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We study nonparametric estimation of convex regression and density functions by methods of least squares (in the regression and density cases) and maximum likelihood (in the density estimation case). We provide characterizations of these estimators, prove that they are consistent and establish their asymptotic distributions at a fixed point of positive curvature of the functions estimated. The asymptotic distribution theory relies on the existence of an \"invelope function\" for integrated two-sided Brownian motion +t4which is established in a companion paper by Groeneboom, Jongbloed and Wellner.

We study the problem of testing for equality at a fixed point in the setting of nonparametric estimation of a monotone function. The likelihood ratio test for this hypothesis is derived in the particular case of interval censoring (or current status data) and its limiting distribution is obtained. The limiting distribution is that of the integral of the difference of the squared slope processes corresponding to a canonical version of the problem involving Brownian motion +t2and greatest convex minorants thereof. Inversion of the family of tests yields pointwise confidence intervals for the unknown distribution function. We also study the behavior of the statistic under local and fixed alternatives.

A clean, closed form, joint density is derived for Brownian motion, its least concave majorant, and its derivative, all at the same fixed point. Some remarkable conditional and marginal distributions follow from this joint density. For example, it is shown that the height of the least concave majorant of Brownian motion at a fixed time point has the same distribution as the distance from the Brownian motion path to its least concave majorant at the same fixed time point. Also, it is shown that conditional on the height of the least concave majorant of Brownian motion at a fixed time point, the left-hand slope of the least concave majorant of Brownian motion at the same fixed time point is uniformly distributed.

A process associated with integrated Brownian motion is introduced that characterizes the limit behavior of nonparametric least squares and maximum likelihood estimators of convex functions and convex densities, respectively. We call this process \"the invelope\" and show that it is an almost surely uniquely defined function of integrated Brownian motion. Its role is comparable to the role of the greatest convex minorant of Brownian motion plus a parabolic drift in the problem of estimating monotone functions. An iterative cubic spline algorithm is introduced that solves the constrained least squares problem in the limit situation and some results, obtained by applying this algorithm, are shown to illustrate the theory.

In observational studies, potential confounders may distort the causal relationship between an exposure and an outcome. However, under some conditions, a causal dose–response curve can be recovered by using the G-computation formula. Most classical methods for estimating such curves when the exposure is continuous rely on restrictive parametric assumptions, which carry significant risk of model misspecification. Non-parametric estimation in this context is challenging because in a non-parametric model these curves cannot be estimated at regular rates. Many available non-parametric estimators are sensitive to the selection of certain tuning parameters, and performing valid inference with such estimators can be difficult.We propose a non-parametric estimator of a causal dose–response curve known to be monotone. We show that our proposed estimation procedure generalizes the classical least squares isotonic regression estimator of a monotone regression function. Specifically, it does not involve tuning parameters and is invariant to strictly monotone transformations of the exposure variable. We describe theoretical properties of our proposed estimator, including its irregular limit distribution and the potential for doubly robust inference. Furthermore, we illustrate its performance via numerical studies and use it to assess the relationship between body mass index and immune response in human immunodeficiency virus vaccine trials.

A general non-asymptotic framework, which evaluates the performance of any procedure at individual functions, is introduced in the context of estimating convex functions at a point. This framework, which is significantly different from the conventional minimax theory, is also applicable to other problems in shape constrained inference. A benchmark is provided for the mean squared error of any estimate for each convex function in the same way that Fisher Information depends on the unknown parameter in a regular parametric model. A local modulus of continuity is introduced and is shown to capture the difficulty of estimating individual convex functions. A fully data-driven estimator is proposed and is shown to perform uniformly within a constant factor of the ideal benchmark for every convex function. Such an estimator is thus adaptive to every unknown function instead of to a collection of function classes as is typical in the nonparametric function estimation literature.

Assume we observe a finite number of inspection times together with information on whether a specific event has occurred before each of these times. Suppose replicated measurements are available on multiple event times. The set of inspection times, including the number of inspections, may be different for each event. This is known as mixed case interval censored data. We consider Bayesian estimation of the distribution function of the event time while assuming it is concave. We provide sufficient conditions on the prior such that the resulting procedure is consistent from the Bayesian point of view. We also provide computational methods for drawing from the posterior and illustrate the performance of the Bayesian method in both a simulation study and two real datasets.