Featured Article Natthasurang Yasungnoen* and Pairote

Featured Article

Natthasurang Yasungnoen* and Pairote Sattayatham

Forecasting Thai Mortality by Usingthe Lee-Carter Model

DOI 10.1515/apjri-2014-0042

Abstract: In this paper, we model the mortality rate in Thailand by using theLee-Carter model. Three classical methods, i.e. Singular Value Decomposition(SVD), Weighted Least Square (WLS), and Maximum Likelihood Estimation(MLE) are used to estimate the parameters of the Lee-Carter model. Withthese methods, we investigate the goodness of fit for the mortality rate span-ning the period 2003 to 2012. The fitted models are compared. The autoregres-sive moving average (ARIMA) is used to forecast the general index andmortality rate the time period from 2013 to 2022. As a result, we also forecastThai life expectancy at birth.

Keywords: Mortality rate, Lee-Carter model, Life expectancy at birth, SingularValue Decomposition, Maximum Likelihood Estimation

1 Introduction

At present, technological advances in medicine and public health have causeddecline in the mortality rate. Forecasting the mortality rate is important instudying the trends of population; the death rate can be as important as thebirth rate. In the insurance business, actuaries need to be able to create anaccurate mortality table to determine insurance premiums and reserving. Inaddition, the mortality rate is also useful in other disciplines such as medicineand public health using life tables to find the average lifespan and longevity ofthe population.

*Corresponding author: Natthasurang Yasungnoen, Institute of Science – School ofMathematics, Suranaree University of Technology, Nakhorn Ratchasima, Muang NakhonRatchasima, Thailand, E-mail: [email protected] Sattayatham, Institute of Science – School of Mathematics, Suranaree University ofTechnology, Nakhorn Ratchasima, Muang Nakhon Ratchasima, Thailand,E-mail: [email protected]

APJRI 2015; aop

Authenticated | [email protected] author's copyDownload Date | 9/8/15 5:46 AM

In life insurance, companies have dealt with two fundamental types of riskwhen issuing contracts: financial risk and demographic risk. Uncertainty aboutfuture mortality is a problem for financial businesses involved in selling annu-ities and all other life contingent products. In demography and actuarial science,there have been many attempts to find an appropriate model to representmortality.

Life expectancy at birth in Thailand as show in Figure 1 increased in theperiod between 2003 to 2012. Hence it is important for the government andinsurance companies to be aware of this trend. This situation is called longevityrisk, i.e. the risk that a pension scheme or an insurer’s annuity portfolio willneed to pay out more than expected due to increasing life expectancy.

There are many ways to project the future mortality rate. A benchmark modelwas developed by R.D. Lee and L. Carter (Lee and Carter 1992). Lee and Carterproposed a remarkably simple model for describing the secular change inmortality as a function of a single time index and specifying a log-bilinearform for the force of mortality. In their original paper, Lee and Carter (1992)applied Singular Value Decomposition (SVD) to estimate the parameters of themodel. The Lee-Carter model has become widely used and has been modifiedand extended in different ways of parameter estimation of the model by otherresearchers, such as Wilmonth (1993), who proposed the Weighted Least Square(WLS) approach to obtain an estimation of the Lee-Carter model parameters.Brouhns, Denuit, and Vermunt (2002) implemented Wilmonth’s recommenda-tion for improving the Lee-Carter approach to forecasting the demographiccomponent and applied Maximum Likelihood Estimation (MLE) to estimate theparameters of the Lee-Carter equation under the assumption that the death ratehas a Poisson distribution. The Lee-Carter model has proved to give good resultsfor mortality in diverse countries: Belgium (Brouhns, Denuit, and Vermunt2002), Japan and Taiwan (Yue, Huang, and Yang 2008), China (Zhao, 2011).The model has many variants and extension. For example, Lee and Miller (2001)

Life Expectancy at Birth

64666870727476788082

2003 2004 2005 2006 2007 2008 2009 2010 2011

Year

Ag

e Females

Males

Figure 1: Life expectancyat birth in Thailand, theperiod between 2003 to2012.

2 N. Yasungnoen and P. Sattayatham


purposed that the adjustment of kt involves fitting to life expectancy at birth inyear t rather than death rates probability which was used the basic Lee-CarterModel while Plat (2009) and O’ Hare and Li (2012) purposed multi factor model.However, Plat model do not give good results for a wider age range. So, in thispaper, we still prefer to use the single factor Lee-Carter model which is suitableto the aim of our investigation.

