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The first hitting time of a boundary or threshold by the sample path of a stochastic process is the central concept of threshold regression models for survival data analysis. Regression functions for the process and threshold parameters in these models are multivariate combinations of explanatory variates. The stochastic process under investigation may be a univariate stochastic process or a multivariate stochastic process. The stochastic processes of interest to us in this report are those that possess stationary independent increments (i.e., Lévy processes) as well as the Esscher property. The Esscher transform is a transformation of probability density functions that has applications in actuarial science, financial engineering, and other fields. Lévy processes with this property are often encountered in practical applications. Frequently, these applications also involve a ‘cure rate’ fraction because some individuals are susceptible to failure and others not. Cure rates may arise endogenously from the model alone or exogenously from mixing of distinct statistical populations in the data set. We show, using both theoretical analysis and case demonstrations, that model estimates derived from typical survival data may not be able to distinguish between individuals in the cure rate fraction who are not susceptible to failure and those who may be susceptible to failure but escape the fate by chance. The ambiguity is aggravated by right censoring of survival times and by minor misspecifications of the model. Slightly incorrect specifications for regression functions or for the stochastic process can lead to problems with model identification and estimation. In this situation, additional guidance for estimating the fraction of non-susceptibles must come from subject matter expertise or from data types other than survival times, censored or otherwise. The identifiability issue is confronted directly in threshold regression but is also present when applying other kinds of

n an infectious disease cohort study, individuals who have been infected with apathogenare often recruited for follow up. The period between infection and the onset of symptomatic disease, referred to as the incubation period, is of interest because of its importance on disease surveillance and control. However, the incubation period is often difficult to ascertain due to the uncertainty associated with asymptomatic infection onset time. An additional complication is that the observed infected sub-jects are likely to have longer incubation periods due to the prevalent sampling. In this paper, we demonstrate how to estimate the distribution of the incubation period with the uncertain infection onset, subject to left-truncation and right-censoring. We employ a family of sufficiently general parametric models, the generalized odds-rate class of regression models, for the underlying incubation period and its correlation with covariates. In simulation studies, we assess the finite sample performance of the model fitting and hazard function estimation. The proposed method is illustrated on data from the HIV/AIDS study on injection drug users admitted to a detoxification program in Badalona, Spain, Ministerio de Ciencia e Innovación, PID2019‐104830RB‐I00; Departament d'Economia i Coneixement de la Generalitat de Catalunya, 2017 SGR 622 (GRBIO); Ministerio de Economía y Competitividad, MTM2015‐64465‐C2‐1‐R (MINECO/FEDER); National Cancer Institute, CA016672; CA193878, Peer Reviewed, Postprint (author's final draft)

The incubation period of coronavirus disease 2019 (COVID-19) is not always observed exactly due to uncertain onset times of infection and disease symptoms. In this paper, we demonstrate how to estimate the distribution of incubation and its association with patient demographic factors when the exact dates of infection and symptoms’ onset may not be observed. The findings from analysis of the confirmed COVID-19 cases indicate that age could be associated with the incubation period, and an age-specific quarantine policy might be more efficient than a unified one in confining COVID-19., This research was funded by the Ministerio de Ciencia e Innovación (Spain) (PID2019-104830RB-I00),the Ministerio de Economía y Competitividad (Spain) (MTM2015-64465-C2-1-R (MINECO/FEDER)), and theDepartament d’Economia i Coneixement de la Generalitat de Catalunya (Spain) (2017 SGR 622 (GRBIO))., Peer Reviewed, Postprint (author's final draft)

In this study the interval censoring case 2 model combined with the Cox model is considered. The event-time distribution function is modeled nonparametrically. Two algorithms are proposed to estimate the event-time distribution function together with the Cox coefficients. Kernel smoothing is applied to the non-parametric MLE of the event-time distribution resulting in the smoothed MLE (SMLE). A two-step method for choosing the smoothing bandwidth based on minimising the MSE is introduced. Given the SMLE, the precision of the MLE is tested using bootstrap simulations. New event-times are sampled from the SMLE which are then used to compute bootstrap estimates of the event-time distribution. This is done for multiple sample sizes to observe large sample behaviour. This study suggests that larger sample sizes lead to better estimates. Monte Carlo simulations and the bootstrap simulations agree on the bandwidth and the large sample distribution of pointwise estimates of the event-time distribution., Applied Mathematics

In most experimental and observational studies, participants are not followed in continuous time. Instead, data is collected about participants only at certain monitoring times. These monitoring times are random and often participant specific. As a result, outcomes are only known up to random time intervals, resulting in interval-censored data. In contrast, when estimating variable importance measures on interval-censored outcomes, practitioners often ignore the presence of interval censoring, and instead treat the data as continuous or right-censored, applying ad hoc approaches to mask the true interval censoring. In this article, we describe targeted minimum loss-based estimation (TMLE) methods tailored for estimation of binary variable importance measures with interval-censored outcomes. We demonstrate the performance of the interval-censored TMLE procedure through simulation studies and apply the method to analyze the effects of a variety of variables on spontaneous hepatitis C virus clearance among injection drug users, using data from the "International Collaboration of Incident HIV and HCV in Injecting Cohorts" project.

We study three estimators for the interval censoring case 2 problem, a histogram-type estimator, the maximum likelihood estimator (MLE) and the smoothed MLE, using a smoothing kernel. Our focus is on the asymptotic distribution of the estimators at a fixed point. The estimators are compared in a simulation study., Delft Institute of Applied Mathematics, Electrical Engineering, Mathematics and Computer Science