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    Improvement of mixed predictors in linear mixed models
    (Taylor and Francis Ltd., 2021) Kuran, Özge; Özkale, M. Revan
    In this paper, we introduce stochastic-restricted Liu predictors which will be defined by combining in a special way the two approaches followed in obtaining the mixed predictors and the Liu predictors in the linear mixed models. Superiorities of the linear combination of the new predictor to the Liu and mixed predictors are done in the sense of mean square error matrix criterion. Finally, numerical examples and a simulation study are done to illustrate the findings. In numerical examples, we took some arbitrary observations from the data as the prior information since we did not have historical data or additional information about the data sets. The results show that this case does the new estimator gain efficiency over the constituent estimators and provide accurate estimation and prediction of the data.
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    Model selection via conditional conceptual predictive statistic for mixed and stochastic restricted ridge estimators in linear mixed models
    (John Wiley and Sons Ltd, 2022) Özkale, M. Revan; Kuran, Özge
    In this article, we characterize the mixed (Formula presented.) ((Formula presented.)) and conditional stochastic restricted ridge (Formula presented.) ((Formula presented.)) 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 (Formula presented.) and (Formula presented.) 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 (Formula presented.) and (Formula presented.) statistics.
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    The ridge prediction error sum of squares statistic in linear mixed models
    (Springer Heidelberg, 2023) Kuran, Özge; Özkale, M. Revan
    In case of multicollinearity, PRESS statistics has been proposed to be used in the selection of the ridge biasing parameter of the ridge estimator which is introduced as an alternative to BLUE. This newly proposed PRESS statistic for the ridge estimator, CPRESSk, depends on the conditional ridge residual and can be computed once at a time by fitting the linear mixed model with all the observations. We also define R-RidPred(2) statistic to evaluate the predictive ability of the ridge fit. Since the PRESS statistic for the BLUE is a special CPRESSk statistic, we indirectly also give closed form solution of the PRESS statistic for the BLUE. Then, we compared the predictive performance of the linear mixed model via the statistics CPRESSk, GCV(k) and C-p by considering a real data analysis and a simulation study where the optimal ridge biaisng parameter is obtained by minimizing each statistic. The study shows that the ridge predictors improve the predictive performance of a linear mixed model over BLUE in the presence of multicollinearity and each statistic gives a different optimum ridge biasing value and they show the best predictive performance at their optimum ridge biasing values. In addition, the simulation study has shown that the intensity of variance and multicollinearity is effective in determining the optimum ridge biasing value and this optimum ridge biasing value is effective on the superiority of the predictive performance of ridge estimator over BLUE.
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    Stochastic restricted Liu predictors in linear mixed models
    (Taylor & Francis, 2021) Kuran, Özge; Özkale, M. Revan
    In this article, we propose the stochastic restricted Liu predictors by augmenting the stochastic restrictions to the linear mixed models. The Liu biasing parameter is selected via generalized cross validation (GCV) criterion. Comparisons between the stochastic restricted Liu estimators and several other estimators, namely the BLUE, the mixed and Liu estimators are made through the mean square error matrix criterion. Finally, a numerical example and a simulation study are done to show the performance of the estimators.