, I J ° I 
\ h J f f J V E B S l T Y O F M O R A T U W A . S B I L A (UK 
ESTIMATING PARAMETERS OF MAKEHAM'S LAW OF 
MORTALITY 
H.I.B.Soysa 
(07/8524) 
Dissertation submitted in partial fulfillment of the degree of Master of Science in 
Financial Mathematics 
Department of Mathematics 
Faculty of Engineering 
University of Moratuwa 13 ft ( 0 4 
Sri Lanka 
T V 1 
May 2011 
Univers i ty o f M o r a t u w a / O 4f& S 
102465 
102465 
Dr. M. D. "TAttygalle 
(Co-Supervisor) 
Senior Lecturer 
Department of Statistics 
University of Colombo 
04- n 2.011 
ii 
A C K N O W L E D G E M E N T 
I would like to make this opportunity to express my indebtedness to my supervisor 
Mr. R.A.Dissanayake, senior lecturer of the Department of Mathematics, University of 
Moratuwa for his patience, and his precious t ime that he imparted with subject 
knowledge and generous guidance which steered me throughout this study. 
It is with pleasure that I also thank Dr. M. D. T. Attygalle, Head of the Department of 
Statistics, university of Colombo, for her substantial assistance, in making this effort 
a success. 
I must also thank Mr. T.MJ.A Cooray, Dr.T.S.G. Peiris and Mr. U.C.Jayathilake for their 
support. 
I would like to offer my heartfelt thanks without any hesitation to the staff of the 
Department of Mathematics for strengthening me with knowledge, without which I 
would have not been able to accomplish this feat. 
I would also thank my family for all the support extended to me in enduring this 
task. 
iii 
ABSTRACT 
Actuar ies and d e m o g r a p h e r s have a long t radi t ion of using collateral da ta to improve mortal i ty 
e s t i m a t e s . Two main a p p r o a c h e s have b e e n used t o accompl ish t h e i m p r o v e m e n t , morta l i ty 
laws and mode l life t ab les . 
The m o s t c o m m o n mortal i ty law t h a t used to illustrate t h e life t ab le s is M a k e h a m ' s law. In Sri 
Lanka, Census a n d Statistics D e p a r t m e n t p r e s e n t s t h e life t ab le and calculation of life t ab le had 
b e e n d o n e using sof tware package Mortpak-l i te deve loped by uni ted nat ions and t h e m o s t 
insurance c o m p a n i e s use UK s t anda rd life tab les t o cons t ruc t insurance p r e m i u m s . So my 
p u r p o s e is t o cons t ruc t t h e m o d e l life t ab l e for Sri Lanka, e s t i m a t e s t h e cons t an t s of M a k e h a m ' s 
law of mortal i ty t h a t used t o cons t ruc t life t ab le , and in t roduces a regression mode l t h a t 
i nco rpo ra t e s t h e early s tages (below 20) and also w a n t s t o d e t e r m i n e w h e t h e r t h e r e ' s big 
difference for ma le s and females . 
The M a k e h a m ' s e s t i m a t e s a r e based upon t h e equationpi(x) = A + BC . Two m e t h o d s a re 
used t o e s t i m a t e s t h e p a r a m e t e r s , namely least squa re m e t h o d and using t h e definition of age 
specific d e a t h r a t e . 
The mode l is d e m o n s t r a t e d t h e morta l i ty da ta (year 2001) of Sri Lanka. 
