The Pennsylvania State University The Graduate School TWO TOPICS: A JACKKNIFE MAXIMUM LIKELIHOOD APPROACH TO STATISTICAL MODEL SELECTION AND A CONVEX HULL PEELING DEPTH APPROACH TO NONPARAMETRIC MASSIVE MULTIVARIATE DATA ANALYSIS WITH APPLICATIONS

This dissertation presents two topics from opposite disciplines: one is from a parametric realm and the other is based on nonparametric methods. The first topic is a jackknife maximum likelihood approach to statistical model selection and the second one is a convex hull peeling depth approach to nonparametric massive multivariate data analysis. The second topic includes simulations and applications on massive astronomical data. First, we present a model selection criterion, minimizing the Kullback-Leibler distance by using the jackknife method. Various model selection methods have been developed to choose a model of minimum Kullback-Liebler distance to the true model, such as Akaike information criterion (AIC), Bayesian information criterion (BIC), Minimum description length (MDL), and Bootstrap information criterion. Likewise, the jackknife method chooses a model of minimum KullbackLeibler distance through bias reduction. This bias, which is inevitable in model selection problems, arise from estimating the distance between an unknown true model and an estimated model. We show that (a) the jackknife maximum likelihood estimator is consistent to the parameter of interest, (b) the jackknife estimate of the log likelihood is asymptotically unbiased, and (c) the stochastic order of the jackknife log likelihood estimate is O(log log n). Because of these properties, the jackknife information criterion is applicable to problems of choosing a model from non nested candidates especially when the true model is unknown. Compared to popular information criteria which are only applicable to nested models, the jackknife information criterion is more robust in terms of filtering various types of candidate models to choose the best approximating model. However, this robust method has a demerit that the jackknife criterion is unable to discriminate nested models. Next, we explore the convex hull peeling process to develop empirical tools for statistical inferences on multivariate massive data. Convex hull and its peeling