Page 124 - Contributed Paper Session (CPS) - Volume 3

CPS1965 Chin Tsung R. et al.
() = + () + () + ∑ () (5)
00 01 01 02 02 =0

where () = ( − ) ; i = 1,2 and () = ( − ) , = 0,1, . . . , (6)
3

+
0
0 +

and ( − ) is ( − ) for ≥ and 0 for < . Hence, the cubic

+
spline can be written as a linear combination of some non-linear functions and
that () has + 4 parameters and is cubic below the first knot and
1
above the last knot, . Additionally, Zhao (2012) proposed that further

quadratic and linear restrictions could be applied to points above the last knot.
For quadratic restriction, we set = 0 and for the linear restriction, we set
= = 0 where and are as per the polynomial ()

Under quadratic restriction above the last knot, we have:

Under the linear restriction above the last knot we have (Zhao, 2012):
() = ( − ) ;
0 +
01

The number of parameters corresponding to the quadratic and linear
restrictions is + 3 and + 2 respectively. Furthermore, we can also apply
linear or quadratic restrictions below the first knot. For the quadratic
restriction, we set 0 () = 0 and for the linear restriction we set
() = () = 0.
0
02
To obtain the modified LC model, we assume that (, ) in (2) has a
binomial distribution than we have (, )~((, ), (, )) where (, )
is the mortality rate of an individual of age group in year . Hence the
proposed model is of the form:

where (), () and () are unknown.
We follow the methodology proposed by Zhao (2012) that is to
approximate these functions as linear combination of a cubic spline with
possible restrictions on the left and right tails. Additionally, other additive
functions, () such as 1/, () 1/√ is added to the estimated
function. According to Zhao (2012), for short based period it is sufficient
assume that () = . We let a fixed set of knots and write
() = () and () = () and hence the model can be expressed

as:

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