  The mixture of Dirichlet process (MDP) defines a flexible prior distribution
on the space of probability measures. This study shows that ordinary
least-squares (OLS) estimator, as a functional of the MDP posterior
distribution, has posterior mean given by weighted least-squares (WLS), and has
posterior covariance matrix given by the (weighted) heteroscedastic-consistent
sandwich estimator. This is according to a pairs bootstrap distribution
approximation of the posterior, using a P\'olya urn scheme. Also, when the MDP
prior baseline distribution is specified as a product of independent
probability measures, this WLS solution provides a new type of generalized
ridge regression estimator which can handle multicollinear or singular design
matrices even when the number of covariates exceeds the sample size, and which
shrinks the coefficient estimates of irrelevant covariates towards zero, thus
useful for nonlinear regressions. Also, this MDP/OLS functional methodology can
be extended to methods for analyzing the sensitivity of the
heteroscedasticity-consistent causal effect size over a range of hidden biases
due to missing covariates omitted from the regression, and more generally
extended to a Vibration of Effects analysis. The methodology is illustrated
through the analysis of simulated and real data sets. Overall, this study
establishes new connections between Dirichlet process functional inference, the
bootstrap, consistent sandwich covariance estimation, ridge shrinkage
regression, WLS, and sensitivity analysis, to provide regression methodology
useful for inferences of the mean dependent response.
