Iterative algorithms for integral equations of the first kind with applications to statistics

This dissertation explores the use of a preconditioned Richardson iterative algorithm for the solution of linear and nonlinear ill-posed integral equations of the first kind. The discussion consists of three parts, which can be roughly categorized as: numerical analysis, applications to statistical methodology, and an application to an inverse problem. In the first part, singular matrix equations that result from discretizing ill-posed integral equations of the first kind are considered. Sufficient conditions for the convergence of Richardson's algorithm to a solution are established, and necessary and sufficient conditions are proved for special cases. The inconsistent case is also discussed. A preconditioning for equations with positive kernels leads to the Conditional Expectation algorithm, which is discussed in detail. A notion of 'iterative regularization' is introduced and related to the more usual penalized least squares approach to regularization. In the second part two problems in statistical methodology are considered which involve the solution of nonlinear integral equations of the first kind. The first is the Behrens-Fisher problem. Trickett and Welch (Biometrika, 1954) determined a very nearly similar test for the Behrens-Fisher problem having reasonable power by numerically 'solving' a nonlinear integral equation. The Trickett-Welch method is examined, and a version of the Conditional Expectation algorithm for nonlinear equations is applied to the Behrens-Fisher problem. The second methodological problem that is considered is that a $\beta$-content tolerance limits involving data from a one-way balanced random effects model. The Conditional Expectation algorithm is used to approximately solve a nonlinear equation of the first kind numerically, and to thereby derive a new tolerance limit procedure which is shown to be a substantial improvement over the only other method in the statistics literature. In the third part an inverse problem is discussed in which the right hand side of the integral equation is estimated. In this example, the objective is to infer the probability density of the radii of random spheres in a two-phase medium from radii of circles in cross-sectional slices of this medium. The Conditional Expectation algorithm leads to an effective technique for solving this problem.

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