Estimation of the incubation time distribution in the singly and doubly interval censored model

TL;DR

The paper tackles estimating the incubation time distribution $F$ from singly and doubly interval-censored data using a nonparametric maximum likelihood approach. It discretizes $F$ via the means $ar{F}_0(i)= frac{1}{i} ext{ or } extstyleigl(ar{F}_0(i)igr)=ar{F}_0(i-1)+p_0(i)$ and computes the MLE with a support reduction algorithm that accommodates both censoring types. The authors derive $ oot n$-consistent, asymptotically Gaussian limits for the discretized parameters and provide practical confidence intervals via the inverse observed Fisher information or bootstrap; they also provide explicit procedures for both singly and doubly interval-censored cases. Their results establish a robust, nonparametric alternative to parametric incubation-time modeling and deliver actionable CI methods for deconvolution under interval censoring, with accompanying software tools. Key takeaways include the feasibility and stability of the nonparametric MLE for $ar{F}_0$ and the ability to obtain valid inference without relying on specific parametric families.

Abstract

We analyze nonparametric estimators for the distribution function of the incubation time in the singly and doubly interval censoring model. The classical approach is to use parametric families like Weibull, log-normal or gamma distributions in the estimation procedure. We propose nonparametric estimates which stay closer to the data than the classical parametric methods. We also give explicit limit distributions for discrete versions of the models and apply this to compute confidence intervals. The methods complement the analysis of the continuous model. R scripts for computation of the estimates are provided on https://github.com/pietg/incubationtime.

Figures (7)

• Figure 1: Singly interval censored data. $E$ is the end of the exposure time, $S$ the time of becoming symptomatic, $I$ infection time and $U$ the (length of the) incubation time. We only can observe $E$ and $S$.
• Figure 2: (a) Box plot of $1000$ nonparametric and parametric estimates of $\int_5^{6} F_a(x)\,dx$, where $F_a$ is the truncated exponential distribution function, defined by (\ref{['F_a']}), with $a=6$, and where the time variables of the sample are rounded to the nearest upper integer. (b) Box plot of $1000$ nonparametric estimates of $\int_5^{6} F_a(x)\,dx$ and $1000$ parametric estimates of $F_a(6)$, where the time variables are rounded in the same way in the samples. In both cases, the red line segment shows the value of the real $\int_5^6 F_a(x)\,dx$.
• Figure 3: Doubly interval censored data. $S$ the time of becoming symptomatic, $I$ infection time and $U$ the (length of the) incubation time. We can only observe $E$ and the interval $[S_L,S_R]$, containing $S$.
• Figure 4: Estimates of the variances of the $\hat{F}_n(i)$ by $n$ times the actual variances of 1000 samples (blue, solid) and by means of the inverses of the observed Fisher information matrices over the $1000$ samples (dashed, red), (a) for sample size $n=1000$ and (b) for sample size $n=10,000$. We used linear interpolation between the values at the points $1,2,\dots$.
• Figure 5: (a) 95% confidence intervals in the singly interval censored model, using (\ref{['conf_int_CLS']}), for the values of $\bar{F}_0(i)=\int_{i-1}^{i} F_0(x)\,dx$ (red dots and linearly interpolated dashed red curve) at the points $3,4,\dots,10$ for a sample of size $n=1000$, where $F_0$ is the Weibull distribution function, with parameters $a=3.035$ and $b=0.0026$, truncated at $M_1=15$. The black dots are the values of $\hat{F}_n$ at these points. (b) Coverage percentages of the 95% confidence intervals at the points $3,4,\dots,10$, using (\ref{['conf_int_CLS']}), for sample size $n=1000$.

Theorems & Definitions (10)

• Lemma 1: Consistency of $\hat{F}_n$ for $\bar{F}_0$
• proof
• Theorem 1
• Remark 1
• Remark 2
• Theorem 2
• Lemma 2: Fenchel duality conditions for minimization on a cone
• proof : Proof of Lemma \ref{['lemma:fenchel']}
• proof : Proof of Theorem \ref{['th:local_limit_discrete']}
• proof : Proof of Theorem \ref{['th:local_limit_discrete_double']}