Sample Size Determination for Subsampling in the Analysis of Big Data, Multiplicative Models for Confidence Intervals and Free-Knot Changepoint Models

Abstract

The dissertation consists of three parts. Motivated by subsampling in the analysis of Big Data and by data-splitting in machine learning, sample size determination for multidimensional parameters is presented in the first part. In the second part, we propose a novel approach to the construction of confidence intervals based on improved concentration inequalities. We provide the missing factor for the tail probability of a random variable which generalizes Talagrand’s (1995) result of the missing factor in Hoeffding’s inequalities. We give the procedure for constructing confidence intervals and illustrate it with simulations. In the third part, we study irregular change-point models using free-knot splines. The consistency and asymptotic normality of the least squares estimators are proved for the irregular models in which the linear spline is not differentiable. Simulations are carried out to explore the numerical properties of the proposed models. The results are used to analyze the US Covid-19 data.