Applications and extensions of three statistical models

Details Applications and extensions of three statistical models Applications and extensions of three statistical models

Nov 2003

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2003phdt.........1e&link_type=abstract

Thesis (PhD). HARVARD UNIVERSITY, Source DAI-B 64/05, p. 2255, Nov 2003, 102 pages.

Statistics

Computation

2

Scientific paper

We present a new method for statistically deconvolving a Point Spread Function from a source image. This method includes some regularization, so it is ideally suited to extended source images. The regularization parameters are fit to the statistical model, so little user intervention is required. The output includes error information for the results, so confidence statements can be constructed from the results. A survey of popular deconvolution techniques is first presented. Then a review of the necessary statistical theory to develop our procedure is presented. Our model is then presented, starting with the likelihood, and then describing the regularization priors. The algorithmic details of fitting the model are then explained, and several examples are presented in detail. We develop Generalized Linear models using Morris' NEF6 distribution, the exponential family generated by the generalized hyperbolic secant distribution. First we review some basic properties of these distributions on (-∞, ∞). Then we specify the model and its features: quadratic variance function with no real roots, and tangent canonical link function. The doubly bounded parameter space for the natural exponential parameter presents some issues peculiar to this family of GLMs. This paper describes and develops a procedure for fitting the regression coefficients and the convolution parameter simultaneously via maximum likelihood. Some simulations from the model are used to assess model fits, and to make frequentist evaluations of our fitting procedure and confidence intervals based on Gaussian approximations. Finally, an analysis of stock return data shows the steps needed to improve model fits, illustrating several key features of this unusual model. We present some basic properties of the Pearson type IV distribution, also known as the Skew-t distribution, as well as algorithms for computation of the distribution function and quantiles. This distribution arises as the conjugate prior for Morris' NEF-GHS distribution, and has many possible applications, heretofore largely unexplored, perhaps due to the lack of computational tools. Here we discuss basic properties of the distribution including its characteristic function and schemes for computation of the distribution, and review the existing literature.

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