Concentration Ellipsoids,Their Planes of Support,and the Linear Regression Model

The relationship between the concentration ellipsoid of a random vector and its planes of support is exploited to provide a geometric derivation and interpretation of existing results for a general form of the linear regression model. In particular, the planes of support whose points of tangency to the ellipsoid are contained in the range (or column space) of the design matrix are the source of all linear unbiased minimum variance estimators. The connection between this idea and estimators based on projections is explored, as is also its use in obtaining and interpreting some existing relative efficiency results.     

The asymptotic equivalence between the iterated improved 2sls estimator and the 3sls estimator

In this paper we show that the 3SLS estimator of a system of equations is asymptotically equivalent to an iterative 2SLS estimator applied to each equation, augmented with the residuals from the other equations. This result is a natural extension of Telser (1964).     

Comparison of conditional and unconditional confidence intervals for the double exponential distribution

Conditional confidence intervals for the location parameter of the double exponential distribution based on maximum likelihood estimators conditioned on a set of ancillary statistics and the corresponding unconditional confidence intervals based on the maximum likelihood estimators alone are compared in two ways. Monte Carlo techniques are used and the conditional approach appears to give slightly better results although agreement as n becomes larger is noted     

A simulation study of five biased estimators for straight line regression

Five biased estimators of the slope in straight line regression are considered. For each, the estimate of the “bias parameter”, k, is a function of N, the number of observations, and [rcirc]² , the square of the least squares estimate of the standardized slope, β. The estimators include that of Farebrother, the ridge estimator of Hoerl, Kennard, and Baldwin, Vinod's shrunken estimators., and a new modification of one of the latter. Properties of the estimators are studied for 13 combinations of N and 3. Results of simulation experiments provide empirical evidence concerning the values of means and variances of the biased estimators of the slope and estimates of the “bias parameter”, the mean square errors of the estimators, and the frequency of improvement relative to least squares. Adjustments to degrees of freedom in the biased regression analysis of variance table are also considered. An extension of the new modification to the case of p> 1 independent variables is presented in an Appendix.     

Reliability estimation for the exponential distribution

This paper is concerned with classical statistical estimation of the reliability function for the exponential density with unknown mean failure time θ, and with a known and fixed mission time τ. The minimum variance unbiased (MVU) estimator and the maximum likelihood (ML) estimator are reviewed and their mean square errors compared for different sample sizes. These comparisons serve also to extend previous work, and reinforce further the nonexistence of a uniformly best estimator. A class of shrunken estimators is then defined, and it produces a shrunken quasi-estimator and a shrunken estimator. The mean square errors for both these estimators are compared to the mean square errors of the MVU and ML estimators, and the new estimators are found to perform very well. Unfortunately, these estimators are difficult to compute for practical applications. A second class of estimators, which is easy to compute is also developed. Its mean square error properties are compared to the other estimators, and it outperforms all the contending estimators over the high and low reliability parameter space. Since, for all the estimators, analytical mean square error comparisons are not tractable, extensive numerical analyses are done in obtaining both the exact small sample and large sample results.     

On the estimation of the true probability of a correct selection

For ranking and selection problems, the true probabiIity of a correct selection P(CS) is unknown even if a selection is made under the indifference-zone approach. Thus to estimate the true P(CS) some Bayes estimators and a bootstrap estimator are proposed for two normcal populations with common known variance. Also a bootstrap estimator and a bootstrap confidence interval are proposed for normal populations with common unknown variance. Some comparisons between proposed estimators and some other known estimators are made via Monte Carlo simulations.     

Minimum average risk estimators for coefficients in linear models

We consider the problem of estimating the coefficient vector β of a linear regression model with quadratic loss function. Some biased estimators which utilize the prior information about β are considered. Also studied is the problem of estimating the parameters of an over-identified structural equation from undersized samples.     

A simulation study of ridge and other regression estimators

We consider a number of estimators of regression coefficients, all of generalized ridge, or 'shrinkage' type. Results of a simulation study indicate that with respect to two commonly used mean square error criteria, two ordinary ridge estimators, one proposed by Hoerl, Kennard and Baldwin, and the other introduced here, perform substantially better than both least squares and the other estimators discussed here     

Trimmed and Outer Means and Their Variances

Some properties of trimmed and outer means for the normal situation are considered; in particular, the mean of the outer half of the sample has the same variance as the mean of the inner half. An inequality involving variances of unbiased estimators of location and their complements is derived and some of its consequences are examined.