Abstract:
The Lee-Carter (LC) model was primarily designed for modelling the mortality pattern with Gaussian error structure in most developed countries. Attempts at using LC for describing the pattern of mortality in developing countries have resulted in the violation of the normality assumption, due to its inability to accommodate variability across age-groups. Therefore, this study was aimed at developing a modified LC model to accommodate variability across all ages.
The LC model, ln〖m_xt=a_x+b_x k_t+ε_xt 〗was modified using a gamma generator given as: Gam(y)= {-ln[1-F(y)] }^(α-1)/Γα f(y); where m_xtis a random variable, ln〖 m〗_xt=y,a_x,b_xandk_tare location parameters, ε_xtis an error term, αis a shape parameter, Γα is (α-1)!,f(y)and F(y)are the density and distribution functions of the normal random variable, respectively. The Kolmogorov Smirnov (KS) and Shapiro Wilks’ (SW) procedures were used to test for normality. The Gamma-Normal Lee-Carter (GNLC) model, the LC model with normal error structure and the Brouhns’ (BR) model with poisson error structure were fitted to Nigeria male and female all-cause mortality datasets obtained from the Global Health Observatory for age-groups < 1, 1-4, 5-9, 10-14, 15-19, 20-24, 25-29, 30-34, 35-39, 40-44, 45-49, 50-54, 55-59, 60-64, 65-69, 70-74, 75-79, 80-84 and 85+ for the period 2000-2015. Forecast of the mortality Index k_twas done in an Auto-regressive Integrated Moving Average (ARIMA) framework for a 20 year-period. Simulation study using data of sizes n;t= (15; 5), (20; 20) and (20; 30) was set up, where n and t are the sizes of the age-groups and periods, respectively. Model performance was evaluated using Bayesian Information Criterion (BIC) and the Corrected Akaike Information Criterion (CAIC). Significance was determined at 0.01 level.
The Probability Density Function (PDF), Cummulative Distribution Function (CDF) and Hazard Function (HF) of the GNLC model were derived. The PDF, CDF and HF of the GNLC were of the forms:
g(y)=1/Γα 1/(σ√2π) e^(-1/2 {〖[y-(a_x+b_x k_t )]〗^2/σ^2 } ) (-ln{1-Φ[(y-〖(a〗_x+b_x k_t))/σ] } )^(α-1) ,
G(y)=1/Γα γ(α,-ln{1-Φ[(y-〖(a〗_x+b_x k_t))/σ] } )and
h(y)= (1/(σ√2π) e^(-1/2 {[y-(a_x+b_x k_t ) ]^2/σ^2 } ) (-ln{1-Φ[(y-〖(a〗_x+b_x k_t))/σ] } )^(α-1))/Γ(α, -ln{1-Φ[(y-〖(a〗_x+b_x k_t))/σ] } ) ,respectively. The KS and SW procedures were significant at 0.01, confirming non-normality with P-values less than or equal to (4.35 ×〖10〗^(-13), 2.22 ×〖10〗^(-5)) and (4.18 ×〖10〗^(-13),1.58〖×10〗^(-13)) for males and females, respectively. The BIC and CAIC for the forecast period were: -0.14, -0.55 (LC), -17.19, -17.60 (BR) and -40.99, -41.40 (GNLC) for the male and -7.05, -7.39 (LC), -14.74, -15.04 (BR) and -56.76, -57.06 (GNLC) for the female. For the simulated data, the obtained BIC and CAIC for the LC and GNLC models were for sample size n;t= (15; 5): 285.6405, 263.8951; 281.3217, 257.4478, n;t= (20; 20): 744.1807, 532.7462; 738.1579, 529.9777 and n;t= (20; 30): 997.1916, 726.5509; 990.7245, 713.258, respectively.
The modified Lee-Carter model was able to accommodate variability across all age-groups better than the referenced classical Lee-Carter. Therefore, the modified Lee-Carter model is recommended for modelling mortality data from developing countries.