The objective of this paper is to use the Lee-Carter model to estimate andforecast mortality rate in Thailand and compare three different methods ofestimating the parameters of the Lee-Carter model: Singular ValueDecomposition (SVD), Weighted Least Square (WLS), and Maximum LikelihoodEstimation (MLE) based on the Poisson statistical framework. Furthermore, wewish to seek for a suitable method (in term of minimum MSE) to forecastmortality rate and life expectancy at birth. The plan for this paper is as follows:Section 2 presents the Lee-Carter model. Section 3 briefly describes the threemethodologies for deriving the parameters of the model. In Section 4, the fittedmodel derived is presented. Section 5 is devoted to studying the forecastedmortality rate and life expectancy at birth. Section 6 gives a summary of theresults.

2 Data and Lee-Carter Model

2.1 Notation

We analyze the changes in mortality as a function of both age x and calendartime t. Henceforth,– mx;t is the mortality force at age x during calendar year t.– ETRx;t is the exposure-to-risk at age x last birthday during year t, that is, the

number of people living of x years olds during the calendar year t.– Dx;t is the number of deaths recorded at age x during year t, from an

exposure-to-risk ETRx;t.

2.2 The Data

The Ministry of the Interior is the main source for data for the number ofage-specific people in the population. The number of deaths was obtainedfrom the Bureau of Policy and Strategy, Ministry of Public Health. (Age rangesfrom < 1, 1, 2,...., > 100.)

Forecasting Thai Mortality 3


We assume that the remaining lifetimes of the individuals aged on January 1of year t are independent and identically distributed. The m̂x;t is the nonpara-metric estimation of mx;t. This is given by the ratio of the observed number ofdeaths Dx;t for age x and year t to ETRx;t:

m̂x;t ¼ Dx;t

ETRx;t: ½1�

2.3 The Lee-Carter Model

The Lee-Carter model is described as the logarithm of the mortality rate as alinear combination of parameter as expressed by the following equation

ln mx;t� � ¼ αx þ βxkt þ "x;t; ½2�

where x ¼ x1; . . . ; xp represent ages and t ¼ t1; . . . :; tn represent the calendaryears. Here kt represents the time series for the general level of mortality andmeasures the trend in mortality over time, while αx describes the age profileaveraged overtime, βx represents how much, at each age, the mortality rateresponds to a change in kt; and "x;t is the error term with mean zero and varianceσ2". In order to make the model identifiable, one needs to impose the followingconstraints on the parameters, i.e.

Xtnt¼t1

kt ¼ 0; andXxpx¼x1

βx ¼ 1: ½3�

3 Estimation Parameter Approaches

3.1 Singular Value Decomposition (SVD)

In its original version, the Lee-Carter model cannot fit into ordinary regres-sion methods because there are no given regressors on the right side of theequation. We have only parameters to be estimated and the unknown indexkt. The estimators of αx, βx, and kt will be denoted by α̂x, β̂x; and k̂trespectively. Firstly, we shall define the estimator α̂x as the averagingln m̂x;t� �

over time t:

α̂x ¼ 1tn � t1 þ 1

Xtnt¼t1

ln m̂x;t� �

: ½4�



The parameters βx and kt are computed by applying singular value decomposi-tion (SVD) to the matrix Z where

Z ¼ ln m̂x;t� �� α̂x; ½5�

i.e.

Z¼

ln m̂< 1;2003� �� α̂< 1 ln m̂< 1;2004

� �� α̂< 1 . . . . . . ln m̂< 1;2012� �� α̂< 1

ln m̂1;2003� �� α̂1 ln m̂1;2004

� �� α̂1 ln m̂1;2012� �� α̂1

..

. . .. ..

.

..

. . .. ..

.

ln m̂100;2003� �� α̂100 ln m̂100;2004

� �� α̂100 . . . . . . ln m̂100;2012� �� α̂100

ln m̂> 100;2003� �� α̂ > 100 ln m̂ > 100;2004

� �� α̂ > 100 . . . . . . ln m̂> 100;2012� ��α̂> 100

2666666666666666664

3777777777777777775

½6�By applying SVD, one obtains the decomposition

Z ¼Xri¼1

ρiUx;iVt;i; ½7�

where r ¼ rank[Z], and ρi (i ¼ 1,2,…,rÞ are the ordered (increasing) singularvalues with Ux;i and Vt;i as the corresponding left and right singular vectors.These give the approximations βx and kt as follows:

β̂x ¼ Ux;1 and k̂t ¼ ρ1Vt;1:

3.1.1 Adjustment of the k̂t by Re-estimating to the Total Observed Deaths

In the first estimation, the estimation of kt do not provide an adequate fit to theobserved data.

