Diane I. Gibbons

A Simulation Study of Estimators for the 2-Parameter Weibull Distribution

Seven estimators for the scale (¿) and shape (ß) parameters and percentiles of the Weibull distribution were compared by Monte Carlo methods. The evaluated estimators include the maximum likelihood estimator (MLE), linear estimators, least squares estimators, and a moment estimator. The performance of these estimators with respect to mean square error was studied in complete and Type II censored samples of sizes 10 and 25. No estimator outperformed all the others in all situations. One estimator, however, consistently performed worse than one of the others. The following summarizes the results. 1. The MLEs performed very well in the simulation study for all parameters when estimating from complete samples of size 25. For smaller samples and/or censored samples, they still performed very well as estimators of 1/ß and the upper percentiles of the distribution. 2. The best linear unbiased estimator (BLUE) was generally better than the best linear invariant estimator (BLIE) for estimating ß and the 10-th percentile. The BLIE was generally better than the BLUE for estimating 1/ß, ¿, and the 90-th percentile. The overall performance of both of these linear estimators was similar to that of the MLEs. A choice between the linear estimators and the MLEs for a specific application can be based on such considerations as the availability of tables and ease of computation. No overriding superiority of the linear estimators over the maximum likelihood estimator was demonstrated, and vice versa. 3.

Estimators for the 2-Parameter Weibull Distribution with Progressively Censored Samples

In many life tests, the initial censoring of items results in withdrawing a portion of the survivors while some remain on test until failure or until a subsequent stage of censoring. If the censoring is progressive through several stages, the resulting sample consists of censored items intermingled with failed ones. The maximum likelihood estimator (MLE) and a least squares median ranks estimator (LSMRE) apply in this situation. Using Monte Carlo methods, the statistical properties of these estimators for the parameters and percentiles of the 2-parameter Weibull distribution are determined. The results are: 1. The MLE performs well in estimating the parameters and percentiles for complete samples of moderate to large size (25 and 100). For small sample size (10) and/or censored samples it performs relatively well in estimating the scale parameter and the upper percentiles of this distribution. 2. The LSMRE was generally less reliable than the MLE in estimating the scale parameter and the upper percentiles of the distribution. It performed relatively well when estimating the shape parameter and the lower percentiles.

A Rational Interpretation of the Ridge Trace

The ridge regression estimator may be written as a linear combination of the least squares estimators derived from all possible subset regressions. This article delineates the relationship between the ridge estimator and the subset regression estimators and highlights the implications of this relationship for ridge trace interpretation. Ridge-regression examples are provided, illustrating how the interpretation of a ridge trace is enhanced.

The complementary use of regression diagnostics and robust estimators

Regression modeling for prediction or forecasting purposes is critically dependent on the quality of the data which are used to estimate the model parameters. Extreme response or predictor-variable values can substantially influence least-squares estimates and disproportionately affect predictions. Robust alternatives to least-squares are less sensitive to extreme observations and can provide more precise predictions. In this article diagnostic displays are used to identify extreme observations and to assess the sensitivity of least-squares parameter estimates and predictions to the inclusion of these observations in a data set. The displays are shown to aid in the interpretation of weights which robust estimators assign to influential observations.

CONSTRAINED REGRESSION ESTIMATES OF TECHNOLOGY EFFECTS ON FUEL ECONOMY

Practical aspects of linear regression modeling are reviewed, explicitly incorporating knowledge about the signs and ordering of the regression parameters. Statistical properties of equality and inequality constrained estimators are presented. This co..

Small-Sample Estimation for the Lognormal Distribution With Unit Shape Parameter

Let X be a random variable such that In [(X - ¿)/¿] has a s-normal distribution with mean zero and variance one. Then X has a 3-parameter lognormal distribution with the third parameter, the shape parameter, fixed at unity. This paper presents the coefficients required to construct the best linear unbiased estimators (BLUEs) of ¿ and ¿ for samples of size fifteen and less. The variances and covariances of these estimators are provided. These estimators yield the BLUEs of the mean, standard deviation, and percentiles of X since these quantities are linear functions of ¿ and ¿. Bloms estimators and maximum likelihood estimators compare favorably with the BLUEs.

An Evaluation of Two Model Specification Techniques for a Lognormal Distribution

Two frequently used statistical estimation techniques are applied to the lognormal distribution with unit shape-parameter. The first technique involves comparing sample estimates of skewness and kurtosis with their corresponding population values in order to determine the suitability of the distribution as a model for a set of empirical data. The sample skewness and kurtosis for samples of moderate size provide biased downward estimates of the population values. This bias can be considered if this technique is applied to a data set by using results presented in this paper. The second technique is probability plotting. The recommended plotting position is (i -0.5)/n. An evaluation of the graphical estimators shows that they are substantially inferior to the best linear unbiased estimators and Bloms estimators. The results are applied to a data set consisting of automotive emissions.

Some characterizations of the ribge trace

Ridge regression is a popular method for estimating linear regression coefficients when the explanatory variables are highly correlated. Ridge regression defines a class of estimators from which a unique estimator may be chosen based on the behavior of the ridge trace, a plot of the regression coefficient estimates versus the nonnegative scalar ridge parameter. In this paper characteristics of the ridge trace are determined algebraically when the correlation matrix of the explanatory variables is assumed to have a special structure (so-called Toeplitz). Special structure was required because quantitative characteristics are intractable in general. The Toeplitz structure was chosen because of its tractability and because it differs from the restrictive cases which have already been studied. Specific properties characterized include: (i) the order of the individual ridge estimates; (ii) the order of the rate-of-change of these estimates; and (iii) the number of sign changes in an individual estimate.

Illustrating regression diagnostics with an air pollution and mortality model

Statistical techniques have been developed recently which facilitate the systematic exploration of the sensitivity of regression models to individual observations in the data base. In this article, the utility of these techniques is demonstrated with a multiple linear regression model (published by L. Lave and E. Seskin) relating air pollution, specifically sulfates and particulates, to human mortality based on data from 117 Standard Metropolitan Statistical Areas (SMSAs). These techniques identify SMSAs which are widely separated from the remainder with repect to the independent (explanatory) variables as well as those which strongly affect the estimated regression coefficients and/or the predicted mortality rates. The problem of collinear explanatory variables is investigated and quantified using ridge regression and variable deletion techniques. It is shown that the estimated mortality benefits from air pollution reduction can be altered by as much as 23% by removing one SMSA from the data base. Overall estimates of the elasticity of air pollution on human mortality are higher using least squares methods than with robust, bounded-influence, or ridge methods.