Filter
Top Products
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
−=
∑
=
x
g()
μ
g()
f Σ
g()
S
g()
;()
C
KL
F
KL
() () () ()
()
() () ()
()
() ()
()
() () ()
()
gggggggggggg
KL
f
μμμ
−Σ
′
−+Σ+Σ=Σ
−−
xxSS,x;,
11
trlog
f
KL
C
KL
F
KL
F
KL
aa,()0≠
C
ML
F
ML