Thus, Lee and Carter (1992) re-estimate kt to get the observed number ofdeaths equal to the fitted number of deaths, i.e.

Dt ¼Xx

ETRx;t exp αx þ βxktð Þ; ½8�

where Dt is the total number of deaths in year t and ETRx;t is the population ofage � in year t. There is need for a second-stage estimation for k̂t of SVDmethod.



3.2 The Weighted Least Square

Fitting the Lee-Carter model into the Weighted Least Square (WLS) can be doneby minimizing the equation

Xxpx¼x1

Xtnt¼t1

Dx;t ln mx;t� �� αx � βxkt

� �� 2: ½9�

To minimize (9), it is obtained by equating to 0 the first derivatives with respectto αx, βx and kt. The updating for the parameters can also be obtained recur-sively using normal equation. The required parameters αx; βx and kt will beobtained numerically according to the following algorithm

α̂x ¼P

t Dx;tðln mx;t� �� β̂xk̂tÞPt Dx;t

;

β̂x ¼P

t Dx;t k̂tðln mx;t� �� α̂xÞP

t Dx;t k̂2t;

k̂t ¼P

t Dx;t β̂xðln mx;t� �� α̂xÞP

t Dx;t β̂2x:

½10�

This process continues until successive computation yields little or no change inparameter value.

3.3 Maximum Likelihood Estimation

We now assume that

Dx;t ,Poisson ETRx;tmx;t� �

: ½11�with

mx;t ¼ exp αx þ βxktð Þ; ½12�where the parameters are still subject to the constraints (3). The model assumesthat the log of mortality has a log-bilinear structure, that is, ln mx;t

� � ¼exp αx þ βxktð Þ. Additionally, the expected number of death is given byλx;t ¼ E Dx;t

� � ¼ ETRx;t exp αx þ βxktð Þ:Instead of resorting to SVD for estimating αx, βx and kt, we now deter-

mine these parameters by maximizing the log-likelihood of model (11) whichis given by



L α; β; kð Þ ¼ lnYtnt¼t1

Yx¼xp

x¼x1

ETRDx;tx;t exp �λx;t

� �Dx;t� �

!

!( )

¼Xtnt¼t1

Xxpx¼x1

Dx;t ln λx;t � λx;t � ln ðDx;t� �

!��

¼Xtnt¼t1

Xxpx¼x1

Dx;t lnETRx;t þ Dx;t αx þ βxktð Þ � ETRx;t exp αx þ βxktð Þ�� ln Dx;t

� �!

� ��

¼Xtnt¼t1

Xxpx¼x1

Dx;t αx þ βxktð Þ � ETRx;t exp αx þ βxktð Þ� �þ constant:

½13�Goodman (1979) proposed the iterative method for estimating the log-linearmodel for the maximum likelihood estimation. In iteration step νþ 1, a singleset of parameters is updated and the other of parameters are fixed at theircurrent estimate by using the following update scheme

θ̂ νþ1ð Þ ¼ θ̂ νð Þ �@L νð Þ@θ

@2L νð Þ@θ2

; ½14�

where L νð Þ ¼ L νð Þ θ̂νð Þ�

:

In our application, there are three sets of parameters to be estimated, i.e.{αxg, fβxg, and ktf g. The updating scheme is as follows: starting withα̂ 0ð Þx ¼ 0; β̂ 0ð Þ

x ¼ 1; and k̂ 0ð Þt ¼ 0 (random values can also be used).

α̂ νþ1ð Þx ¼ α̂ νð Þ

x �P

t Dx;t � D̂ νð Þx;t

� �Pt D̂

νð Þx;t

; β̂ νþ1ð Þx ¼ β̂ νð Þ

x ; k̂ νþ1ð Þt ¼ k̂ νð Þ

t ;

k̂ νþ2ð Þt ¼ k̂ νþ1ð Þ

t �P

t Dx;t � D̂ νð Þx;t

� β̂ νþ1ð Þx

�Px D̂νð Þx;t β̂ νþ1ð Þ

x

� 2 ; α̂ νþ2ð Þx ¼ α̂ νþ1ð Þ

x ; β̂ νþ2ð Þx ¼ β̂ νþ1ð Þ

x ;

β̂ νþ3ð Þx ¼ β̂ νþ2ð Þ

x �P

t Dx;t � D̂ νþ2ð Þx;t

� k̂ νþ2ð Þt

�Pt D̂νþ2ð Þx;t k̂ νþ2ð Þ

t

� 2 ; α̂ νþ3ð Þx ¼ α̂ νþ2ð Þ

x ; k̂ νþ3ð Þt ¼ k̂ νþ2ð Þ

t ;

½15�

where D̂ νð Þx;t ¼ ETRx;t exp α̂ νð Þ

x þ β̂ νð Þx k̂ νð Þ

t

� , and ν ¼ 0; 1; 2; 3; . . ..



