Kolmogorov-Smirnov-type test for dependently double-truncated durations: A copula approach.

Abstract

Given a double-truncated sample of lifespans, we test the hypothesis of a parametric distribution family for the lifespan. Demography typically certifies the life expectancy to be nonstationary in time. We model the resulting dependence between a lifespan and the birthday of an individual with a copula. Our main example is the Farlie-Gumbel-Morgenstern copula. The asymptotic null distribution of the test is based on Donsker-class arguments and the functional delta method for empirical processes. One assumption for the test is consistency in the estimation of the parameter for the model under the null hypothesis. Requirements for consistency are fulfilled by our set of assumptions. The test is carried out in two stages, the computation of the test statistics and the simulation of the critical value. The statistic can be reached after finitely many computation steps, whereas the critical value can only be an approximation. With the exponential distribution as an example for the lifespan distribution, and for the application to 55,000 German double-truncated enterprise lifespans, the constructed Kolmogorov-Smirnov test rejects clearly an age-homogeneous closure hazard.