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 likelihoodapproach to statistical model selection and the second one is a convexhull peeling depth approach to nonparametric massive multivariatedata analysis. The second topic includes simulations and applicationson massive astronomical data.First, we present a model selection criterion, minimizing the Kullback-Leiblerdistance by using the jackknife method.Various model selection methods have been developed to choosea 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 Kullback-Leibler distancethrough bias reduction. This bias, which is inevitable in model selectionproblems, arise from estimating the distance between an unknown true modeland an estimated model.We show that (a) the jackknife maximum likelihood estimator isconsistent to the parameter of interest, (b) the jackknife estimate of thelog 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 amodel from non nested candidates especially when the true model is unknown.Compared to popular information criteria which areonly applicable to nested models, the jackknifeinformation criterion is more robust in terms of filtering various types ofcandidate models to choose the best approximating model. However,this robust method has a demerit that the jackknife criterion is unableto discriminate nested models.Next, we explore the convex hull peeling process to developempirical tools for statistical inferences on multivariate massive data.Convex hull and its peeling process has intuitive appeals for robustlocation estimation. We define the convex hull peeling depth,which enables to order multivariate data.This ordering processbased on data depth provides ways to obtain multivariate quantiles includingmedian. Based on the generalized quantile process, we definea convex hull peeling central region, a convex hull level set, anda volume functional, which lead us to invent one dimensional mappings,describing shapes of multivariate distributions along data depth.We define empiricalskewness and kurtosis measures based on the convex hull peeling process.In addition to these empirical descriptive statistics, we finda few methodologies to find multivariate outliers in massive data sets.Those outlier detection algorithmsare (1) estimating multivariate quantiles up to the level $alpha$,(2) detecting changes in a measure sequence of convex hull level sets,and (3) constructing a balloon to exclude outliers. The convex hull peelingdepth is a robust estimator so thatthe existence of outliers do not affect properties of inner convex hulllevel sets. Overall, we show all these good characteristics of the convex hullpeeling process through bivariate synthetic data sets to illustratethe procedures. We prove these empirical procedures are applicable toreal massive data set by employing Quasars and galaxies from SloanDigital Sky Survey. Interesting scientific results from the convexhull peeling multivariate data analysis are also provided.
