## Open Access Theses & Dissertations

2013-01-01

#### Degree Name

Master of Science

#### Department

Mathematical Sciences

Ori Rosen

#### Abstract

Density estimation has a long history in statistics. There are two main approaches to density, estimation parametric and nonparametric. The first approach requires specification of a family of densities f and estimation of the unknown parameter $\theta$ using a suitable estimation method, for example, maximum likelihood estimation. This approach may be prone to bias that arises from either estimation of the parameter or from incorrect specification of the probability distribution. The second approach, does not assume a specific parametric family.

In this thesis, we implement three density estimation methods that use Bayesian nonparametric approaches utilizing Markov Chain Monte Carlo methods. Specifically, these methods are the Dirichlet process prior, a method that converts density estimation to a regression problem, and a mixture of normal densities with known means and variances whose mixing weights are logistic with unknown parameters.

We briefly review two traditional methods that are used to obtain density estimates. The first is the density histogram which is one of the simplest and oldest methods. The second method is kernel estimation. In addition, we compare the three nonparametric methods by simulation and use them to estimate the density underlying the 1872 Mexican Hidalgo Stamp. The thesis concludes with a summary.

en

82 pages

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