Abstract: The distribution of the incubation period of the novel coronavirus disease that emerged in 2019 (COVID-19) has crucial clinical implications for understanding this disease and devising effective disease-control measures. Our study is based on a cross-sectional and forward follow-up study that collected the duration times between a specific observation time and the onset of COVID-19 symptoms for individuals. The original study further proposed a mixture forward-incubation-time epidemic model, which is a mixture of an incubation-period distribution and a forward time distribution, to model the collected duration times and to estimate the incubation-period distribution of COVID-19. In this study, we provide sufficient conditions for the identifiability of the unknown parameters in the aforementioned epidemic model when the incubation period follows a two-parameter distribution. Under the same setup, we propose a likelihood ratio test (LRT) for testing the null hypothesis that the mixture forward-incubation-time epidemic model is a homogeneous exponential distribution. The testing problem is nonregular because a nuisance parameter is present only under the alternative. We establish the limiting distribution of the LRT and identify an explicit representation for it, and obtain the limiting distribution of the LRT under a sequence of local alternatives. Our simulation results indicate that the LRT exhibits desirable type-I errors and power. Lastly, we analyze a COVID-19 outbreak data set from China to illustrate the usefulness of the LRT.

Key words and phrases: Identifiability, likelihood ratio test, non-regularity.