Confidence Intervals for Discrete Approximations to Ill-Posed Problems

, and . Journal of Computational and Graphical Statistics 3 (1): pp. 67-96 (1994)


We consider the linear model Y = Xβ + ε that is obtained by discretizing a system of first-kind integral equations describing a set of physical measurements. The n vector β represents the desired quantities, the m × n matrix X represents the instrument response functions, and the m vector Y contains the measurements actually obtained. These measurements are corrupted by random measuring errors ε drawn from a distribution with zero mean vector and known variance matrix. Solution of first-kind integral equations is an ill-posed problem, so the least squares solution for the above model is a highly unstable function of the measurements, and the classical confidence intervals for the solution are too wide to be useful. The solution can often be stabilized by imposing physically motivated nonnegativity constraints. In a previous article (O'Leary and Rust 1986) we developed a method for computing sets of nonnegatively constrained simultaneous confidence intervals. In this article we briefly review the simultaneous intervals and then show how to compute nonnegativity constrained one-at-a-time confidence intervals. The technique gives valid confidence intervals even for problems with m < n. We demonstrate the methods using both an overdetermined and an underdetermined problem obtained by discretizing an equation of Phillips (Phillips 1962).

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