Marginal Conceptual Predictive Statistic for Mixed Model Selection

Abstract

Abstract

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

“…The parameter vector 𝛽 is chosen as 𝛽 = (𝛽 0 , 𝛽 1 , 𝛽 2 , 𝛽 3 , 𝛽 4 ) T = (2, 9, 2, −2, 4) T via Kuran and Özkale, 17 Wenren, 8 and Wenren et al 9 so that results can be comparable with that of Kuran and Özkale, 17 Wenren, 8 and Wenren et al 9 The underlying true model takes the form…”

“…The parameter vector $$$ \beta $$$ is chosen as $$$ \beta ={\left({\beta}_0,{\beta}_1,{\beta}_2,{\beta}_3,{\beta}_4\right)}^T={\left(2,9,2,-2,4\right)}^T $$$ via Kuran and Özkale, 17 Wenren, 8 and Wenren et al 9 so that results can be comparable with that of Kuran and Özkale, 17 Wenren, 8 and Wenren et al 9 The underlying true model takes the form $$$$ {y}_{ij}={\beta}_0+{\beta}_1{x}_{ij1}+\cdots +{\beta}_4{x}_{ij p}+{u}_{i1}{z}_{ij1}+{\varepsilon}_{ij},\kern1em i=1,\dots, m,\kern1em j=1,\dots, n, $$$$ where $$$ {z}_{ij1}=1 $$$ is the random effect. Equivalently, we can rewrite model () as …”

“…Wenren 8 extended the application of $$$ {C}_p $$$ from linear regression models to LMMs. Wenren 8 and Wenren et al 9 suggested marginal $$$ {C}_p $$$ ($$$ {\mathrm{MC}}_p $$$) statistic based on the expected marginal Gauss discrepancy as a model selection criterion in LMMs. Then, they also proposed an improvement of $$$ {\mathrm{MC}}_p $$$ ($$$ {\mathrm{IMC}}_p $$$) and it was proven an asymptotically unbiased estimator of the expected marginal Gauss discrepancy.…”

“…When multicollinearity problems that mentioned in the above paragraph are arose, one or more variables related to the fixed effects usually are deleted, but this could cause some not irrelevant consequences: the fitted candidate model could be misspecified and so, the underfitted or the overfitted candidate models result in large variances for the BLUE as well as a large variance for $$$ {\hat{y}}_i $$$. For this reason, Kuran and Özkale 17,18 enlarged Wenren, 8 Wenren and Shang 10 and Wenren et al's 9 studies for the ridge estimator and the ridge predictor under multicollinearity.…”

“…For this reason, Kuran and Özkale 17,18 enlarged Wenren, 8 Wenren and Shang 10 and Wenren et al's 9 studies for the ridge estimator and the ridge predictor under multicollinearity.…”