In this thesis we address the problem of estimating a curve of interest (which might be a probability density, a failure rate or a regression function) under monotonicity constraints. The main concern is investigating large sample distributional properties of smooth isotonic estimators, which have a faster rate of convergence and a nicer graphical representation compared to standard isotonic estimators such as the constrained nonparametric maximum likelihood and the Grenander-type estimator. In the first part, we focus on the pointwise behavior of estimators for the hazard rate in the right censoring and Cox regression models, while the second part is dedicated to global errors of estimators in a general setup, which includes estimation of a probability density, a failure rate, or a regression function. We provide central limit theorems and assess the finite sample performance of the estimators by means of simulation studies for constructing confidence intervals and goodness of fit tests. @en

We study the asymptotic behavior of the marginal expected shortfall when the two random variables are asymptotic independent but positively associated, which is modeled by the so-called tail dependent coefficient. We construct an estimator of the marginal expected shortfall, which is shown to be asymptotically normal. The finite sample performance of the estimator is investigated in a small simulation study. The method is also applied to estimate the expected amount of rainfall at a weather station given that there is a once every 100 years rainfall at another weather station nearby.

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We investigate the asymptotic behavior of the Lp-distance between
a monotone function on a compact interval and a smooth estimator
of this function. Our main result is a central limit theorem for the Lp-error
of smooth isotonic estimators obtained by smoothing a Grenander-type
estimator or isotonizing the ordinary kernel estimator. As a preliminary result
we establish a similar result for ordinary kernel estimators. Our results
are obtained in a general setting, which includes estimation of a monotone
density, regression function and hazard rate. We also perform a simulation
study for testing monotonicity on the basis of the L2-distance between the
kernel estimator and the smoothed Grenander-type estimator.@en

We consider Grenander-type estimators for a monotone function (Formula presented.), obtained as the slope of a concave (convex) estimate of the primitive of λ. Our main result is a central limit theorem for the Hellinger loss, which applies to estimation of a probability density, a regression function or a failure rate. In the case of density estimation, the limiting variance of the Hellinger loss turns out to be independent of λ.

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We consider the smoothed maximum likelihood estimator and the smoothed Grenander-type estimator for a monotone baseline hazard rate 0 in the Cox model. We analyze their asymptotic behaviour and show that they are asymptotically normal at rate nm=.2mC1/, when 0 is m 2 times continuously differentiable, and that both estimators are asymptotically equivalent. Finally, we present numerical results on pointwise confidence intervals that illustrate the comparable behaviour of the two methods.

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We consider the process Λ̂_n−Λ_n, where Λ_n is a cadlag step estimator for the primitive Λ of a nonincreasing function λ on [0,1], and Λ̂_n is the least concave majorant of Λ_n. We extend the results in Kulikov and Lopuhaä (2006, 2008) to the general setting considered in Durot (2007). Under this setting we prove that a suitably scaled version of Λ̂_n−Λ_n converges in distribution to the corresponding process for two-sided Brownian motion with parabolic drift and we establish a central limit theorem for the L_p-distance between Λ̂_n and Λ_n.

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We consider two isotonic smooth estimators for a monotone baseline hazard in the Cox model, a maximum smooth likelihood estimator and a Grenander-type estimator based on the smoothed Breslow estimator for the cumulative baseline hazard. We show that they are both asymptotically normal at rate n^m∕(2m+1), where m≥2 denotes the level of smoothness considered, and we relate their limit behavior to kernel smoothed isotonic estimators studied in Lopuhaä and Musta (2016). It turns out that the Grenander-type estimator is asymptotically equivalent to the kernel smoothed isotonic estimators, while the maximum smoothed likelihood estimator exhibits the same asymptotic variance but a different bias. Finally, we present numerical results on pointwise confidence intervals that illustrate the comparable behavior of the two methods.

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We investigate the problems of drift estimation for a shifted Brownian motion and intensity estimation for a Cox process on a finite interval [0,T], when the risk is given by the energy functional associated to some fractional Sobolev space . In both situations, Cramér–Rao lower bounds are obtained, entailing in particular that no unbiased estimators (not necessarily adapted) with finite risk in exist. By Malliavin calculus techniques, we also study super-efficient Stein type estimators (in the Gaussian case).@en

We consider kernel smoothed Grenander-type estimators for a monotone hazard rate and a monotone density in the presence of randomly right censored data. We show that they converge at rate n2/5 and that the limit distribution at a fixed point is Gaussian with explicitly given mean and variance. It is well known that standard kernel smoothing leads to inconsistency problems at the boundary points. It turns out that, also by using a boundary correction, we can only establish uniform consistency on intervals that stay away from the end point of the support (although we can go arbitrarily close to the right boundary).@en