The criterion used to stop the procedure is a very small increase in the log-likelihood function.

4 Fitting the Model

The Lee-Carter model is mx;t ¼ exp αx þ βxkt þ "xtð Þ; x ¼ < 1,1,2,…,100, > 100,t ¼ 2003,...,2012.

To fit the logarithm of mortality, we use the following equation

ln m̂x;t� � ¼ α̂x þ β̂xk̂t; ½16�

where the mortality rate m̂x;t is given by

m̂x;t ¼ expðα̂x þ β̂xk̂tÞ: ½17�

Figure 2 shows comparison of α̂x, β̂x and k̂t with SVD, WLS, and MLE. Figure 2(a)shows graph of the estimations of the shape parameter αx for females. The graphis high at the beginning of life, i.e. when x < 1, and it declines in 1–14 interval ofage. Moreover, it continues to rise until it reaches the 97 age level. The graph ofthe estimations of the shape parameter αx show the same trend for males asappeared in Figure 2(b). Figure 2(c) shows graph the estimations of the parameterβx for females. The graphs of SVD and WLS exhibit a similar trend but MLE isdifferent from the others in the 16–26 intervals of age and over 90 age level.Figure 2(d) shows the estimations of the parameter βx for males with those threemethods and all of them show the same trend. Figure 2(e) shows graph of kt forfemales and Figure 2(f) shows graph of kt for males.

Figure 3 compares the mortality rate for the three methods in 2007 and 2011.Figure 3(a) and 3(b) show the mortality rate for females in 2007 and 2011respectively. The graphs of all methods are closed to the observed mortalityrate except the MLE method is different from the observed data when the age ishigh. Figure 3(c) and 3(d) show the mortality rate for males in 2007 and 2011respectively. The graphs show that the mortality rate of all methods are closed tothe graph of observed data.

We consider the fitted mortality rate. The data is span from 2003 to 2012.The results appear in Table 1. It can be seen that, for females, the MSE of theSVD has the smallest value. For males, the MSE of the WLS gives the smallestvalue. This indicates that the SVD and the WLS are suitable methods forforecasting mortality rate for females and males respectively.



Alp

ha

(x)

for

fem

ales

−9−8−7−6−5−4−3−2−10

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

9510

0

Ag

e(a

)(b

)

(c)

(d)

(e)

(f)

Alpha (x)S

VD

WLS

MLE

Alp

ha

(x)

for

mal

es

−10−8−6−4−20

<0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100

Ag

e

Alpha (x)

SV

D

WLS

MLE

Bet

a (x

) fo

r fe

mal

es

−0.0

6

−0.0

4

−0.0

20

0.02

0.04

0.06

0.080.

1

0.12

<1

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100

Ag

e

Beta (x)

SV

D

WLS

MLE

Bet

a (x

) fo

r m

ales

−0.0

6

−0.0

4

−0.0

2

0

0.02

0.04

0.06

<1

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100

Ag

e

Beta (x)

SV

D

WLS

MLE

Kap

pa

(t)

for

fem

ales

−15

−10−505101520

2003

2004

2005

2006

2007

2008

2009

2010

2011

2012

Yea

r

Kappa (t)

SV

D

WLS

MLE

Kap

pa

(t)

for

mal

es

−15

−10

−50510152025

2003

2004

2005

2006

2007

2008

2009

2010

2011

2012

Yea

r

Kappa (t)

SV

D

WLS

MLE

Figu

re2:

Lee-Ca

rter

parameter

estim

ations

ofα x,β

xan

dk t

bythethreemetho

ds(SVD

,WLS

,and

MLE).(a)E

stim

ations

ofα x,fem

ales.(b)

Estimations

ofα x,

males.(c)

Estimations

ofβ x,fem

ales.(d)

Estim

ations

ofβ x,m

ales.(e)

Estim

ations

ofk t,fem

ales.(f)Estimations

ofk t,m

ales.



m (

x,t)

, fem

ales

, 200

7

0

0.050.1

0.150.2

0.250.3

<1

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100

Ag

e

(a)

(c)

(d)

(b)

m (x,t)

SV

D

WLS

MLE

Obs

erve

d

m (

x,t)

, fem

ales

, 201

1

0

0.050.

