Fit of the 2011 Indonesian Mortality Table to Gompertz's and Makeham's Law using Maximum Likelihood Estimation
Abstract
This research discusses the estimation of the parameters for Gompertz’s law and Makeham’s law using the Maximum Likelihood Estimation method. A numerical approach to estimate the parameters of Gompertz’s law is the Newton-Raphson method. In the Makeham’s law, we use the Lagrange multiplier method to solve constraints of 0.001<A<0.003, 10^(-6)<B<10^3 and 1.075<C<1.115, and Broyden as a method to estimate the parameter numerically. The estimation result shows that parameter B converges to 0.005749 and parameter C converges to 1.024738 in the Gompertz’s law. In the Makeham’s law, the estimated parameters that satisfied the constraints are A converges to 0.00300344, B converges to 0.0002716465, and C converges to 1.113395. Based on the Average Relative Error (ARE) that calculated from the estimated for px, the 2011 Indonesian Mortality Table (the 2011 TMI) for men and for women are more accurate when approached using the Gompertz’s law than the Makeham’s law. The estimated for px uses the Gompertz’s law are very close to the px at the 2011 TMI (with Absolute Percentage Errors of less than 1%) at age intervals, for men: 0 – 10 years, 10 – 20 years, 20 – 30 years, and 60 – 70 years, and for women: 0 – 10 years, 10 – 20 years, and 70 – 80 years.
Keywords: parameter estimation; Newton-Raphson method; Broyden method; Lagrange Multiplier method.
Abstrak
Penelitian ini membahas mengenai estimasi parameter hukum mortalitas Gompertz’s dan hukum mortalitas Makeham’s menggunakan metode Maximum Likelihood Estimation. Pendekatan numerik untuk estimasi parameter hukum mortalitas Gompertz dilakukan menggunakan metode Newton-Raphson. Untuk mengatasi syarat batas 0.001<A<0.003, 10^(-6)<B<10^3 dan 1.075<C<1.115, pada estimasi parameter hukum mortalita Makeham digunakan metode pengali Lagrange dan pendekatan numerik metode Broyden. Hasil estimasi menunjukkan bahwa parameter B konvergen ke 0,005749 dan parameter C konvergen ke 1,024738 pada hukum mortalitas Gompertz. Pada hukum mortalitas Makeham’s, hasil estimasi parameter yang memenuhi syarat batas adalah nilai A konvergen ke 0,00300344, B konvergen ke 0,0002716465, dan C konvergen ke 1,113395. Berdasarkan nilai Average Relative Error (ARE) yang dihitung untuk estimasi , Tabel Mortalita Indonesia (TMI 2011) untuk pria dan untuk wanita lebih sesuai jika didekati menggunakan hukum Gompertz daripada hukum Makeham. Estimasi menggunakan pendekatan hukum Gompertz berada sangat dekat dengan nilai pada TMI 2011 (dengan Mean Absolute Percentage Error kurang dari 1%) pada interval usia, untuk pria: 0 – 10 tahun, 10 – 20 tahun, 20 – 30 tahun, dan 60 – 70 tahun, dan untuk wanita: 0 – 10 tahun, 10 – 20 tahun, dan 70 – 80 tahun.
Kata kunci: estimasi parameter; metode Newton-Raphson; metode Broyden; metode Pengali Lagrange.
References
N. Bowers, JR., H. Gerber, D. Jones dan C. Nesbitt, Actuarial Mathematics, Illinois: The Society of Actuaries, 1997.
J. Cohen, C. Bohk-Ewald dan R. Rau, “Gompertz, Makeham, and Siler models explainTaylor’s law in human mortality data,” Demographic Research, vol. 38, no. 29, pp. 773-842, 2018.
T. B. L. Kirkwood, “Deciphering death : a commentary on Gompertz (1825) 'On The Nature of The Function Expressive of The Law of Human Mortality, and on A New Mode of Determining The Value of Life Contingencies',” Philosophical Transactions Of The Royal Society B, vol. 370, no. 1666, 2015.
E. L. Turner dan J. A. Hanley, “Cultural imagery and Statistical Models of The Force of Mortality : Addison, Gompertz, and Pearson,” Journal of The Royal Statistical Society, vol. 173, no. 3, pp. 483-500, 2010.
W. M. Makeham, “On the Law of Mortality and the Construction of Annuity Tables,” Journal Of The Institute Of Actuaries, vol. 8, no. 6, pp. 301-310, 1860.
C. W. Jordan, Society of Actuaries' Textbook on Life Contingencies, Chicago: The Society of Actuaries, 1991.
S. J. Olshansky dan B. A. Carnes, “Ever Since Gompertz,” Demography, vol. 34, no. 1, pp. 1-15, 1997.
T. H. Tai dan A. Noymer, “Models For Estimating Empirical Gompertz Mortality : With An Application To Evolution Of Gompertzian Slope,” The Society Of Population Ecology, vol. 60, no. 1-2, pp. 171-184, 2018.
M. L. Garg, B. R. Rao dan C. K. Redmond, “Maximum Likelihood Estimation Of The Parameters Of The Gompertz Survival Function,” Journal Of The Royal Statistical Society Series C, vol. 19, no. 2, pp. 152-159, 1970.
A. Lenart, “The Moments of The Gompertz Distribution and Maximum Likelihood Estimation of Its Parameters,” Scandinavian Actuarial Journal, vol. 2014, no. 3, pp. 255-277, 2014.
X. Feng, G. He dan Abdurishit, “Estimation Of Parameters Of The Makeham Distribution Using The Least Squares Method,” Mathematics and Computers in Simulation, vol. 77, no. 1, pp. 34-44, 2008.
V. Huang dan F. Kristiani, “Analisis Kesesuaian Hukum Mortalita Gompertz dan Makeham Terhadap Tabel Mortalita Amerika Serikat dan Indonesia,” Prosiding Seminar Nasional Matematika Unpar, vol. 7, no. 2, pp. 63-69, 2012.
R. V. Hogg dan A. T. Craig, Intoduction to Mathematical Statistics, New Jersey: Prentice-Hall, 2013.
A. Ben-Israel, “A Newton-Rhapson Method for the Solution of systems of Equations,” Journal of Mathematical Analysis and Applications, vol. 15, pp. 243-252, 1996.
C. G. Broyden, “A Class of Methods for Solving Nonlinear Simultaneous Equations,” Mathematics of Computation, vol. 19, no. 92, pp. 577-593, 1965.
S. Zhou, X. S. Qian dan P. C. Yi, Probability Theory and Mathematical Statistics, Zhejiang: Higher Education Press, 1988.
P. Andreopoulos, B. G. Fragkiskos, A. Tragaki dan A. Rovolis, “Mortality modeling using probability distributions,” Communications in Statistics - Theory and Methods, vol. 48, no. 1, pp. 127-140, 2018.
DOI: 10.15408/inprime.v1i2.13276
Refbacks
- There are currently no refbacks.
.
