Showing papers on "U-statistic published in 1976"

Invariant Quadratic Unbiased Estimation for Two Variance Components

Abstract: For a normally distributed mixed model with two unknown variance components $\theta_1$ and $\theta_2$, a tractable characterization is given for the admissible estimators within the class $\tilde{\mathscr{N}}_\delta$ of invariant quadratic unbiased estimators for $\delta_1\theta_1 + \delta_2\theta_2$. Here the term admissible is used with reference only to the class $\tilde{\mathscr{N}}_\delta$. This characterization is based on a result for general linear models which characterizes the admissible estimators within the class of linear unbiased estimators. The admissibility of MINQUE estimators and the usual analysis of variance estimators is considered.

Best Linear Unbiased Estimation of Missing Observations in an Economic Time Series

Abstract: The best linear unbiased estimator which we proposed previously for interpolating, distributing, and extrapolating a time series by related series is applied to the estimation of missing observations. Under special assumptions, the problem reduces to the one treated in Doran [2]. Our estimator is compared with his and is shown to be more efficient.

Asymptotic Behavior of Minque-Type Estimators of Variance Components

Abstract: The limiting distributions are obtained for two estimators of variance components: C. R. Rao's MINQUE, and an estimator produced by an iterative procedure, referred to as I-MINQUE. Limits are taken as the number of independent and identically distributed vector observations on the model assumed gets large. This approach provides the asymptotics of interest when an experiment with a large number of observations can be thought of as independent replications of a smaller experiment, a condition applying to some common experimental designs. The main result, from which the limiting distributions are obtained, is essentially an extension of a theorem due to T. W. Anderson (1973), who provides an application in time series. Both estimators considered here are consistent, and require only modest assumptions on the sampled distribution. The I-MINQUE has a limiting distribution which is functionally independent of the choice of norm; when it is further assumed that the sampled distribution is normal, the estimator is asymptotically equivalent to the m.1.e. and asymptotically efficient. The MINQUE itself is less robust in the sense that these two properties do not always apply, the conditions being dependent on the choice of design.

A note on minque for normal models

Abstract: Consider a normal linear model where e has covariance matrix with c = (c1 … cp) ′ unknown. Restricting estimators to the class of functions for all β we prove, that Minque is locally minimum variance unbiased estimator. This result is extended to the class of all unbiased estimators if the range of X is assumed to be an invariant subspace with respect to all linear mappings given by V 1 … Vp . Furthermore necessary and sufficient conditions for the existence of uniformly best unbiased estimators (BUE) or only uniformly best invariant unbiased estimators (BIUE) are developed. They are small extensions of earlier results published by C.R. Rao [7] and J. Seely [8]. A sufficient condition for a BUE t exist given in 8, is shown to be not necessary.

The robust estimation of the density of a forest stand using a new conditioned distance method

Abstract: SUMMARY Estimators for the density of points in a plane may be unbiased for the spatial pattern for which they were designed but are usually biased for most others. Two new estimators are proposed which are approximately unbiased for a wide range of patterns, and are used on data in the form of the positions of the centres of trees in a forest stand.

Unbiased Estimation in Fixed Cost Sequential Sampling Schemes

Abstract: Fixed cost sequential sampling schemes are introduced in this article. In these schemes units are observed sequentially according to a given sampling method until the total cost reaches a preassigned value; it is assumed that the cost of examining each unit is unknown in advance. It is shown how the notion of sufficiency in sampling can be used to construct unbiased estimators of population parameters under these schemes.

A Comparison of Estimators of Variance Components in the Balanced Three-Stage Nested Random Effects Model Using Mean Squared Error Criterion

Abstract: Various estimators of variance components for the balanced three-stage nested random effects model are compared under standard assumptions of normality and independence of random effects, using MSE as the measure of performance. Several results are proven, demonstrating the inequality relationships on MSE of the unbiased, restricted maximum likelihood and maximum likelihood, and several modifications of the unbiased and maximum likelihood estimators as introduced here, including Stein-type estimators. Some Bayesian estimators are also considered for numerical comparison of the MSE. Computations are carried out to assess the order of difference in the MSE's. A large class of equivariant estimators of the error component is shown to be inadmissible.

Comparison of Some Bounds in Estimation Theory

Abstract: Conditions are given for the attainment of the Hammersley-Chapman-Robbins bound for the variance of an unbiased estimator, in both regular and nonregular cases. Comparisons are made between this bound and the Bhattacharyya system of bounds for a wide class of distributions and parametric functions. Sufficient conditions are provided to determine when one bound is sharper than the other one.

Almost unbiased product-type estimator

Abstract: In sample survey methods the use of product estimators was suggested byMurthy [1964] andSrivastava [1966] and were found to serve good purpose provided the two variables viz. the main variable under study and the auxiliary variable have a very high negative correlation between them. The product estimators suggested by them are biased. In the present paper the author has obtained unbiased product estimators (to the first degree of approximation) with the help of the technique developed byQuenouille [1956] and has established that this new estimator is better than the other product estimator in the mean square error sense.

On a Criterion for Simultaneous Extrapolation in Nonfull Rank Normal Regression

Abstract: In recent work by O'Reilly, a necessary and sufficient condition for the existence of an unbiased estimate of the distribution function of a "future" observation was given. The result was obtained under the condition that the model had full rank. Here, the result is generalized to any number of future observations and the full rank condition is relaxed. The corresponding uniformly minimum variance unbiased estimator is identified from the density estimates given by Ghurye and Olkin.

Universal model for adjusting observed values

Abstract: Fundamental models of the calculus of observations. Regularity and singularity of models. Universal model. Unbiased estimable and unbiased unestimable functions of the model parameters.

On the Computation of Certain Minimum Variance Unbiased Estimators

Abstract: It is shown that the technique of Sathe and Varde for finding MVUE's of reliability functions can be extended to the estimation of other functions of parameters by the introduction of an auxiliary random variable. Examples from reliability theory are considered.

Some Alternatives to Linear Unbiased Estimation and Traditional Pretest Procedures

Abstract: In economics the most widely used statistical technique is based on the linear statistical model and least squares and maximum likelihood concepts. These tools are supported by a body of theory that dates back well over a century to Gauss and Legendre and includes the powerful and beautiful Gauss-Markoff theorem, which says in effect that, out of the class of linear unbiased estimators, least squares is the "best" (minimum variance, minimum mean square error) estimator. The linear unbiased restrictions that are necessary to achieve the Gauss-Markoff result appear to be generally accepted in econometric analysis. However, for the economic researcher who is interested in parameter estimation or prediction appropriate for choice or decision purposes, the property of unbiasedness, which yields an estimator that is right on the average, may have limited usefulness. In addition the best unbiased statistical property only holds if the statistical model is correctly specified, and in economics the sampling process by which the data are generated is seldom known. Consequently, the purpose of this expository note is to point out that over a range of criteria for evaluating estimators, distributional assumptions, and conditions normally fulfilled in practice, simple estimators exist that are uniformly and nontrivially superior over the parameter space to the least squares and pretest estimators used by most applied researchers in economics.

Linear Estimation of the Logistic Parameters for Complete or Tail-Censored Samples

Abstract: In this paper we present the asymptotically best linear unbiased estimators for the location and scale parameters of the logistic distribution based on samples where there may be Type 2 censoring in the tails. Given the amount of censoring, the weights of the estimators can be expressed in simple closed forms. Comparisons of these with the best linear unbiased estimators and the Cramer-Rao lower bounds demonstrate that they have good efficiency even for small samples.