arXiv subjects

Anne-Marie Toparkus

Publications and source records attributed to Anne-Marie Toparkus.

3 recordsLinked to original sources

arXiv2025

A Kolmogorov-Smirnov-Type Test for Dependently Double-Truncated Data

With double-truncated lifespans, we test the hypothesis of a parametric distribution family for the lifespan. The typical finding from demography is an instationary behaviour of the life expectancy, and a copula models the resulting weak dependence of lifespan and the age at truncation. Our main example is the Farlie-Gumbel-Morgenststern copula. The test is based on Donsker-class arguments and the functional delta method for empirical processes. The assumptions also allow parametric inference, and proofs slightly simplify due to the compact support of the observations. An algorithm with finitely many operations is given for the computation of the test statistic. Simulations becomes necessary for computing the critical value. With the exponential distribution as an example, and for the application to 55{,}000 German double-truncated enterprise lifespans, the constructed Kolmogorov-Smirnov test rejects clearly an age-homogeneous closure hazard.

stat.ME↗

arXiv2025

Censored lifespans in a double-truncated sample: Maximum likelihood inference for the exponential distribution

The analysis of a truncated sample can be hindered by censoring. Survival information may be lost to follow-up or the birthdate may be missing. The data can still be modeled as a truncated point process and it is close to a Poisson process, in the Hellinger distance, as long as the sample is small relative to the population. We assume an exponential distribution for the lifespan, derive the likelihood and profile out the unobservable sample size. Identification of the exponential parameter is shown, together with consistency and asymptotic normality of its M-estimator. Even though the estimator sequence is indexed in the sample size, both the point estimator and the standard error are observable. Enterprise lifespans in Germany constitute our example.

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arXiv2024

Testing Truncation Dependence: The Gumbel-Barnett Copula

In studies on lifetimes, occasionally, the population contains statistical units that are born before the data collection has started. Left-truncated are units that deceased before this start. For all other units, the age at the study start often is recorded and we aim at testing whether this second measurement is independent of the genuine measure of interest, the lifetime. Our basic model of dependence is the one-parameter Gumbel-Barnett copula. For simplicity, the marginal distribution of the lifetime is assumed to be Exponential and for the age-at-study-start, namely the distribution of birth dates, we assume a Uniform. Also for simplicity, and to fit our application, we assume that units that die later than our study period, are also truncated. As a result from point process theory, we can approximate the truncated sample by a Poisson process and thereby derive its likelihood. Identification, consistency and asymptotic distribution of the maximum-likelihood estimator are derived. Testing for positive truncation dependence must include the hypothetical independence which coincides with the boundary of the copula's parameter space. By non-standard theory, the maximum likelihood estimator of the exponential and the copula parameter is distributed as a mixture of a two- and a one-dimensional normal distribution. The application are 55 thousand double-truncated lifetimes of German businesses that closed down over the period 2014 to 2016. The likelihood has its maximum for the copula parameter at the parameter space boundary so that the $p$-value of test is $0.5$. The life expectancy does not increase relative to the year of foundation. Using a Farlie-Gumbel-Morgenstern copula, which models positive and negative dependence, finds that life expectancy of German enterprises even decreases significantly over time.

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