Scott Morris and Dr. Scott Grimshaw, Statistics
Introduction
The purpose of our project was to model a person’s life expectancy using lifetimes from their ancestry. Typical mortality models are based on data from large populations, and are not necessarily representative of a specific individual. Our primary objective was to model my personal life expectancy based on the men in my own ancestry by looking at their ages at death. The results from the study are unique to my own life, but the methodology applies to anyone who gathers their own ancestral data. Furthermore, these results can be useful in the medical field and in fnancial and retirement planning.
Weibull Distribution
The core of this study used properties of estimators of the statistical Weibull distribution. Waloddi Weibull was a Swedish physicist in the 1930’s who derived a distribution to model fatigue and the breaking strength of materials. His “Weibull” distribution is now commonly used in time-to-failure applications. Defining human death as a “failure,” this distribution was the perfect choice for our model’s foundation.
The parameters Y and B are respectively known as scale and shape parameters. Scale parameters aff ect the spread, or variability, of the distribution, while shape parameters aff ect the shape of the distribution. We estimated the parameters of the Weibull distribution using four diff erent methods. Our goal was to use these estimated model parameters to ultimately estimate the expected value, or mean, of the distribution. The expected value estimates the age at which I will die.
Methods of Estimation
We studied four different methods of estimation; Maximum Likelihood, Method of Moments, a literature estimator, and my own derivation of estimators for and . The Method of Moments and my own estimators were the least beneficial estimators, so we further discuss the Maximum Likelihood and Literature estimators.
Simulation
We conducted a simulation to characterize the bias and Mean Squared Error (MSE) of the four estimators. The gure below left shows the bias and MSE of the Maximum Likelihood and Literature estimators. The gure below-right shows the distribution of the ages at death of the men in my ancestry. Sample sizes of n = 50; 100 and 150 were tested because it is likely that many people will be able to nd information about 50 to 150 of their ancestors.
Conclusions
The bias and MSE are clearly greater for the B parameters than for the Y parameters. Based on results from plugging in my own ancestry data to the Maximum Likelihood and Literature methods’ estimates, we predicted that I will live between 79 and 80 years! This result is not too surprising, given the distribution of the ages at death of the men in my ancestry.