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Estimating Hazard Functions for Discrete Lifetimes

ABSTRACT Frequently in inference, the observed data are modeled as a sample from a continuous probability model, implying the observed data are precisely measured. Usually, the actual data available to the investigator are discrete–-either because they are rounded, meaning the exact measurement is within an interval defined by some small measurement unit related to the precision of the measuring device, or because the data are discrete, meaning the time periods until the event of interest are countable instead of continuous. This article is motivated by the common practice of testing for duration dependence (non constant hazard function) in economic and financial data using the continuous Weibull distribution when the data are discrete. A simulation study shows that biased parameter estimates and distorted hypothesis tests result when the degree of discretization is severe. When observations are rounded, as in measuring the time between stock trades, it is proper to treat them as interval-censored. When observations are discrete, as in measuring the length of stock runs, a discrete hazard function must be specified. Both cases are examined in simulation studies and demonstrated on financial data.