Browsing by Author "Sarkar, Jyoti"

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    Sample Size Determination for Subsampling in the Analysis of Big Data, Multiplicative Models for Confidence Intervals and Free-Knot Changepoint Models
    (2024-05) Zhang, Sheng; Peng, Hanxiang; Tan, Fei; Sarkar, Jyoti; Boukai, Ben
    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.
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    Sample Size Determination in Multivariate Parameters With Applications to Nonuniform Subsampling in Big Data High Dimensional Linear Regression
    (2021-12) Wang, Yu; Peng, Hanxiang; Li, Fang; Sarkar, Jyoti; Tan, Fei
    Subsampling is an important method in the analysis of Big Data. Subsample size determination (SSSD) plays a crucial part in extracting information from data and in breaking the challenges resulted from huge data sizes. In this thesis, (1) Sample size determination (SSD) is investigated in multivariate parameters, and sample size formulas are obtained for multivariate normal distribution. (2) Sample size formulas are obtained based on concentration inequalities. (3) Improved bounds for McDiarmid’s inequalities are obtained. (4) The obtained results are applied to nonuniform subsampling in Big Data high dimensional linear regression. (5) Numerical studies are conducted. The sample size formula in univariate normal distribution is a melody in elementary statistics. It appears that its generalization to multivariate normal (or more generally multivariate parameters) hasn’t been caught much attention to the best of our knowledge. In this thesis, we introduce a definition for SSD, and obtain explicit formulas for multivariate normal distribution, in gratifying analogy of the sample size formula in univariate normal. Commonly used concentration inequalities provide exponential rates, and sample sizes based on these inequalities are often loose. Talagrand (1995) provided the missing factor to sharpen these inequalities. We obtained the numeric values of the constants in the missing factor and slightly improved his results. Furthermore, we provided the missing factor in McDiarmid’s inequality. These improved bounds are used to give shrunken sample sizes.