On a Statistical Model of Strand and Westwater for the Numerical Solution of a Fredholm Integral Equation of the First Kind

ABSTRACT

A statistical model is presented which is useful in the solution of a Fredholm integral equation of the first kind and equivalent to one proposed by Strand and Westwater. The model and the related problem presented here are familiar to statisticians from the study of regression analysis and are essentially, “(GLM): Find the best linear unbiased estimate of &bgr; given the observation y which satisfies y = H&bgr; + e, e distributed as N (0, &Ggr;).” Here y, &bgr; c are vectors, H is an (m + n) × k matrix, and &Ggr; is a certain (m + n) × (m + n) positive definite, known matrix. The main content of the paper is that (GLM) provides an equivalent way of considering the problem of Strand and Westwater and is to be preferred by virtue of the rich store of results available for the study of (GLM) and its intrinsic geometric nature.

REFERENCES

• 1: AITKEN, A.C. On least squares and linear combinations of observations. Proc. Roy. Soc. Edinburgh 55 (1934), 42-48.
• 2: KaVSKAL, W. When are Gauss-Markov and least squares estimators identical? A coordinatefree approach. Ann. Math. Slat. 39 (1968), 70-75.
• 3: Otto Neall Strand , Ed. R. Westwater, Statistical Estimation of the Numerical Solution of a Fredholm Integral Equation of the First Kind, Journal of the ACM (JACM), v.15 n.1, p.100-114, Jan. 1968 [doi>10.1145/321439.321445]
• 4: STRAND, O. N., AND WESTWATER, E.R. Minimum-RMS estimation of the numerical solution of a Fredholm integral equation of the first kind. SIAM J. Numer. Anal. 5 (1968), 287-295.
• 5: RAo, C. R. Linear Statistical Inference and Its Applications. Wiley, New York, 1965, pp. 192-193.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.9 Integral Equations
          Subjects: Fredholm equations

Additional Classification:
  G. Mathematics of Computing

General Terms:
Performance, Theory, Verification