Showing papers on "U-statistic published in 1999"

Unbiased estimation following a group sequential test

Abstract: It is shown that, in a group sequential test about the drift 0 of a Brownian motion X(t) stopped at time T, the sufficient statistic (T,X(T)) is not complete for 0. There exist infinitely many unbiased estimators of 0 and none has uniformly minimum variance. A truncation-adaptable criterion is proposed, and the uniformly-minimum-variance estimator among all truncation-adaptable unbiased estimators is found. This estimator is identical to estimators of Ferebee (1983) and Emerson & Fleming (1990).

On the equivalence of meta-analysis using literature and using individual patient data.

Abstract: Summary. When data come from several independent studies for the purpose of estimating treatment-control differences, meta-analysis can be carried out either on the best linear unbiased estimators computed from each study or on the pooled individual patient data modelled as a two-way model without interaction, where the two factors represent the different studies and the different treatments. Assuming that observations within and between studies are independent having a common variance, Olkin and Sampson (1998) have obtained the surprising result that the two meta-analytic procedures are equivalent, i.e., they both produce the same estimator. In this article, the same equivalence is established for the two-way fixed-effects model without interaction with the only assumption that the observations across studies be independent. A consequence of the equivalence result is that, regardless of the covariance structure, it is possible to get an explicit representation for the best linear unbiased estimator of any vector of treatment contrasts in a two-way fixed-effects model without interaction as long as the studies are independent. Another interesting consequence is that, for the purpose of best linear unbiased estimation, an unbalanced two-way fixed-effects model without interaction can be treated as several independent unbalanced one-way models, regardless of the covariance structure, when the studies are independent.

Cramer-Von Mises Variance Estimators for Simulations

Abstract: We study estimators for the variance parameter σ2 of a stationary process. The estimators are based on weighted Cramer-von Mises statistics, and certain weightings yield estimators that are "first-order unbiased" for σ2. We derive an expression for the asymptotic variance of the new estimators; this expression is then used to obtain the first-order unbiased estimator having the smallest variance among fixed-degree polynomial weighting functions. Our work is based on asymptotic theory; however, we present exact and empirical examples to demonstrate the new estimators' small-sample robustness. We use a single batch of observations to derive the estimators' asymptotic properties, and then we compare the new estimators among one another. In real-life applications, one would use more than one batch; we indicate how this generalization can be carried out.

U-Statistics and imperfect ranking in ranked set sampling

Abstract: Ranked set sampling has attracted considerable attention as an efficient sampling design, particularly for environmental and ecological studies. A number of authors have noted a gain in efficiency over ordinary random sampling when specific estimators and tests of hypotheses are applied to rank set sample data. We generalize such results by deriving the asymptotic distribution for random sample U-statistics when applied to ranked set sample data. Our results show that the ranked set sample procedure is asymptotically at least as efficient as the random sample procedure, regardless of the accuracy of judgement ranking. Some errors in the ranked set sampling literature are also revealed, and counterexamples provided. Finally, application of majorization theory to these results shows when perfect ranking can be expected to yield greater efficiency than imperfect ranking.

Best linear unbiased estimators for the simple linear regression model using ranked set sampling

Abstract: When sample observations are expensive or difficult to obtain, ranked set sampling is known to be an efficient method for estimating the population mean, and in particular to improve on the sample mean estimator. Using best linear unbiased estimators, this paper considers the simple linear regression model with replicated observations. Use of a form of ranked set sampling is shown to be markedly more efficient for normal data when compared with the traditional simple linear regression estimators.

Improved nonnegative estimation of multivariate components of variance

Abstract: In this paper,we consider a multivariate one-way random effect model with equal replications. We propose nonnegative definite estimators for “between” and “within” components of variance. Under the Stein loss function, it is shown that the proposed estimators of the “within” component dominate the best unbiased estimator. Restricted maximum likelihood, truncated and order-preserving minimax estimators are also proposed. A Monte Carlo simulation is carried out to choose among these estimators. For estimating the “between” component, we consider the Stein loss function for jointly estimating the two positive definite matrices (“within” and “within” plus “between”) and obtain estimators for the “between” component dominating the best unbiased estimator. Other estimators as considered for “within” are also proposed. A Monte Carlo simulation is carried out to choose among these estimators.

