One of the main objectives of this study is the estimation of parameters of

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1 C H A P T E R 3 M E T H O D O L O G Y 31 Introduction One of the main objectives of this study is the estimation of parameters of Makeham's law of mortality I believe that it is mandatory to explain briefly the actuarial and statistical theories that were applied here before describing the rest of the study 32 Definitions All the definitions and theorems which used through out this study given as follows 321 Force off Mortality The force of mortality, /u x,as a measure of the death rate at specific age x Mathematically, the force of mortality simply defined as u r = hm, (321) = lim/j q r To develop the definition, take the number of survivors in a closed group, attaining age x (l x ), and plot with respect to age Noting the smooth continuous curve we feel that fi x can be described by this graph So the force mortality defines by the slop of this l x function The ratio of this instantaneous rate of decrease of / t to the corresponding value of l x \s called the force of mortality fi x and it can be express -dl x as fj, x - From the above definition can get that force of mortality is nothing more than that an instantaneous death rate Age specific death rate defines as the ratio of total deaths to total population in a specified community or area over a specified period of time known as death rate 23

2 When death rate is calculated for individuals of a specified age (or age range) then the rate is described as being age-specific It should be emphasized that the "age specific death rate" specified here is discrete in the sense that it describes the probability of death within a short but fixed interval of time - in this case, one year In practice, the term "force of mortality" is used to describe the limiting probability of death within a continuous interval, as the interval's length shrinks to zero So, in this study force of mortality consider as age specific death rate Here considered only the value of l x for integral values of agex, but it is evident that deaths occur at all times through each year of age, so that l x may be considered as a continuously varying function So first of all have to calculate the age specific death rate for total population, males and for females separately 322 Population ratio Population ratios are used to describe the degree of balance between two elements of the population Sex ratio The ratio of males to females in the population (normalized to 100) The primary sex ratio is the ratio at the time of conception, secondary sex ratio is the ratio at time of birth, and tertiary sex ratio is the ratio of mature organisms 323 Mathematical expression of age specific death rate number of deaths in the age group(x, x + n) m x ~ 7- population at that specific age group nm x denotes age specific death rate for the age group (x,x+n) In this study time interval n=l Using the above definition, age specific death rate can be calculated for all 24

3 324 Makeham's law The mortality hazard rate can be written as the sum of a constant term A that has been interpreted as capturing the accident hazard and the term BC" as capturing the hazard of aging Mix) = A + BC X ; Where B > 0,A> B,C > \and x>0 In this study two methods are used to estimate Makeham's parameters Namely Direct method Least squares method 33 Direct method Average number dying per annum over a period of n years define as the rate of decrease of l x at any particular age is the limit when n vanishes of the function n Expressing the limit in language of infinitesimal calculus: dx The ratio of the above instantaneous rate of decrease of l x to the corresponding value of l x is called the force of mortality fx x ~dl x 331 Deriving a formula for age specific death rate From the definition of force of mortality X = Z^2isL (33!!, dx From expression of Makeham's formula 25

4 fi(x) = A + BC X By putting A = -\og e m and B --log, g log, C; where m>0,g>0 M,=~ log, m - (log, g log, C)C (3312) Equating equation (3311) and (3312) ~ d ] 0 g * 1 ' = - log, m - (log, g log, C)C* dx Rearranging the above formula -d\ogj x = (-log, m-(log, g log, C)C*)cft (3313) By integrating the formulae (3313), get that log, /, = log, k + x log, m + C x log, g Where log, k is the constant of integration, or /, = km'g c ' The simplest way of determining the constants in Makeham's formula is from four equidistant values of natural logarithm log, k + x log, m + C x log e g = \og e l x log, k + 'x + t) log, m + C x+ ' log, g = log, l x+l log, k + (x + It) log, m + C x+21 log, g = log, l x+2l log, k + (x + 30 log, m + C* +3 ' log, g = log, l x+3r Taking the differences, / log, m+ C x (C -1) log, g = log, l x+l - log, / log, m + C x+i (C -1) log, g = log, /, +2, - log, /, / log, m + C x+21 (C -1) log, g = log, l x+3l - log, I X+2I 26