1

0.150.

2

0.250.

3

<1

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100

Ag

e

m (x,t)

SV

D

WLS

MLE

Obs

erve

d

m (

x,t)

, mal

es, 2

007

0

0.050.

1

0.150.

2

0.25

<1

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100

Ag

e

m (x,t)

SV

D

WLS

MLE

Obs

erve

d

m (

x,t)

, mal

es, 2

011

0

0.050.

1

0.150.

2

<1

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100

Ag

e

m (x,t)

SV

D

WLS

MLE

Obs

erve

d

Figu

re3:

Thefitof

themortalityrate

bythreeestimationmetho

ds(SVD

,WLS

,and

MLE)a

scompa

rewiththeob

served

data

intheyear

2007an

d20

11.

(a)Fitted

mx;t,females,20

07.

(b)Fitted

mx;t;females,20

11.(c)Fitted

mx;t;males,20

07.

(d)Fitted

mx;t;males,20

11.



5 Forecasting Mortality Rate and Life Expectancyat Birth ARIMA Model for Estimation kt

We model the forecast index by using the time series model. Box-Jenkinsmethodology is used to estimate and forecast kt with the appropriate ARIMAtime series model. From the values of AIC and BIC in Tables 2 and 3, we

Table 1: MSE of the fitted mortality rate m̂x;t .

MSE SVD WLS MLE

Females . . .Males . . .

Note: The bolded entries show the smallest value of MSE for each gender.

Table 2: ARIMA-models fitted to kt for females.

(,,) (,,) (,,) (,,) (,,) (,,)ARIMA

with drift with drift with drift with drift with drift

SVD AIC . . . . . .BIC . . . . . .

WLS AIC . . . . . .BIC . . . . . .

MLE AIC . . . . . .BIC . . . . . .

Note: The bolded entries show the smallest value of AIC and BIC.

Table 3: ARIMA-models fitted to kt for males.

(,,) (,,) (,,) (,,) (,,) (,,)ARIMA

with drift with drift with drift with drift with drift

SVD AIC . . . . . .BIC . . . . . .

WLS AIC . . . . . .BIC . . . . . .

MLE AIC . . . . . .BIC . . . . . .

Note: The bolded entries show the smallest value of AIC and BIC.



conclude ARIMA(0,1,0) with drift to be the most appropriate model based on thelower value of AIC and BIC. Other models have a higher AIC-value and BIC-value. The estimated model ARIMA(0,1,0) with drift is described by

k̂t � k̂t�1 ¼ θ þ "t; where "t ,N 0; σ2"� �

: ½18�

The estimation of ARIMA(0,1,0) with drift parameters are showed in Table 4:

The forecasted k̂2012þs, s ¼ 1,2,…,10 are inserted into the formulas giving theforce of mortality and provide

lnðm̂x;2012þsÞ ¼ α̂x þ β̂xk̂2012þs: ½19�

We note that the life expectancy at x, ex, is defined as follows: ex ¼ Txlxwhere Tx

is the cumulative number of years lived by the cohort population in the ageinterval and all subsequent age intervals. The lx value represents the number ofliving people at the beginning of the x age interval from a population of l0newborn babies. This is usually defined as l0 ¼ 100;000:

Firstly, we consider the problem of fitted life expectancy at birth. The data isspan from 2003 to 2012. The results appear in Table 5. It can be seen that, forfemales, the MSE of the SVD gives the smallest value. For males, the MSE of theWLS has the smallest value. This indicates that the SVD and the WLS aresuitable methods for forecasting life expectancy for females and malesrespectively.

Next, we investigate the problem of forecasting life expectancy at birth. Ourplan is to forecast from 2013 to 2022. As the results of our analysis (see Table 5),

Table 4: Show the parameters of ARIMA(0,1,0) with drift which were obtained bythe three estimation methods (SVD, WLS, and MLE).

