Mini-max-risk and mini-mean-risk inferences for a partially piecewise regression

We consider a partially piecewise regression in which the main regression coefficients are constant in all subdomains, but the extraessential regression function is variable in different pieces and is difficult to be estimated. Under this situation, two new regression methodologies are proposed under the criteria of mini-max-risk and mini-mean-risk. The resulting models can describe the regression relations in maximum-risk and mean-risk environments, respectively. A two-stage estimation procedure, together with a composite method, is introduced. The asymptotic normality of the estimators is established, the standard convergence rate and efficiency are achieved. Some unusual features of the new estimators and predictions, and the related variable selection are discussed for a comprehensive comparison. Simulation studies and a real-financial example are given to illustrate the new methodologies.