Model selection via conditional conceptual predictive statistic for mixed and stochastic restricted ridge estimators in linear mixed models

In this article, we characterize the mixed Cp$$ {C}_p $$ ( CMCp$$ {\mathrm{CMC}}_p $$ ) and conditional stochastic restricted ridge Cp$$ {C}_p $$ ( CSRRCp$$ {\mathrm{CSRRC}}_p $$ ) statistics that depend on the expected conditional Gauss discrepancy for the purpose of selecting the most appropriate model when stochastic restrictions are appeared in linear mixed models. Under the known and unknown variance components assumptions, we define two shapes of CMCp$$ {\mathrm{CMC}}_p $$ and CSRRCp$$ {\mathrm{CSRRC}}_p $$ statistics. Then, the article is concluded with both a Monte Carlo simulation study and a real data analysis, supporting the findings of the theoretical results on the CMCp$$ {\mathrm{CMC}}_p $$ and CSRRCp$$ {\mathrm{CSRRC}}_p $$ statistics.