∗ Signatures are on file in the Graduate School. 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 Kullback-Leibler distance through bias reduction. This bias, which is inevitable in model

We consider the model selection problem in the class of stationary variable length Markov chains (VLMC) on a nite space. The processes in this class are still Markovian of higher order, but with memory of variable length. Various aims in selecting a VLMC can be formalized with dierent non-equivalent risks, such as nal prediction error or expected Kullback-Leibler information. We consider the asymptotic behavior of dierent risk functions and show how they can be generally estimated with the same resampling strategy. Such estimated risks then yield new model selection criteria. In particular, we obtain a data-driven tuning of Rissanen's tree structured context algorithm which is a computationally feasible procedure for selection and estimation of a VLMC. Key words and phrases. Bootstrap, zero-one loss, nal prediction error, nite-memory source, FSMX model, Kullback-Leibler information, L 2 loss, optimal tree pruning, resampling, tree model. Short title: Selecting variable length Mar...

The Akaike information criterion, AIC, is a widely known and extensively used tool for statistical model selection. AIC serves as an asymptotically unbiased estimator of a variant of Kullback's directed divergence between the true model and a fitted approximating model. The directed divergence is an asymmetric measure of separation between two statistical models, meaning that an alternate directed divergence may be obtained by reversing the roles of the two models in the definition of the measure. The sum of the two directed divergences is Kullback's symmetric divergence. Since the symmetric divergence combines the information in two related though distinct measures, it functions as a gauge of model disparity which is arguably more sensitive than either of its individual components. With this motivation, we propose a model selection criterion which serves as an asymptotically unbiased estimator of a variant of the symmetric divergence between the true model and a fitted approximating model. We examine the performance of the criterion relative to other well-known criteria in a simulation study. Keywords: AIC, Akaike information criterion, I-divergence, J-divergence, Kullback-Leibler information, relative entropy. Correspondence: Joseph E. Cavanaugh, Department of Statistics, 222 Math Sciences Bldg., University of Missouri, Columbia, MO 65211. y This research was supported by NSF grant DMS--9704436. 1.

We propose a new GARCH model with tree-structured multiple thresholds for volatility estimation in nancial time series. The approach relies on the idea of a binary tree where every terminal node parameterizes a (local) GARCH model for a partition cell of the predictor space. Fitting of such trees is constructed within the likelihood framework for non-Gaussian observations: it is very dierent from the well-known CART procedure for regression which is based on residual sum of squares. Our strategy includes the classical GARCH model as a special case and allows to increase model-complexity in a systematic and exible way. We derive a consistency result and conclude with simulations and real data analysis that the new method has better predictive potential in comparison with other approaches. Keywords: Conditional variance; Financial time series; GARCH model; Maximum likelihood; Threshold model; Tree model; Volatility. 1 Introduction We propose a new method for estimating volatility in...

We propose a model selection criterion for regression applications where resistant fitting is appropriate. Our criterion gauges the adequacy of a fitted model based on the median squared error of prediction. The criterion is easily computed using the bootstrap "bumping" algorithm of Tibshirani and Knight (1999), which provides a convenient method for obtaining least median of squares model parameter estimates. We present an example to illustrate the merit of the criterion in instances where the underlying data set contains influential values. Additionally, we present and discuss the results of a simulation study which illustrates the effectiveness of the criterion under a wide range of error distributions.