Females Males

θ̂ σ̂" θ̂ σ̂"

SVD −. . −. .WLS −. . −. .MLE −. . −. .



we shall use SVD and WLS to forecast the life expectancy at birth of females andmales respectively. The forecasted results can be found in Table 6. We notefurther that, for females, the forecasted values of the life expectancy at birth bySVD increase from 79.12 year in 2013 to 81.01 year in 2022. For males, theforecasted values of the life expectancy at birth by WLS increase from 72.66year in 2013 to 74.53 year in 2022.

Figure 4 shows the fitted and forecasted life expectancy at birth for females bySVD and for males by WLS. It shows that the life expectancy at birth tends toincrease from the past to the future.

Table 6: Forecasted valued of life expectancy at birth, for femalesby SVD and males by WLS respectively, 2013–2022.

Year Females Males

SVD WLS

. . . . . . . . . . . . . . . . . . . .

Table 5: MSE of the fitted life expectancy at birth.

MSE SVD WLS MLE

Females . . .Males . . .

Note: The bolded entries show the smallest value of MSE for each gender.



6 Conclusion

This paper compares the estimation method of the parameters of the Lee-Cartermodel for mortality data in Thailand. The parameters of the model were esti-mated with the Singular Value Decomposition (SVD), the Weighted Least Square(WLS), and the Maximum Likelihood Estimate (MLE) methods. Estimation ofparameters αx, βx; and kt using each of these three methods show the same trendfor males. For females, the estimation of βx and kt by MLE is different from theothers. Moreover, the general mortality index kt is a time series which shows adecreasing trend. There are some variations in the estimates of the time-depen-dent mortality index kt.

For the comparison of fitted mortality rate, the results of the MSE show thatthe SVD is the best fit of parameter estimation for females and WLS is the best fitfor males. Hence, we can conclude from our data for Thailand that the SVDmethod is appropriate for the estimation of the parameters of the Lee-Cartermodel for females. For males, WLS is appropriate methods for the estimation ofthe parameters of the Lee-Carter model.

By using SVD method for forecasting, it can be seen from Table 6 that lifeexpectancy at birth for females increases 2.38% from 2013 to 2022. Analogously,for males, WLS method gives 2.57% increase from 2013 to 2022.

Life Expectancy at Birth of females, SVD

75

76

77

78

79

80

81

82

2004

2005

2006

2007

2008

2009

2010

2011

2012

2013

2014

2015

2016

2017

2018

2019

2020

2021

2022

Year

(a)

(b)

Ag

e

Life Expectancy at Birth of males, WLS

68

69

70

71

72

73

74

75

2005

2006

2007

2008

2009

2010

2011

2012

2013

2014

2015

2016

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Figure 4: Fit (2004–2012) and forecast (2013–2022) of life expectancy at birth for femalesby SVD and males by WLS respectively. (a) Life expectancy at birth of females, SVD.(b) Life expectancy at birth of males, WLS.



Acknowledgement: The main source for data about population was provided by theMinistry of the Interior. The number of deathswas obtained from the Bureau of Policyand Strategy, Ministry of Public Health. These data are gratefully acknowledged.

Funding: We would like to express our sincerely thank to the two referees fortheir valuable comments that help us a lot in improving this paper.

References

Brouhns, N., M. Denuit, and J. Vermunt. 2002. “A Poisson Log-Bilinear Regression Approach tothe Construction of Projection Life Table.” Insurance: Mathematics & Economics 31:373–93.

Goodman, L. 1979. “Simple Models for the Analysis is Association in Cross ClassificationHavingorders Categories.” Journal of the American Statistical Association 74:357–552.

Lee, R.D., and L. Carter. 1992. “Modelling and Forecast the Time Series of US Mortality.” Journalof the American Statistical Association 87:659–71.

Lee, R.D., and T. Miller. 2001. “Evaluating the Performance of the Lee Carter Mortality Forecast.”Demography 38:537–49.

O’ Hare, C., and Y. Li. 2012. “Explaining Young Mortality.” Insurance: Mathematics andEconomics 50:12–25.

Plat, R. 2009. “On Stochastic Mortality Modeling.” Insurance: Mathematics and Economics,Elsevier 45:393–404.

Wilmonth, J.R. 1993. “Computational Methods for Fitting and Extrapolating the Lee-Carter Modelof Mortality Change Technical Report.” University of California, Berkeley, USA.

Yue, C.J., H. Huang, and S. Yang. 2008. “An Empirical Study of Mortality Model in Taiwan.” Asia-Pacific Journal of Risk and Insurance 3:150–64.

Zhao, B.B. “A Modified Lee-Carter Model for Analyzing Short Base Period Data.” PopulationStudies. Accessed December 16, 2011.



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