A non-parametric visual test of mixed hazard models

Mortality models are increasingly used to answer a number of pension related questions. They have always been of importance when calculating appropriate prices of risk products depending on individuals' survival. More recently, mortality models are being used in complex models that assess the value of financial products incorporating survival in a variety of ways. Therefore we see that actuaries are not the only financial users of mortality models nowadays. Investors looking for opportunities in survival bonds and other packages of survival risks are also using them, and with different purposes of mortality models we see different measures of quality.

In this paper we develop a visualisation technique useful for the individual assessment of the quality of a mortality model. This could be used for forecasting of mortalities, which is a basic building block for the financial pricing of survival, but also as a useful tool in asset liability management of pension portfolios. Relatively simple parametric mortality models including calendar effects are typically used as starting point for mortality forecasts. The calendar effect is the explicit tool for the forecast and is often isolated and estimated through standard time series methodology. A perfect historical fit of the past is therefore not always what the mortality modeler is looking for. Often it is more important to have an overall good fit, without too systematic deviations giving reliable and meaningful forecasts. These latter objectives are not easy to generalise to some quantitative model that can be tested. Often simple mortality models are rejected, simply because mortality data is sufficiently abundant to inform relatively complex underlying parametric structures. Therefore, a test rejecting our simple model is often not what we want. We do know that our simple model is not accurate but we do not want an excessive fit. What we want is a good, intuitive and reliable forecast.

When modelling mortality of a population, there are a variety of potentially suitable lifetime data models available. Potential models differ in levels of complexity and in the features of data they to capture. Specific parametric life tables combined with time series forecasts are omnipresent in the actuarial and demographic literature. The stability of the forecast depends crucially on the choice of the parametric form. Generally, a complex model with many parameters is not a good choice. even though such models might be selected from classical mathematical statistical model selection designed for in-sample prediction. Models with many parameters generally fit data better than models with fewer parameters. On the other hand, a large number of parameters are harder to forecast than fewer parameters. Forecasting uncertainty increases dramatically with the number of parameters. To obtain reliable forecasts therefore we want models which describe the key features of data with as few parameters as possible.

The purpose of this paper is to introduce a visual diagnostic tool which can be used as a guide when choosing a parametric model. A good parametric model is a simple model without obvious systematic errors. That model could be chosen by the well informed statistician working with the particular mortality forecast application in mind. Our visual diagnostic tool will be just one helpful tool in the overall mathematical statistical toolbox. Our method is inspired from recent developments in extreme value estimation, where transformations of data give visual information on the quality of the distributional fit in the tail. This recent methodology has found its way into insurance pricing and also the related field of operational risk.

The transformation based method can compare the performance of several candidate models for a data set at hand. If we had definitive knowledge of what the exact true distribution is, we would transform our data using this information so that it would exactly originate from an uniform distribution. Obviously this knowledge is not available. However, if we take some estimated parametrically fitted survival distribution as defining our transformation, then any detectable deviance on the transformed scale from the uniform distribution implies deviances of the parametric distribution used in the transformation step from the underlying true distribution. Our methodology uses a nonparametric smooth kernel estimator on the transformed scale. One difficulty we meet here is that our data is classical survival data that is not independent identically distributed. We therefore use a recent local linear kernel density estimator that is adjusted for the truncation and censoring pattern we meet in our data. Comparison between different underlying suggested parametric models are carried out by first estimating these parametric models and then to investigate through visual inspection, whether the density of the transformed data indeed looks uniform. Specifically we use a local linear density estimator that should be working well at our survival type of data that can be viewed as truncated and censored samples approximating a uniform distribution.

If the underlying parametric model under investigation would be true, the estimated density should be close to one over the unit interval. Therefore different underlying parametric models can be visualised and compared on the transformed scale. In principle, the densities could also be estimated and compared on the original scale. However, there are several visual and estimational advantages to working on the transformed scale. One of these is that our method makes maximal use of sparse and volatile data and is thus particularly well suited to explore how potential models describe the mortality at advanced ages where exposure is invariably limited. We test our method in the one-dimensional space using data from nations of different size: USA, United Kingdom, Denmark and Iceland.

The complete executive summary for this research is available for download below. The complete research paper is now available for download from SORT (Statistics and Operational Research Transactions).