Asymptotic Theory for Multiple-Sample Semiparametric Density Ratio Models and Its Application to Mortality Forecasting

Abstract

A multiple-sample semiparametric density ratio model, which is equivalent to a generalized logistic regression model, can be constructed by multiplicative exponential distortions of a reference distribution. Distortion functions are assumed to be nonnegative and of a known finite-dimensional parametric form, and the reference distribution is left as nonparametric. The combined data from all the samples are used in the semiparametric large sample problem of estimating each distortion and the reference distribution. The large sample behavior for both the parameters and the unknown reference distribution are studied. The estimated reference cumulative distribution function is proved to converge weakly to a zero-mean Gaussian process, whose covariance structure provides confidence bands for the reference distribution function. A Kolmogorov-Smirnov type statistic for a goodness-of-fit test of the density ratio model is also studied.

In the second part, an approach to modeling and forecasting age-specific mortality in the United States is provided. The approach is based on an extension of a class of semiparametric models to time series. The method combines information from several time series and estimates their predictive distributions conditional on past data. The conditional expectation, the most common predictor, is obtained as a by product from the first moment of the predictive distribution. The confidence band of the predictor is obtained by applying the asymptotic results of the semiparametric density ratio model. A comparison of short term prediction is made between the semiparametric method and the well known method of Lee and Carter \cite{LC(1992)}. Judging by the mean square error (MSE) of prediction for all ages, the semiparametric method reduces the overall MSE appreciably.