Computing Science and Statistics pp 63-72 | Cite as

# Nonnegatively Constrained Confidence Interval Estimation for Ill-Posed Problems

Conference paper

## Abstract

Let the be obtained by discretizing a system of first-kind integral equations with known functions where

*m*x*n*linear model$$\hat{y}=K\bar{x}+\hat{\epsilon}$$

$$\hat{y}_i=\int_{a}^{b}K_i(\xi)x(\xi)d\xi+\hat{\epsilon}_i,i+1,...m,$$

*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$$\phi=w^T\bar{x}\approx \int_{a}^{b}w(\xi)x(\xi)d\xi,$$

*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

Assure Convolution Lution Estima Lawson## Preview

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### References

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© Springer-Verlag New York, Inc. 1992