| The model averaging method plays a very important role in dealing with model uncertainty.Its main idea is to consider the weighted average of multiple candidate models for statistical prediction and inference,which can solve the uncertainty problem caused by using only one model to a certain extent.Among them,the methods and theoretical research on frequency school model averaging have achieved rapid development in the past decade.A large number of model averaging methods such as Smoothed AIC and Smoothed BIC(SAIC and SBIC),Mallows model averaging(MMA),and the cross validation model averaging(JMA)have been proposed,and a series of theoretical results such as asymptotic optimality have also been established.However,there are many aspects that require further research.Firstly,most existing model averaging methods assume that there is no missing data,and in practical applications,missing data is very common.The first research content of this article is a model averaging method for fragmentary data that is prevalent in many fields such as economics,medicine,and social sciences.The main challenge for averaging fragmentary data models is that the sample used for fitting candidate models is inconsistent with the weight selection.If MMA is directly applied,it will introduce bias.Therefore,this paper proposes a linear model average estimation that is effective mallows model averaging(EMMA)based on fragmentary data,and considers using the”effective model size” to correct the deviation of the weight selection criteria.We prove the asymptotic optimality of EMMA and verify its effectiveness through a large number of numerical simulations and actual data analysis.Secondly,the current weight selection methods for generalized linear model averaging are mainly KL loss with penalty and cross validation,but they are not unbiased estimates of Kullback-Leibler(KL)divergence Even when the data are completely observed,existing methods cannot guarantee unbiased estimates of KL divergence.The second research content of this article is that we propose a novel generalized linear model averaging method: effective model averaging(EMA).Starting from finding an asymptotically unbiased estimate of KL divergence,we derive an ”effective model size” that can characterize the degree of model misclassification similar to the EMMA methodology.Theoretically,we have discussed two situations:(1)all candidate models are misclassified,and(2)the correct model is included.In case(1),we prove the asymptotic optimality when both the number of candidate models and the number of covariates are allowed to diverge.In case 2,we prove that the weight of the wrong candidate model converges to 0,and the weighted regression coefficient estimators obtained are consistent.A large number of numerical simulation results and actual data analysis confirm our theory.Finally,in theory,model averaging research mainly focuses on the asymptotic optimality and asymptotic distribution of model average estimators.However,even the most basic least square model averaging,there are many theories that have not yet been resolved.The third content of this article is to further improve some theoretical results of the least squares model average estimation under nested models.Two cases are also discussed:(1)all candidate models are under fitted,and(2)the correct model is included in the candidate model.Under certain conditions,we prove that in Case(1),the weights will be concentrated on the largest candidate model,so the estimation of the regression coefficients has asymptotic normality.In Case 2,when the penalty terms diverge by a certain order,the resulting weights will all be concentrated on the true model,so that the average estimator of the model is asymptotically optimal.When the penalty term is constant,MMA does not have asymptotic optimality.Overall,this study has effectively innovated and complemented the field of model averaging from both methodological and theoretical perspectives. |
