593
Appendix
B
Discrepancy Functions
Amos minimizes discrepancy functions (Browne, 1982, 1984) of the form:
(D1)
Different discrepancy functions are obtained by changing the way f is defined. If
means and intercepts are unconstrained and do not appear as explicit model
parameters, and will be omitted and f will be written .
The discrepancy functions and are obtained by taking f to be:
Except for an additive constant that depends only on the sample size, is –2 times
the Kullback-Leibler information quantity (Kullback and Leibler, 1951). Strictly
speaking, and do not qualify as discrepancy functions according to
Browne’s definition because .
For maximum likelihood estimation (ML), , and are obtained by taking f to be:
()
[]
() () () () ()
()
[]
(
)
a
S,x;,
a,, 1
α
μ
α
FrN
N
fN
rNC
G
g
ggggg
−=
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎝
⎛Σ
−= ∑
=
xg()
μ
g()
fΣg() Sg()
;()
CKL
FKL
() () () ()
()
() () ()
()
() ()
()
() () ()
()
gggggggggggg
KL
f
μμμ
−Σ
′
−+Σ+Σ=Σ −− xxSS,x;, 11
trlog
fKL
CKL
FKL
CML
FML