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 for functions of the observations, 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 in Groeneboom (2021, 2023). R scripts for computation of the estimates are provided in Groeneboom (2020).

We construct bootstrap confidence intervals for a monotone regression function. It has been shown that the ordinary nonparametric bootstrap, based on the nonparametric least squares estimator (LSE) (Formula presented.), is inconsistent in this situation. We show that an (Formula presented.) -consistent bootstrap can be based on the smoothed (Formula presented.), to be called the SLSE (Smoothed Least Squares Estimator). The asymptotic pointwise distribution of the SLSE is derived. The confidence intervals, based on the smoothed bootstrap, are compared to intervals based on the (not necessarily monotone) Nadaraya Watson estimator and the effect of Studentization is investigated. We also give a method for automatic bandwidth choice, correcting work in Sen and Xu (2015). Analogous methods for constructing confidence intervals in the current status model are discussed, improving on work in Groeneboom and Hendrickx (2018).

We consider least squares estimators of the finite regression parameter α in the single index regression model Y = ψ(α^T X) + ε, where X is a d-dimensional random vector, E(Y|X) = ψ(α^T X), and ψ is a monotone. It has been suggested to estimate α by a profile least squares estimator, minimizing ±∑ⁿ_i=1(Y_i - ψ(α^T X_i))² over monotone ψ and α on the boundary S_d-1 of the unit ball. Although this suggestion has been around for a long time, it is still unknown whether the estimate is √n-convergent. We show that a profile least squares estimator, using the same pointwise least squares estimator for fixed α, but using a different global sum of squares, is √n-convergent and asymptotically normal. The difference between the corresponding loss functions is studied and also a comparison with other methods is given.

Let (Formula presented.) be the nonparametric maximum likelihood estimator of a decreasing density. Grenander characterized this as the left-continuous slope of the least concave majorant of the empirical distribution function. For a sample from the uniform distribution, the asymptotic distribution of the L₂-distance of the Grenander estimator to the uniform density was derived in an article by Groeneboom and Pyke by using a representation of the Grenander estimator in terms of conditioned Poisson and gamma random variables. This representation was also used in an article by Groeneboom and Lopuhaä to prove a central limit result of Sparre Andersen on the number of jumps of the Grenander estimator. Here we extend this to the proof of the main result on the L₂-distance of the Grenander estimator to the uniform density and also prove a similar asymptotic normality results for the entropy functional. Cauchy's formula and saddle point methods are the main tools in our development.

We consider smooth nonparametric estimation of the incubation time distribution of COVID-19, in connection with the investigation of researchers from the National Institute for Public Health and the Environment (Dutch: RIVM) of 88 travelers from Wuhan: Backer et al. (2020). The advantages of the smooth nonparametric approach with respect to the parametric approach, using three parametric distributions (Weibull, log-normal and gamma) in Backer et al. (2020) is discussed. It is shown that the typical rate of convergence of the smooth estimate of the density is n^2/7 in a continuous version of the model, where n is the sample size. The (nonsmoothed) nonparametric maximum likelihood estimator itself is computed by the iterative convex minorant algorithm (Groeneboom and Jongbloed (2014)). All computations are available as R scripts in Groeneboom (2020a).

