
Mathematical and Statistical Aspects of Inverse Problems 63
for θ , i.e., solving (3.38). However, we note the GLS relations hip (3.38) does
not follow fr om minimizing the weighted least squares with weights chosen as
w
j
= f
−2γ
(t
j
; θ) (see p. 89 of [35]).
Another motivation for the GLS estimating equations (3.37) and (3.38) can
be found in [18]. In that text, Carroll and Ruppert claim that if the data are
distributed according to the gamma distribution, then the maximum-likelihood
estimate for θ (a standard approach when one assumes that the distribution
for the measurement error is co mpletely known—to be discussed later) is the
solution to
N
X
j=1
f
−