Page 123 - Contributed Paper Session (CPS) - Volume 3

CPS1965 Chin Tsung R. et al.
(1992) estimated the parameters of (1) using a two-stage method with
̂
̂
restrictions ∑ () = 0 and ∑ () = 1 to ensure a unique solution for the

system of equations of the model. The singular value decomposition (SVD)
approach was applied to the matrix of centered age profiles ( ) − ̂(),
,
which allows a first estimation of parameters () and (). A second step
̂
̂
based on the refitting of () on the number of deaths is usually suggested to
̂
assure a better convergence between the estimated and observed deaths. The
aim is to find the () such that:
̂

̂
̂
∑ (, ) = ∑ (, ) (̂() + () ()), (2)
= 1 = 1

where (, ) is the number of deaths of age in time, and (, ) is the
exposure to risk of age in time . For this method, () and () are fixed.
The adjusted () is then extrapolated using autoregressive integrated
̂
moving average (ARIMA) models. The LC model uses a random walk with drift
model, which can be expressed as:

() = ( − 1) + + (), (3)

where is known as the drift parameter and measures the average annual
change in the series, and () is an uncorrelated error.
Zhao (2012) introduced a modified approach to LC model for analysing
short base period age-specific data using linearized cubic splines and other
additive functions to estimate the unknown functions, () and (). Zhao
observed that predicted mortality curves were not smooth and fluctuate
significantly over the age range. In this study, we applied the modified model
by Zhao (2012) for the Malaysian mortality data (Department of Statistics
Malaysia) for the period 1991-2015 for males and females.

2. Methodology
According to Zhao (2012), a piecewise cubic spline function with knots
= { , , … , } can be expressed as

2
1

() = () + () +∙∙∙ + () (4)
0 0 ≤≤ 1 1 1 ≤≤ 2 ≤≤ +1

2
3
where () = ( − ) + ( − ) + ( − ) + , = 0,1, … and
≤≤ +1
is the indicator function with value 1 on the interval [, +1] and 0 elsewhere.
It is assumed that () and its first and second derivatives are continuous

and therefore we have −1() = (), −1() = (), −1() = () for
′′
′′
′
′
= 1,2,… , . Hence () can be expressed as:

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