$d$-Posterior approach in regression

In this paper, we have used the $d$-posterior approach in regression. Regression predictions are a sequence of similarly made decisions. Thus, $d$-risk can be helpful to estimate the quality of such decisions. We have introduced a method to apply the $d$-posterior approach in regression models. This method is based on posterior predictive distribution of the dependent variable with the given novel input of predictors. In order to make $d$-risk of the prediction rule meaningful, we have also considered adding probability distribution of the novel input to the model.
The method has been applied to simple regression models. Firstly, linear regression with Gaussian white noise has been considered. For the quadratic loss function, estimates with uniformly minimal $d$-risks have been constructed. It appears that the parameter estimate in this model is equal to the Bayesian estimate, but the prediction rule is slightly different. Secondly, regression for the binary dependent variable has been investigated. In this case, the $d$-posterior approach is used for the logit regression model. As for the $0$–$1$ loss function, the estimate with uniformly minimal $d$-risk does not exist, we suggested a classification rule, which minimizes the maximum of two $d$-risks. The resulting decision rules for both models are compared to the usual Bayesian decisions and the decisions based on the maximum likelihood principle.