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The article revisits univariate and multivariate linear regression models. It is shown that least-square estimators (LSEs) are minimum risk estimators in general class of linear unbiased estimators under some general divergence loss. This amounts to the loss robustness of LSEs. [ABSTRACT FROM AUTHOR]

The Dirichlet process has been used extensively in Bayesian non parametric modeling, and has proven to be very useful. In particular, mixed models with Dirichlet process random effects have been used in modeling many types of data and can often outperform their normal random effect counterparts. Here we examine the linearmixed model with Dirichlet process random effects from a classical view, and derive the best linear unbiased estimator (BLUE) of the fixed effects. We are also able to calculate the resulting covariancematrix and find that the covariance is directly related to the precision parameter of the Dirichlet process, giving a new interpretation of this parameter. We also characterize the relationship between the BLUE and the ordinary least-squares (OLS) estimator and show how confidence intervals can be approximated. [ABSTRACT FROM AUTHOR]

This article gives a nonlinear version of the Gauss-Markov theorem. It is shown that in a linear regression model y = X beta + u, the lower bound for the risk matrix E[beta - beta][beta - beta]' of a nonlinear estimator of beta belonging to a certain class is the covariance matrix of the Gauss-Markov estimator, provided the distribution of error term u belongs to the class of elliptically symmetric distributions with second moments. [ABSTRACT FROM AUTHOR]