Improved Nonnegative Estimation of Multivariate Components of Variance

Abstract: In this paper, we consider a multivariate one-way random effect model with equal replications. We propose non-negative definite estimators for 'between' and 'within' components of variance. Under the Stein loss function/Kullback-Leibler distance function, these estimators are shown to be better than the corresponding unbiased estimators. In particular, it is shown that the proposed restricted maximum likelihood estimator performs better than the unbiased as well as the truncated estimators proposed in this paper. Minimax and order-preserving minimax estimators are also proposed.

One-Term Edgeworth Expansion for Finite Population U-Statistics of Degree Two

Abstract: By means of Hoeffding"s decomposition, we represent a finite population U-statistic of degree two by the sum of a linear and a quadratic part. Assuming that the linear part is nondegenerate, we prove the validity of one-term Edgeworth expansion for the distribution function of the statistic under the optimal (minimal) conditions on the linear part and 2 + δ moment condition on the quadratic part. No condition is imposed on the ratio N / n, where N, respectively n, denotes the sample size respectively the population size.

On the Rao-Blackwell Theorem for fuzzy random variables

Abstract: In a previous paper, conditions have been given to compute iterated expectations of fuzzy random variables, irrespectively of the order of integration. In another previous paper, a generalized real-valued measure to quantify the absolute variation of a fuzzy random variable with respect to its expected value have been introduced and analyzed. In the present paper we combine the conditions and generalized measure above to state an extension of the basic Rao–Blackwell Theorem. An application of this extension is carried out to construct a proper unbiased estimator of the expected value of a fuzzy random variable in the random sampling with replacement from a finite population.

Constructing an unbiased estimator of population mean in finite populations using auxiliary information

Abstract: The objective of this paper is to construct an unbiased estimator (up to order 0(1/n)) of the population mean\(\bar Y\) of the study variatey which is more efficient than the sample mean\(\bar y\) of the ‘n’ obsrvedy-values. In particular, the unbiased estimators are discussed for the cases of positive and negative correlations of the study variatey and the auxiliary variatex.

Greedy Importance Sampling

Abstract: I present a simple variation of importance sampling that explicitly searches for important regions in the target distribution. I prove that the technique yields unbiased estimates, and show empirically it can reduce the variance of standard Monte Carlo estimators. This is achieved by concentrating samples in more significant regions of the sample space.

Classes of almost unbiased ratio and product type estimators in two phase sampling

Abstract: This paper suggests classes of almost unbiased ratio type and product -type estimators for estimating the population mean in two phase (or double) sampling and analyses their properties. Empirical studies, are carried out to demonstrate the performances of the suggest estimators over others.

Survey estimation for highly skewed data

Abstract: Estimation of the population total of a highly skewed survey variable from a small sample is problematic if straightforward methods are used since (i) when there are no extreme values in the sample, too small estimates will be obtained (ii) if extreme values are sampled, the estimates will become grotesquely large. Traditional methods for outlier treatment will usually compensate for outliers in the sample, thereby avoiding (ii), whereas the small negative bias of (i) will persist. Here, a lognormal superpopulation model is proposed. A particular strength of the lognormal model estimator is that even in the absence of extremely large values in the sample, the assumed lognormal structure of the survey variable is used for estimating the population total. Two estimators based on a lognormal superpopulation distribution are proposed: (i) one estimator applicable if the shape parameter of the assumed lognormal superpopulation distribution is known (ii) one estimator applicable if the shape parameter is unknown. For both estimators, any number of auxiliary variables can be utilized. Estimator (i) is of little practical importance, but has the advantage that it is model unbiased, and that a model unbiased estimator of its estimation error variance also easily can be derived. Estimator (ii), although only approximately model unbiased, is more practically applicable, because of the more realistic assumption of unknown shape parameter. Both estimators (i) and (ii) are applicable only for variables that are strictly positive. A third estimator, based on a combined lognormal-logistic superpopulation model is therefore proposed; this estimator can be applied to situations in which the survey variable, while highly skewed, may assume the value zero for a number of units. The three model-based estimators are compared to a number of alternative estimators (design-based estimators as well as estimators specifically constructed for outlier treatment) in a simulation study, using random populations as well as real survey populations. The simulation results give at hand that the model-based estimators constitute a sensible alternative to the alternative estimators, in particular when the sample size is small and when the distribution of the survey variable is close to the assumed superpopulation distribution.

