
152 Modeling and Inverse Problems in the Presence of Uncertainty
cost function
J
N
WLS
(q; Y) =
1
N
N
X
k=1
Y
k
− f(t
k
; q)
w(t
k
)
2
,
where Y = (Y
1
, Y
2
, . . . , Y
N
)
T
. In other words, the weighted least squares
estimator q
N
WLS
is obtained by solving
q
N
WLS
= arg min
q∈Ω
q
J
N
WLS
(q; Y),
and its corresponding realization is obtained by solving
ˆ
q
N
WLS
= arg min
q∈Ω
q
J
N
WLS
(q; y),
where y is a realization of Y. Similarly, the restricted weighted least squares
estimator q
N,H
WLS
(or estimate
ˆ
q
N,H
WLS
) can be obtained by minimizing J
N
WLS
(q; Y)
(or J
N
WLS
(q; y)) over Ω
H
q
, as we did for the ordinary least s quares case.
Using assumptions (A1
′
a), (A1
′
b), (A2)–(A6) and (A7
′
) given in Section ...