In carcinogenicity experiments with animals where the tumor is not palpable it is common to observe only the time of death of the animal, the cause of death (the tumor or another independent cause, as sacrifice) and whether the tumor was present at the time of death. These last two indicator variables are evaluated after an autopsy. Defining the non-negative variables T₁ (time of tumor onset), T₂ (time of death from the tumor) and C (time of death from an unrelated cause), we observe (Y,Δ1,Δ2), where Y = min{T₂,C},Δ₁ =1 {T₁≤C}, and Δ₂ =1 {T₂≤C}. The random variables T₁ and T₂ are independent of C and have a joint distribution such that P(T₁ ≤ T₂) = 1. Some authors call this model a “survival-sacrifice model”. [20] (generally to be denoted by LJP (1997)) proposed a Weighted Least Squares estimator for F₁ (the marginal distribution function of T₁), using the Kaplan-Meier estimator of F₂ (the marginal distribution function of T₂). The authors claimed that their estimator is more efficient than the MLE (maximum likelihood estimator) of F₁ and that the Kaplan-Meier estimator is more efficient than the MLE of F₂. However, we show that the MLE of F₁ was not computed correctly, and that the (claimed) MLE estimate of F₁ is even undefined in the case of active constraints. In our simulation study we used a primal-dual interior point algorithm to obtain the true MLE of F₁. The results showed a better performance of the MLE of F₁ over the weighted least squares estimator in LJP (1997) for points where F₁ is close to F₂. Moreover, application to the model, used in the simulation study of LJP (1997), showed smaller variances of the MLE estimators of the first and second moments for both F₁ and F₂, and sample sizes from 100 up to 5000, in comparison to the estimates, based on the weighted least squares estimator for F1, proposed in LJP (1997), and the Kaplan-Meier estimator for F₂. R scripts are provided for computing the estimates either with the primal-dual interior point method or by the EM algorithm. In spite of the long history of the model in the biometrics literature (since about 1982), basic properties of the real maximum likelihood estimator (MLE) were still unknown. We give necessary and sfficient conditions for the MLE (Theorem 3.1), as an element of a cone, where the number of generators of the cone increases quadratically with sample size. From this and a self-consistency equation, turned into a Volterra integral equation, we derive the consistency of the MLE (Theorem 4.1). We conjecture that (under some natural conditions) one can extend the methods, used to prove consistency, to proving that the MLE is √n consistent for F₂ and cube root n convergent for F₁, but this has presently not yet been proved.

Single-index models are popular regression models that are more flexible than linear models and still maintain more structure than purely nonparametric models. We consider the problem of estimating the regression parameters under a monotonicity constraint on the unknown link function. In contrast to the standard approach of using smoothing techniques, we review different "non-smooth" estimators that avoid the difficult smoothing parameter selection. For about 30 years, one has had the conjecture that the profile least squares estimator is an n-consistent estimator of the regression parameter, but the only non-smooth argmin/argmax estimators that are actually known to achieve this n-rate are not based on the nonparametric least squares estimator of the link function. However, solving a score equation corresponding to the least squares approach results in n-consistent estimators. We illustrate the good behavior of the score approach via simulations. The connection with the binary choice and current status linear regression models is also discussed.

We construct n-consistent and asymptotically normal estimates for the finite dimensional regression parameter in the current status linear regression model, which do not require any smoothing device and are based on maximum likelihood estimates (MLEs) of the infinite dimensional parameter. We also construct estimates, again only based on these MLEs, which are arbitrarily close to efficient estimates, if the generalized Fisher information is finite. This type of efficiency is also derived under minimal conditions for estimates based on smooth nonmonotone plug-in estimates of the distribution function. Algorithms for computing the estimates and for selecting the bandwidth of the smooth estimates with a bootstrap method are provided. The connection with results in the econometric literature is also pointed out.

We discuss a new way of constructing pointwise confidence intervals for the distribution function in the current status model. The confidence intervals are based on the smoothed maximum likelihood estimator, using local smooth functional theory and normal limit distributions. Bootstrap methods for constructing these intervals are considered. Other methods to construct confidence intervals, using the non-standard limit distribution of the (restricted) maximum likelihood estimator, are compared with our approach via simulations and real data applications.

We consider estimation in the single-index model where the link function is monotone. For this model, a profile least-squares estimator has been proposed to estimate the unknown link function and index. Although it is natural to propose this procedure, it is still unknown whether it produces index estimates that converge at the parametric rate. We show that this holds if we solve a score equation corresponding to this least-squares problem. Using a Lagrangian formulation, we show how one can solve this score equation without any reparametrization. This makes it easy to solve the score equations in high dimensions. We also compare our method with the effective dimension reduction and the penalized least-squares estimator methods, both available on CRAN as R packages, and compare with link-free methods, where the covariates are elliptically symmetric.

We give a direct derivation of the distribution of the maximum and the location of the maximum of one-sided and two-sided Brownian motion with a negative parabolic drift. The argument uses a relation between integrals of special functions, in particular involving integrals with respect to functions which can be called "incomplete Scorer functions". The relation is proved by showing that both integrals, as a function of two parameters, satisfy the same extended heat equation, and the maximum principle is used to show that these solutions must therefore have the stated relation. Once this relation is established, a direct derivation of the distribution of the maximum and location of the maximum of Brownian motion minus a parabola is possible, leading to a considerable shortening of the original proofs.