Fit of the 2011 Indonesian Mortality Table to Gompertz Law and Makeham Law using Maximum Likelihood Estimation

Dino Agustin Putra, Nina Fitriyati, Mahmudi Mahmudi

Abstract


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.


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DOI: 10.15408/inprime.v1i2.13276

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