The estimation of standard errors is essential to statistical inference. Statistical variability is inherent within data, but is usually of secondary interest; still, some options exist to deal with this variability. One approach is to carefully model the covariance structure. Another approach is robust estimation. In this approach, the covariance structure is estimated from the data. White (1980) introduced a biased, but consistent, robust estimator. Long et al. (2000) added an adjustment factor to White's estimator to remove the bias of the original estimator. Through the use of simulations, this project compares restricted maximum likelihood (REML) with four robust estimation techniques: the Standard Robust Estimator (White 1980), the Long estimator (Long 2000), the Long estimator with a quantile adjustment (Kauermann 2001), and the empirical option of the MIXED procedure in SAS. The results of the simulation show small sample and asymptotic properties of the five estimators. The REML procedure is modelled under the true covariance structure, and is the most consistent of the five estimators. The REML procedure shows a slight small-sample bias as the number of repeated measures increases. The REML procedure may not be the best estimator in a situation in which the covariance structure is in question. The Standard Robust Estimator is consistent, but it has an extreme downward bias for small sample sizes. The Standard Robust Estimator changes little when complexity is added to the covariance structure. The Long estimator is unstable estimator. As complexity is introduced into the covariance structure, the coverage probability with the Long estimator increases. The Long estimator with the quantile adjustment works as designed by mimicking the Long estimator at an inflated quantile level. The empirical option of the MIXED procedure in SAS works well for homogeneous covariance structures. The empirical option of the MIXED procedure in SAS reduces the downward bias of the Standard Robust Estimator when the covariance structure is homogeneous.



College and Department

Physical and Mathematical Sciences; Statistics



Date Submitted


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Selected Project




standard errors, robust estimators, white estimator, empirical option