sampling plan. Explicit expressions for the variance of class of estimators is obtained to the first order of approximation. Minimum variance unbiased estimators (optimum estimators) in the class are also found. The class of estimators proposed by Kushwaha and Singh (1989) comes out to be a particular case of this class estimators.
TL;DR: In this paper, two new unbiased point estimates of an unknown population variance are introduced, which are compared to three known estimates using the mean-square error (MSE), and a computer program is developed for performing calculations for the estimates.
Abstract: Two new unbiased point estimates of an unknown population variance are introduced. They are compared to three known estimates using the mean-square error (MSE). A computer program, which is available for download at http://program.20m.com, is developed for performing calculations for the estimates.
Journal Article•
TL;DR: In this paper, the existence problem of uniformly minimum variance unbiased estimators of the reliability function and its inverse function for the population of exponential distribution has been investigated and the consistency of the estimators has been proved.
Abstract: In the reliability theory,reliability function and its inverse function are usually unknown and need to be estimated by using sample.For the population of exponential distribution,existence problem of uniformly minimum variance unbiased estimators of the reliability function and its inverse function are investigated.The uniformly minimum variance unbiased estimators of the reliability function and its inverse function for the exponential distribution are obtained by the theories of sufficient statistic and complete statistic.Furthermore,consistency of the estimators is discussed,and the uniformly minimum variance unbiased estimators of the reliability function and its inverse function are proved to be strong consistent.
TL;DR: This paper considers the minimum variance unbiased estimation for the maximum entropy of the transformed inverse Gaussian random variable by and the properties of the derived UMVU estimator is investigated.
Abstract: The concept of entropy, introduced in communication theory by Shannon (1948) as a measure of uncertainty, is of prime interest in information-theoretic statistics. This paper considers the minimum variance unbiased estimation for the maximum entropy of the transformed inverse Gaussian random variable by . The properties of the derived UMVU estimator is investigated.
TL;DR: In this paper, the variance of a linear unbiased estimator of unknown mean value for some correlation models of homogeneous and isotropic random fields is derived for the first time.
Abstract: We obtain explicit formulas for the variance of a linear unbiased estimator of unknown mean value for some correlation models of homogeneous and isotropic random fields.
Posted Content•
TL;DR: In this paper, a condition for improvement on the unbiased estimators of the precision matrices is derived under a quadratic loss function, similar to the superharmonic condition established by Stein (1981).
Abstract: This paper treats the problem of simultaneously estimating the precision matrices in multivariate normal distributions. A condition for improvement on the unbiased estimators of the precision matrices is derived under a quadratic loss function. The improvement condition is similar to the superharmonic condition established by Stein (1981). The condition allows us not only to provide various alternative estimators such as shrinkage type and enlargement type estimators for the unbiased estimators, but also to present a condition on a prior density under which the resulting generalized Bayes estimators dominate the unbiased estimators. Also, a unified method improving upon both the shrinkage and the enlargement type estimators is discussed.
01 Jan 2006
TL;DR: In this paper, the authors have employed ms type sampling scheme to estimate the population variance and compared with certain known ones, and necessary and sufficient conditions have been obtained for their superior performance as compared to the known ones.
Abstract: summary We have employed ms type sampling scheme to propose two unbiased strategies for estimating the population variance. These strategies have been compared with certain known ones, and necessary and sufficient conditions have beenobtained for their superior performance as compared to the known ones. An unbiased variance estimator of the population variance has also been worked out. Real-life data are shown to yield substantial gains via these strategies.
TL;DR: In this paper, a formula for an unbiased estimator for the variance of a survey population total as well as for its variance based on sampling in two-stages following Rao et al. (1962) was presented.
Abstract: A formula is presented for an unbiased estimator for the variance of an unbiased estimator of a survey population total as well as for an unbiased estimator of its variance based on sampling in two-stages following Rao et al. J Roy Stat Soc B 24: 482–491 (1962) scheme in both stages when the originally selected units in both stages cannot be fully covered in the survey but are to be randomly sub-sampled. The development is helpful to tackle non-responses if assumed to have occurred at random in either or both the stages
01 Jan 2006
TL;DR: In this article, the authors consider a random-design regression model with vector-valued observations and develop nonparametric estimation of smooth conditional moment functions in the predictor variable, including estimation of higher order mixed moments and also functionals of the moments.
Abstract: We consider a random-design regression model with vector-valued observations and develop nonparametric estimation of smooth conditional moment functions in the predictor variable. This includes estimation of higher order mixed moments and also functionals of the moments, such as conditional covariance, correlation, variance, and skewness functions. Our asymptotic analysis targets the limit distributions. We find that some seemingly reasonable procedures do not reproduce the identity or other linear functions without undesirable bias components, i.e., they are linearly biased. Alternative linearly unbiased estimators are developed which remedy this bias problem without increasing the variance. A general linearly unbiased estimation scheme is introduced for arbitrary smooth functionals of moment functions.