iv 
T A B L E O F C O N T E N T S 
Declaration of the candidate i 
Declaration of the supervisor ii 
Acknowledgements iii 
Abstract iv 
Table of contents v 
List of Figures viii 
List of Tables ix 
List of abbreviations x 
Chapter 1 : Introduction 1 
1.1 Background 1 
1.2 Focus of the research 5 
1.3 Objectives 5 
1.4 Overview of chapters 5 
1.5 Summary 6 
Chapter 2: Literature Review 7 
2.1 Introduction 7 
2.2 Mortality laws 7 
2.3 Life tables 10 
2.4 Summary 22 
Chapter 3: Methodology 23 
3.1 Introduction 23 
3.2 Definitions 23 
3.2.1 Force of mortality 23 
3.2.2 Sex ratio 24 
3.2.3 Mathematical expression of age specific death rate 24 
3.2.4 Makeham's law 25 
3.3 Direct method 25 
3.3.1 Deriving a formula for age specific death rate 25 
3.4 Least squares method 28 
3.4.1 Linear least squares method 28 
3.4.2 Application of linear least squares method 28 
3.4.3 Residual analysis 30 
3.4.3.1 Mean of residuals 31 
3.4.3.2 Constant variance 31 
v 
Chapter 4 : 
Chapter 5: 
Chapter 6: 
Chapter 7: 
3.4.3.3 Autocorrelation 31 
3.4.3.4 Normality of error terms 32 
3.6 Life tables 33 
3.7 Summary 34 
Descriptive Analysis of Mortality Data 35 
4.1 Introduction 35 
4.2 Graphical interpretation 35 
4.3 calculation of age specific death rate 36 
4.4 Summary 45 
Direct Method 46 
5.1 Introduction 46 
5.2 Estimation of parameters using derived formula 46 
5.2.1 For Total population 47 
5.2.2 For males 49 
5.2.3 For Females 51 
5.3 Summary 53 
Estimation of parameters using least squares method 54 
6.1 Introduction 54 
6.2 Estimation of parameters of Makeham's law for 
total population 54 
6.2.1 Residual Analysis 58 
6.3 Estimation of parameters of Makeham's law for Males 59 
6.3.2 Residual Analysis 62 
6.4 Estimation of parameters of Makeham's law for Females 63 
Chapter 8 
6.4.2 Residual Analysis 66 
6.5 Summary 67 
Life Tables for Sri Lanka 68 
7.1 Introduction 68 
7.2 Construction of life tables 68 
7.2.1 General life table 69 
7.2.2 Life table for males' 75 
7.2.3 Life table for females' 81 
7.3 Summary 86 
Conclusions and Recommendations 87 
8.1 Introduction 87 
vi 
8.2 Analysis of the results of the direct method 87 
8.3 Analysis of the results of least squares method 88 
8.4 Conclusions 88 
8.5 Recommendations 89 
8.6 Further analysis 89 
Reference List 90 
Appendix A l 92 
Appendix A2 100 
vii 
LIST OF Figures Page 
Figure 4.1 Graph of deaths in 2001 35 
Figure 4.2 Graph of deaths for Male and female 36 
Figure 4.3 Graph of age specific death rate with exponential curve 38 
Figure 4.4 Graph of age specific death rate with exponential curve for age 
Group 23-80 39 
Figure 4.5 Graph of specific death rate of Males wi th exponential curve 44 
Figure 4.6 Graph of specific death rate of Females with exponential curve 44 
Figure 4.7 Graph of age specific death rate for Males and Females 45 
Figure 6.1 Fitted line plot of dependant variable 55 
Figure 6.2 Unstandardized residual plot 58 
Figure 6.3 Fitted line plot of dependant variable for Males 59 
Figure 6.4 Unstandardized residual plot 62 
Figure 6.5 Fitted line plot of dependant variable for Females 63 
Figure 6.6 Unstandardized residual plot 66 
viii 
L I S T O F T A B L E S Page 
Table 2.1 Dr. Halley's life table 11 
Table 2.2 Northampton life table 12 
Table 2.3 Carlisle life table 14 
Table 4.1 Death rates for specific ages 37 
Table 4.2 Death rates of specific ages for males and females 39 
Table 7.1 General life table 69 
Table 7.2 Life table for males 75 
Table 7.3 Life table for females 81 
ix 
L I S T O F A B B R E V I A T I O N S 
Abbreviation Description 
Instantaneous death rate 
x Exact age 
(x, x + n) Age group with initial age x with the length of interval n 
n m x Death rate for the age group (x,x+n) 
p x Probability that a person aged (x) survived in (x, x+1) 
q x Probability that a person aged (x) died in (x, x+1) 
l x Number of survivors at age x in a life table with radix (starting 
population) of 100,000 persons 
l x + 1 Expected number of survivors of age (x ) 
d x Expected number of deaths in age group (x, x+1) 
Lx Total expected number of years lives between ages x and x+1 
T x Total expected number of years lived beyond age x, by survivorship group 
with l 0 initial numbers 
e x Expected number of years of future lifetime of an individual of the l x 
survivors of the group at age x 
AD Anderson Darling statistics 
R2 Coefficient of determination 
R2adj Adjusted Coefficient of determination 
SSfieg Regression sum of squares 
SSiotai T o t a ' s u m ° f squares 
SSR Residual sum of squares 
n Sample size 
p Number of parameters 
x