A cohort-based extension to the Lee

Report
死亡率模型
Lee-Carter的延伸
徐健勳
Lee-Carter Model
• Lee-Carter Model明顯易懂
• 各國效果不同,例如:英格蘭與威爾斯男
性死亡資料
• Renshaw and Haberman 修正模型,加入世
代效應
cohort-effect
• 世代效應(cohort-effect)
– 意指橫跨不同出生年代所產生的變化,即不同
世代的人在某些特質會有差異存在,亦反應累
積一生所暴露的風險
二維資料形勢
Lee-Carter Model(age-period)
 xt  exp(  x   x  t )
or log(  xt )   x   x  t
tn
subject to
åk = 0
t
;
t1
åb
x
x
 xt : x 年齡在年代 t 的中央死亡率
 x : x 年齡的死亡率平均曲線
 x : x 年齡相對死亡率的變化
 t : 年代 t 的死亡率變化量
速度
=1
Mortality Reduction Factor
• Mortality Reduction Factor F
F(x, t) = exp(b t + b k )
(0)
x
z
(1)
x
t
where z = t - x Î [t1 - xk , tn - x1 ]
b , b
(0)
x
(1)
:為x年齡,相對死亡率的變化速度
x
tz
:為在世代z之下,死亡率的變化量
Lee-Carter Model
m xt = exp(a x + log F(x, t))
or m xt = exp(a x )× F(x, t)
age - period
LC : b
(0)
x
= 0 (b
(1)
x
º bx )
(0)
x
º bx )
age - cohort
AC : b
(1)
x
= 0 (b
參數估計
•
•
•
•
LC Model
APC Model
AC Model
Method A : Wilmoth and Brouhns
– SVD(奇異值分解)
• Method B : James and Segal
Singular Value Decomposition
A = USV
T
where A m ´ n matrix
U m ´ m unitary matrix
S m ´ n diagonal matrix
V n ´ n matrix 模型分配
• 死亡率服從Gaussian(常態)分配
– 原始提出的模型即是Gaussian
• 死亡人口數服從Poisson分配
– Brouhns (2002) 提出死亡人口數服從Poisson分
配的假設
LC Model
• Method A (Gaussian)
– 先用SVD求
bˆx , kˆt 再用MLE求 aˆ x並算 yˆxt
1. update aˆ x ; compute yˆxt
tn
2. update kˆt , adjust such that åk t = 0, compute yˆxt
t1
3. update bˆ, ; compute yˆxt
4. compute D(yxt , yˆxt )
重複上述 1. 2. 3. 4. 的步驟,直到 D(yxt , yˆxt ) 收斂
where yxt = log mxt , yˆxt = aˆ x + bˆxkˆt
ì1, ext > 0
D(yxt , yˆxt ) = åw xt (yxt - yˆxt ) with w xt = í
î0, ext = 0
x,t
2
LC Model
• Method A (Poisson)
– 先用SVD求
bˆx , kˆt 再用MLE求 aˆ x並算 yˆxt
1. update aˆ x ; compute yˆxt
tn
2. update kˆt , adjust such that åk t = 0, compute yˆxt
t1
3. update bˆ, ; compute yˆxt
4. compute D(yxt , yˆxt )
重複上述 1. 2. 3. 4. 的步驟,直到 D(yxt , yˆxt ) 收斂
where
y xt  d xt , yˆ  dˆ xt  e xt exp( ˆ x  ˆ x ˆ t )
 y xt
D ( y xt , yˆ xt )    xt { y xt log 
x ,t
 yˆ xt

1, e xt  0
  ( y xt  yˆ xt )} with  xt  

 0 , e xt  0

LC Model
• Method B (Gaussian)
Set starting values ˆ x
1 . given ˆ x , update ˆ x , ˆ t
2 . given ˆ t , update ˆ x , ˆ x
3 . compute D ( y xt , yˆ xt )
repeating
the updating
cycle ; stop when D ( y xt , yˆ xt ) converges
where yxt = log mxt , yˆxt = aˆ x + bˆxkˆt
ì1, ext > 0
D(yxt , yˆxt ) = åw xt (yxt - yˆxt ) with w xt = í
î0, ext = 0
x,t
2
LC Model
• Method B (Poisson)
Set starting values ˆ x
1 . given ˆ x , update ˆ x , ˆ t
2 . given ˆ t , update ˆ x , ˆ x
3 . compute D ( y xt , yˆ xt )
repeating
where
the updating
cycle ; stop when D ( y xt , yˆ xt ) converges
y xt  d xt , yˆ  dˆ xt  e xt exp( ˆ x  ˆ x ˆ t )
 y xt
D ( y xt , yˆ xt )    xt { y xt log 
x ,t
 yˆ xt

1, e xt  0
  ( y xt  yˆ xt )} with  xt  

 0 , e xt  0

LC Parameter updating relationships
APC Model
• Three factors are constrained by the
relationship
Cohort=Period-Age
• Method A
– 先用SVD方法求初估計值,再用MLE求 ˆ x
• Method B
–
x
(0)
 x
(1 )
 1/ k
F ( x , t )  exp( 
z  
(0)
x
(1 )
x
t)
APC Parameter updating relationships
AC Model
• Method A
– 先用SVD方法求初估計值,再用MLE求 ˆ x
• Method B
–
ˆ x  1
F ( x , t )  exp(  x z )
AC Parameter updating relationships
Mortality Rate Projection
mx,tn +s = mˆ x,tn F(x, tn + s), s > 0
where F(x, tn + s) = exp(bˆx(0) (t tn -x+s - tˆtn -x ) + bˆx(1) (k tn +s - kˆtn )), s > 0
ìïtˆtn -x+s , 0 < s £ x - x1
t tn -x+s = í
ïît tn -x+s , s > x - x1

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