5 Taking differences again, C*(C -1) 2 log, g = log, l x+2l - 2log e l x+l + log, C c +' (C -1) 2 log, g = log, /, +3, - 2 log, /, +2, + log, l x+l Dividing the above two equations, c, _ lq ge 1*+ 3I ~ 2 l Q g e h+i, + l Q g e K» (3314) log/, + 2,-21og r / J r + l +log,/ J t From this equation C is determined and then in succession In g, In m, and In A: Consider the equations, C'(C -1) 2 log, g = log, l x+2, -2log, /, +, + log, l x = p(say) C* +l (C -1) 2 log, g = log, /, +3, - 2 log, /, +2, + log, /, +, = cy (say) Taking the natural logarithms, A- log, C + 2 log,(c -1) + log log, g = log, p (3315) O + /) log, C + 2 log, (C - 1) + log log, g = log Taking difference, / log, C = log, 9 - log, p l o g < c =logg-log/> Substituting C'aro/log,C values in equation (3315) log log, g = log, /? - x log, C - 2 log, (C - 1) log g = ev &,p-* lo z c - 1,0 S'( c '- l» Using equation (3315) x log, C + log, (C -1) + log log, g = log, p - log, (C -1) C> \og e g(c -]) 2 = p C*log,g(C'-l) = - / (C'-l) 27

6 But t\og e m + C x (C -l)log,g = log,/, +, - log,/, So, / log, m = log, /, +, - log, /, - ( C, /? _ 1 ) - log, m = ' o g, - log,/, - x log, C + log log, g = log, p - 2 log, (C -1) Substituting values in very first equation log, * + x log, in + C x log, g = log, l x log A: = log, l x -x log, m - C* log, g Using the determined values parameter B can be determined 34 Least squares method For this investigation method of least square is also applies to estimate the Makeham's parameters The goal of Least-Squares Method is to find a good estimation of parameters that fits a function, f(x), of a set of data, Xix n The Least-Squares Method requires that the estimated function has to deviate as little as possible from f(x) Generally, Least-Squares Method has two categories, linear and non-linear We can also classify these methods further: ordinary least squares (OLS), weighted least squares (WLS), and alternating least squares (ALS) and partial least squares (PLS) To fit a set of data best, the least-squares method minimizes the sum of squared residuals (it is also called the Sum of Squared Errors, SSE) m 28

7 With, r the residual, which is the difference between the actual points and the regression line, and is defined as n=y,-f( x i) Where the m data pairs are^v,,^), and the model function is fix,) Here, we can choose n different parameters for /(x), so that the approximated function can best fit the data set 341 Linear Least-Squares Method Linear Least-Squares (LLS) Method assumes that the data set falls on a straight line Therefore, /(x) = ax + b, where a and b are constants However, due to experimental error, some data might not be on the line exactly There must be error (residual) between the estimated function and real data Linear Least-Squares Method defined the best-fit function as the function that minimizes S = f j (y i -(ax,+b)) 2 The advantages of LLS: 1 If we assume that the errors have a normal probability distribution, then minimizing S gives us the best approximation of a and b 2 We can easily use calculus to determine the approximated value of a and b 342 Application of linear least squares method Force of Mortality is defined to be, ju x - ^ x in a closed group, attaining age x dx, where / is the number of lives Makeham's law, force of mortality expressed as//(x) = A + BC X ; whereb > 0,A> B,C >\andx > 0 The above formula can be rearranged as follows, ( Mx -A) = BC X Taking the logarithm of above equation log, ( Mx -A) = log, B + (log, C)x (3421) 29

8 Straight line is getting, hence linear least squares method is applicable In least squares the dependent variable is y i = \og E (/J X -A) = l x + e r t xis the predicted value of the dependent variable and e,is the error or discrepancy So equation (3421) should be change as \og e (ju x -A) - log (, B + (\og e C)x + log, cf Here log, shows the error term Because of that Makeham's law should be modified as /u(x) = A + EJBC* The least squares estimates of 0 O and 0 X are ZCv, -x) 9= (3422) Z(*,-30 2 /=i 9 0 = y - 9 t x (3423), Where x and y are sample means The residuals are defined to bee, = y,-& 0-0 { x, and the residual sum of squares, ESS = ^e t 2 is minimized with respect to t9 0 and <9, by using (3422) and (3423) For any value of A, getting different ESS Varying A systematically and choose the one with minimal ESS as our least squares estimate of A, namely 4 By reference to modified version of (3421) we see that B = e 0 " and C = e 6 ' To test whether the data set is normal, normal probability test can be used By displaying the Anderson- Darling statistics (AD) on the plot, which can indicates whether the data are normal If the p value is lower than the chosen level, the data do not follow a normal distribution The error term e,or log c should have normal distribution so has lognormal distribution 343 Residual Analysis The examination of the residuals produced by fitting a particular response function will be of utmost importance Its main purpose will be to examine and test the adequacy of the presumed model The assumptions which we must put under scrutiny are Errors are independent 30

