Nonnegatively Constrained Confidence Interval Estimation for Ill-Posed Problems

Bert W. Rust ()
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Bert W. Rust: National Institute of Standards and Technology, Center for Computing and Applied Mathematics

A chapter in Computing Science and Statistics, 1992, pp 63-72 from Springer

Abstract: Abstract Let the m x n linear model $$\hat{y}=K\bar{x}+\hat{\epsilon}$$ be obtained by discretizing a system of first-kind integral equations $$\hat{y}_i=\int_{a}^{b}K_i(\xi)x(\xi)d\xi+\hat{\epsilon}_i,i+1,...m,$$ with known functions K i(ξ), unknown function x(ξ) and an m-vector ŷ of measurements corrupted by random errors є drawn from a distribution with zero mean vector and known variance matrix. Consider the problem of estimating linear functions of the form $$\phi=w^T\bar{x}\approx \int_{a}^{b}w(\xi)x(\xi)d\xi,$$ where w(ξ) is an averaging function designed to elicit some desired information about x(ξ). For such problems, the least squares solution is a highly unstable function of the measurements, and the classical confidence intervals are too wide to be useful. The solution can often be stabilized by imposing physically motivated, a priori non-negativity constraints on x. This paper will show how to extend the classical confidence interval estimation technique to accommodate these nonnegativity constraints and how to compute the resulting much-improved confidence intervals.

Keywords: Duality Theorem; Ridge Regression; Optimal Interval; Confidence Interval Estimation; Nonnegativity Constraint (search for similar items in EconPapers)
Date: 1992
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4612-2856-1_8

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DOI: 10.1007/978-1-4612-2856-1_8

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