The problem of interest is to predict y epsilon¸ {0, 1}, based on some characteristics x. The approach of the Gibbs Posterior distribution, which is constructed from an empirical classification risk and aims to minimizing a risk function without modeling the data probabilistically, is a new direction in dealing with the problem of classification and prediction, even in presence of high dimensionality of x.;In this dissertation, we study panel data binary response model, which includes n mutually independent sequences of some dependent process, such as strong mixing or even more general than strong mixing, with time period T and the response variable is of binary choice, either 0 or 1. Firstly, we extend two inequalities: Bosq inequality (1993) and Triplex inequality (2009), to this data structure to construct an upper probability bound for the pointwise deviation and the uniform deviation between the empirical risk and its expectation.;Then the Gibbs Posterior is applied in panel data binary response model to generate some parameters b to construct some linear classifiers so that we can make classification and prediction on the response variable y. The asymptotic properties of the risk function for the proposed b in two kinds of scenarios, either under T → infinity or under n → infinity but with not large T, are discussed. The near optimal performance of the risk of the Gibbs Posterior has been achieved only when T → infinity, while a new marginalized risk has been proposed in the other situation and the relation between the two risk measure is demonstrated as well.;We also study the convergence rate of the risk minimization with variable selection in high dimension either from frequentist and Bayesian approaches. The Bayesian treatment is the Gibbs Posterior, constructed directly from an empirical classification risk, which has a robust property rather than classical Bayesian Posterior. The risk function converges to the optimal risk at near parametric rate, only dependent on the sample size, despite the high dimensionality.;A simulation study has been conducted to study the classification and prediction performance of the Gibbs Posterior distribution in the panel data binary response model with a random individual effect. Comparison with the classical Bayesian likelihood method confirms that the Gibbs Posterior performs as well as the Bayesian method when the generating process is correctly modeled. While the data is generating from the model, which is misspecified, the classification performance of the Gibbs Posterior, which doesn't depend on the model assumption, is much better. On the other hand, we find that increasing T helps to reduce the prediction error more effectively compared to increasing n. We also illustrate the method with Gibbs Posterior in a real data application on the brand choice of yogurt purchases.
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