RW I:Discussion Papers - Rheinisch-Westfälisches Institut für ...

Therefore, the contribution to the likelihood is
P (∆w o i |F 1) =
1
√
σ 2 w + σ 2 m
φ
(
∆w o i − X iα
√
σ 2 w + σ 2 m
)
(23)
given one measurement error, and
P (∆w o i |F 2) =
(
)
1 ∆wi o √ φ √ − X iα
σ 2 w +2σm 2 σ 2 w +2σm
2
. (24)
given two measurement errors.
We have now all ingredients to build up the likelihood function to be maximized.
Using indicator functions I(·) taking the value of unity if the condition in the brackets is
satisfied and zero otherwise, we get
L(∆wi o|Ω,X i)= P (N)P (M0)P (∆wi o ∈ C|i ∈ N0)I(∆wo i =0)+
P (N)P (M0)P (∆wi o ∈ U|i ∈ N0)I(∆wo i > 0)+
[P (N)P (M1)[P (∆wi o ∈ C|i ∈ N1) + P (∆wo i ∈ U|i ∈ N1)]+
P (R)P (M0)[P (∆wi o ∈ C|i ∈ R0) + P (∆wo i ∈ U|i ∈ R0)]+
P (R)P (M1)[P (∆wi o ∈ C|i ∈ R1) + P (∆wo i ∈ U|i ∈ R1)]+ (25)
P (R)P (M2)[P (∆wi o ∈ C|i ∈ R2) + P (∆wo i ∈ U|i ∈ R2)]+
P (F )P (M0)P (∆wi o |i ∈ F 0)+
P (F )P (M1)P (∆wi o |i ∈ F 1)+
P (F )P (M2)P (∆wi o|i ∈ F 2)] I(∆wo i ≠0),
where Ω = (α, β rn ,β rf ,γ,σ 2 m,σ 2 w,σ 2 r,P m ) is the vector of parameters to be estimated.
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