Penalty, non-penalty shrinkage and ensemble methods for low and high dimensional data

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This proposal is an initiative to i) provide non-penalty, penalty, and post penalty shrinkage methods and ii) combine models to enable a more robust prediction for low and high dimensional data. These data arise in various fields of sciences including genomics and health sciences. A statistical model is affected by problems associated with improved estimation of parameters when the datasets contain information on many predictors, some of which are not actively associated with the response or not of primary interest to the researcher. The pretest and non-penalty shrinkage methods (P-NPS) consider such inactive predictors as auxiliary information and incorporate them into the estimation procedure to improve the classical estimator of the regression parameters. In low dimensional data, my plan is to work on the P-NPS for joint modeling (JM) of longitudinal and survival data. I then plan to extend the work by simply choosing different longitudinal and survival models in a JM setting. The application of JM has a long history of being used in cancer clinical trials. I also have a plan to develop the P-NPS in generalized partially linear models for independent and longitudinal data as these models allow modeling the predictors nonparametrically along with the linear part. We define high dimensional data (HDD) when the number of predictors is larger than the sample size. The analysis of HDD is an important feature in a host of research areas such as social media, artificial intelligence, MRI and biomedical research. Most of the existing methods for dealing with HDD begin with model selection for further investigation. Penalty methods are unstable unless very stringent conditions are imposed. I consider the post penalty shrinkage strategies for generalized linear models (GLMs) to combat some of the issues inherited from penalty methods. These strategies will improve the prediction errors based on the selected submodels. I also propose to focus on investigating ensemble methods for GLMs to analyze HDD. These are very powerful techniques for improving the accuracy of predictions. I will then calculate aggregated predictions from the models based on the ensemble methods and compare them with the predictions from the model based on the existing penalty methods in the literature. Ensemble methods work well in various applications like image recognition, medicine, network security, and others. This proposal emphasizes that statisticians play an important role in solving problems in the analysis of non-normal (e.g., binary or count), longitudinal and survival data. It will also provide opportunities for training highly qualified personnel at all levels. This training has three components: methodology, computation, and analysis of real life data. Outcomes of this proposal are expected to aid stakeholders in health care services, and in other areas of applications where such data arise, in making proper decisions based on correct statistical inferences.