9 Residual have constant variance Autocorrelation Normality of the error terms For checking all four assumptions, graphical techniques and statistical values can be used 3431 Mean of Residuals To test the unbiasedness of the error term, there are two simple techniques, one is arithmetic and the other one is graphical The arithmetic method illustrates the fact that minimizing the sum of square errors Hence, expects the sum of all residuals to be closed to zero In general graphical method will be the major tool to test the unbiased ness of the error term Using the residual versus fitted values plot can be check the independence of the error terms 3432 Residual has constant variance Residuals versus fits plot will be used for assumption This plot should show a random pattern of residuals on both sides of 0 If a point lies far from the majority of points, it may an outlier Also, there should not be any recognizable patterns in the residual plot The following may indicate error that is not random: a series of increasing or decreasing points a predominance of positive residuals, or a predominance of negative residuals patterns, such as increasing residuals with increasing fits 3433 Autocorrelation (Residuals are independent of one another) Minitab provides two methods to determine if residuals are correlated: A graph of residuals versus data order ( n) can provides a means to visually inspect residuals for autocorrelation A positive correlation is indicated by a clustering of residuals with the same sign A negative 31

10 correlation is indicated by rapid changes in the signs of consecutive residuals The Durbin-Watson statistic tests for the presence of autocorrelation in regression residuals by determining whether or not the correlation between two adjacent error terms is zero The test is based upon an assumption that errors are generated by a first-order autoregressive process If there are missing observations, these are omitted from the calculations, and only the non missing observations are used To reach a conclusion from the test, it needs to compare the displayed statistic with lower and upper bounds in a table If D > upper bound, no correlation exists; if D < lower bound, positive correlation exists; if D is in between the two bounds, the test is inconclusive 3434 Normality of the error terms A normal probability plot and histogram of residual will be used to establish normality of the error terms Normal plot of residuals The points in this plot should generally form a straight line if the residuals are normally distributed If the points on the plot depart from a straight line, the normality assumption may be invalid If your data have fewer than 50 observations, the plot may display curvature in the tails even if the residuals are normally distributed As the number of observations decreases, the probability plot may show substantial variation and nonlinearity even if the residuals are normally distributed Histogram of residuals An exploratory tool to show general characteristics of the data, including: Typical values, spread or variation, and shape Unusual values in the data 32

11 Long tails in the plot may indicate skewness in the data If one or two bars are far from the others, those points may be outliers Because the appearance of the histogram changes depending on the number of intervals used to group the data, use the normal probability plot to assess the normality of the residuals 35 Life tables A published life table usually contains tabulations, by individual ages, of the basic functionsq x, l x,d x and, possibly, additional derived functions such as L x,t x ande x Makeham's law survival function equation is given by S(x) = cxp(-ax - m{c -1)); where m = log e C As in above equation of survival function S(x +1) can be written as follows S(x +1) = exp(-a(x +1) - m(c* +l -1)) When x = 110,S(x +1) = 0 (Since there is no more survival) Following are the equations that were used to construct the life table S(x + \) S(x) <Jx = ] -Px> /, + =',/>,; d=l-l X X x+i Uniform distribution or, perhaps more properly, a uniform distribution of deaths assumption within each year of age, tq* is a linear function Ie tq x = tq x Under the assumption of Uniformity of deaths in a unit interval, L = 'x+\ J T x =L x +T x+i 33

12 36 Summary In this chapter, all actuarial and statistical theories and methods that were used to estimate Makeham's parameters are explained Also the formulas used to construct life tables are properly explained In the next chapter, descriptive measures that were used for data analysis and interpretation are discussed 34