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8. Model Mtbh (Chao et al., 2001):
z
Estimating Equation: Mtbh(EE)
Equation for N :
t
∑
j =1
Equation for φ :
t
∑
j =1
αˆ j = αˆ j (φ , N ) =
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Mˆ *j (φ u j + m j ) − Nm j
= 0,
2
(1 − Cˆ j −1 )[1 + (φ − 1)Cˆ j −1 − φ (1 + γˆtbh
) αˆ j ]
Mˆ *j (φ u j + m j ) − Nm j
= 0.
2
[1 + (φ − 1)Cˆ j −1 − φ (1 + γˆtbh
)αˆ j ]
2
A j − [ A 2j − 4Nφn j (1 + γˆtbh
)]1/ 2
2
2Nφ (1 + γˆ tbh
)
, where
2
2
A j = A j (φ , N ) = N + φn j (1 + γˆ tbh
) + (φ − 1)[NCˆ j −1 − (1 + γˆ tbh
)m j ].
j −1
2
Cˆ j −1 = Cˆ j −1 (φ ) = 1 − u j /(u j + m j / φ ) , Mˆ *j = M j + [∑k =1 ρˆ k , j −1 ] u j −1 γˆtbh
,
γˆ
2
tbh
⎧ Nˆ bh ∑t [ j ( j − 1)f jt + 2(φ − 1)( j − 1)f jt ]
⎫
⎪
⎪
j =1
= γˆ (φ ) = max ⎨ t
1
,
0
−
⎬, where
t
2
2
[
(
)]
(
)
+
−
+
m
φ
u
m
φ
u
⎪⎩ ∑ j =1 j
⎪⎭
∑ j =1 j
j
j
2
tbh
Nˆ bh is a simple estimator valid under model Mbh. Here, ρ k , j −1 = e k / e j −1
denotes the unknown relative time effect of sample k. A convenient estimator of
ρ k , j −1 = e k / e j −1 is a function of φ and can be presented as
ρˆ k , j −1 = ρˆ k , j −1(φ ) = (u k + mk / φ ) /(u j −1 + m j −1 / φ ) .
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