On the small-sample optimality of multiple-regeneration estimators

Abstract: We describe a simulation output analysis methodology suitable for stochastic processes that are regenerative with respect to multiple regeneration sequences. Our method exploits this structure to construct estimators that are more efficient than those that are obtained with the standard regenerative method. We illustrate the method in the setting of discrete-time Markov chains on a countable state space, and we present a result showing that the estimator is the uniform minimum variance unbiased estimator for finite-state-space discrete-time Markov chains. Some empirical results are given.

Generalized L-statistics: Correction to the Form of the Asymptotic Variance Presented by Helmers et al. (1990)

Abstract: We show that the asymptotic variance of a “generalized L-statistic” is a function of the difference between the conditional and unconditional cumulative distribution functions of the kernel used to form the statistic.

Characterization of an optimal matrix estimator under convex loss function

Abstract: Characterization of an optimal vector estimator and an optimal matrix estimator are obtained. In each case appropriate convex loss functions are considered. The results are illustrated through the problems of simultaneous unbiased estimation, simultaneous equivariant estimation and simultaneous unbiased prediction. Further an optimality criterion is proposed for matrix unbiased estimation and it is shown that the matrix unbiased estimation of a matrix parametric function and the minimum variance unbiased estimation of its components are equivalent.

Theory of quadratic estimates of variance

Abstract: A class of quadratic estimates is constructed for the second-order moment and variance of a random variable. These estimates are found on the basis of sample values obtained by simple sampling. The best quadratic estimates are found for the second-order moment and variance in the case of known mathematical expectation. The exactness of biased and unbiased estimates of variance is investigated in the case of unknown mathematical expectation.

Unbiased estimators of a lattice mixing distribution and the characteristic function of a general mixing distribution

Abstract: SUMMARY. Let f(xjµ) be a known parametric family of probability density functions with respect to a ae-finite measure „. The density function f(x) of a random variable X belongs to a mixture model if f(x) = R f(xjµ)dG(µ). We derive unbiased estimators of the characteristic functions of the mixing distribution G under some integrability conditions on G and the probability mass function of G when G is a lattice distribution. Upper bounds for the variances of these unbiased estimators are provided. Three types of exponential families and a location-type model are considered, including the Poisson and gamma families.

An empirical investigation of the distribution of modified liml estimators for equations with more than two endogenous variables

Abstract: SUMMARY. In this paper we consider the general problem of estimation and inference in stochastic simultaneous equations. We discuss a class of modified limited information maximum likelihood estimators called scaled down LIML estimators. One of these estimators is asymptotically unbiased to order ae2 and a second one is minimum mean square to order ae4. Based on the unbiased estimator, asymptotic tests of hypotheses are proposed. Simulation results indicate that these tests perform better compared to traditional tests.

Stepwise Estimation of Random Processes

Abstract: The problem of estimating a continuous-time random process from its observations at appropriately designed sampling points is considered. The quality of an estimator is measured by its integrated mean square error (IMSE). Here, sampling points are designed stepwisely to minimize the IMSE and the best linear unbiased estimator (BLUE) is so determined that the earlier calculations do not have to be repeated with addition of one or more new samples. For random processes whose covariance has a sharp corner at the diagonal, it is shown that essentially, an optimal one-step forward sampling location is one of the midpoints of intervals determined by the current and previous sampling points. Both analytical and numerical examples are considered.