Explore the vast range of titles available.

We consider nonparametric estimation of a mixed discrete-continuous distribution under anisotropic smoothness conditions and a possibly increasing number of support points for the discrete part of the distribution. For these settings, we derive lower bounds on the estimation rates. Next, we consider a nonparametric mixture of normals model that uses continuous latent variables for the discrete part of the observations. We show that the posterior in this model contracts at rates that are equal to the derived lower bounds up to a log factor. Thus, Bayesian mixture of normals models can be used for (up to a log factor) optimal adaptive estimation of mixed discrete-continuous distributions. The proposed model demonstrates excellent performance in simulations mimicking the first stage in the estimation of structural discrete choice models.

This paper considers Bayesian nonparametric estimation of conditional densities by countable mixtures of location-scale densities with covariate dependent mixing probabilities. The mixing probabilities are modeled in two ways. First, we consider finite covariate dependent mixture models, in which the mixing probabilities are proportional to a product of a constant and a kernel and a prior on the number of mixture components is specified. Second, we consider kernel stick-breaking processes for modeling the mixing probabilities. We show that the posterior in these two models is weakly and strongly consistent for a large class of data-generating processes. A simulation study conducted in the paper demonstrates that the models can perform well in small samples.

The dissertation consists of three distinct essays on Bayesian nonparametric methods for conditional density estimation. In the first chapter, I propose a novel way to consistently estimate both regression coefficients and conditional error distributions in a linear regression model with a conditional moment restriction. In the second chapter, posterior consistency theorems for conditional density estimation by covariate dependent mixtures of general location-scale densities are provided and proved. Given practical popularity of such approaches this chapter provides theoretical justification for practical applications. In the third chapter, a Bayesian approach for flexible modeling of joint continuous and discrete distributions is proposed and the performance of the proposed method is compared to existing alternative estimation methods. In the first chapter I propose a new way of modeling flexible error distributions in a linear regression model. Such an approach is desired as consistent estimation of both linear coefficients and error distributions is obtained. Conditional prediction intervals can be obtained from such a model given consistent estimation of error distributions. Furthermore, one can hope for a more efficient estimation of regression coefficients given flexible modeling of error distributions. Monte Carlo experiments show that the proposed estimator outperforms alternative Bayesian estimation methods for regression coefficients. The second chapter, written jointly with Andriy Norets, is a theoretical piece on Bayesian nonparametric estimation of conditional densities using flexible mixtures of location scale densities. Predictor dependent mixing probabilities are modeled in two alternative ways. First alternative is to use finite covariate dependent mixtures and the second alternative is to use kernel stick breaking process to model covariate dependent mixing probabilities. The main theoretical contribution of this chapter is to show that the posterior in these two models is weakly and strongly consistent for a large class of data generating processes. The results in this chapter fill the niche in the literature as it provides theoretical justification for the use of flexible conditional density models encountered in the literature. The third chapter, written jointly with Andriy Norets, proposes to model joint distributions of both continuous and discrete variables as finite mixtures of Gaussian distributions. Conditional distributions of interest can then be extracted from the estimated joint distribution. Theoretical justification of such an approach is provided as it is shown that the proposed Bayesian estimator of the density is consistent in total variation distance for a large class of data generating processes. The method can be used as a heteroscedasticity and non-linearity robust regression model with both discrete and continuous variables. The proposed method is compared with alternative frequentist and Bayesian estimation methods in a small sample of separate data studies and is shown to perform at least as well or better than the alternatives.

We consider nonparametric estimation of a mixed discrete-continuous distribution under anisotropic smoothness conditions and possibly increasing number of support points for the discrete part of the distribution. For these settings, we derive lower bounds on the estimation rates in the total variation distance. Next, we consider a nonparametric mixture of normals model that uses continuous latent variables for the discrete part of the observations. We show that the posterior in this model contracts at rates that are equal to the derived lower bounds up to a log factor. Thus, Bayesian mixture of normals models can be used for optimal adaptive estimation of mixed discrete-continuous distributions.

We consider nonparametric estimation of a mixed discrete-continuous distribution under anisotropic smoothness conditions and possibly increasing number of support points for the discrete part of the distribution. For these settings, we derive lower bounds on the estimation rates in the total variation distance. Next, we consider a nonparametric mixture of normals model that uses continuous latent variables for the discrete part of the observations. We show that the posterior in this model contracts at rates that are equal to the derived lower bounds up to a log factor. Thus, Bayesian mixture of normals models can be used for optimal adaptive estimation of mixed discrete-continuous distributions.

We consider Bayesian estimation of restricted conditional moment models with linear regression as a particular example. The standard practice in the Bayesian literature for semiparametric models is to use flexible families of distributions for the errors and assume that the errors are independent from covariates. However, a model with flexible covariate dependent error distributions should be preferred for the following reasons: consistent estimation of the parameters of interest even if errors and covariates are dependent; possibly superior prediction intervals and more efficient estimation of the parameters under heteroscedasticity. To address these issues, we develop a Bayesian semiparametric model with flexible predictor dependent error densities and with mean restricted by a conditional moment condition. Sufficient conditions to achieve posterior consistency of the regression parameters and conditional error densities are provided. In experiments, the proposed method compares favorably with classical and alternative Bayesian estimation methods for the estimation of the regression coefficients.

This paper considers Bayesian nonparametric estimation of conditional densities by countable mixtures of location-scale densities with covariate dependent mixing probabilities. The mixing probabilities are modeled in two ways. First, we consider finite covariate dependent mixture models, in which the mixing probabilities are proportional to a product of a constant and a kernel and a prior on the number of mixture components is specified. Second, we consider kernel stick-breaking processes for modeling the mixing probabilities. We show that the posterior in these two models is weakly and strongly consistent for a large class of data generating processes.