Functional Cramer-Rao bounds and Stein estimators in Sobolev spaces, for Brownian motion and Cox processes

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 $H^1_0\subset W^{\alpha,2}\subset L^2$. In both situations, Cramer-Rao lower bounds are obtained, entailing in particular that no unbiased estimators with finite risk in $H^1_0$ exist. By Malliavin calculus techniques, we also study super-efficient Stein type estimators (in the Gaussian case)

Malliavin Calculus with Applications to Statistical Inference

The purpose of this thesis is to investigate the use of Malliavin Calculus in both parametric and nonparametric statistical inference. It is of interest to see how classical statistical results, which rely on the integration by parts formula, can be established in more general settings using Malliavin Calculus techniques. This study essentially consists of two parts: the probabilistic theory of Malliavin Calculus and its applications in statistics. We first provided a collection of the main results concerning Malliavin Calculus on the canonical Wiener space, using the stochastic calculus of variations approach. It is worth mentioning the integration by parts formula, which plays a major role in this context, the Clark-Ocone-Karatzas formula and the derivability in Malliavin sense of the solutions of stochastic differential equations. There are several books and papers which extend the classical Malliavin Calculus to the Poisson space. Using the experience with the Brownian motion and the Poisson process, we introduced Malliavin Calculus for doubly stochastic processes, following again the stochastic calculus of variations approach. In particular we showed that the properties we were interested in, such as the integration by parts formula, the chain rule and an explicit representation of the divergence for deterministic processes, are still valid. Once we had established the basics of Malliavin Calculus, we turned our attention towards exploring its applications in a parametric statistical model relying on a recent paper by José M. Corcuera and A. Kohatsu-Higa. Making use of Malliavin Calculus we derived expressions of the score function as a conditional expectation involving Skorohod integral. Then one can immediately obtain the Fisher Information and the Cramer-Rao lower bound. In most classical models the calculations are straightforward, as the expression of the density is available. The goal was to show that in some cases we can derive such expressions without knowing the likelihood explicitly. In particular we find out that this method is appropriate to study asymptotic properties of continuous time models considering discrete observations of diffusion processes where the driving process is a Brownian motion. Finally, the last part relies on some results established by N. Privault and A. Réveillac but we focused on two particular problems of nonparametric functional estimation: drift estimation for the Brownian motion and intensity estimation for the Cox process. We provided Cramer-Rao bounds and extended Stein's argument for superefficient estimators to an infinite dimensional setting using Malliavin Calculus. In addition we discussed the estimation in fractional Sobolev spaces of the unknown function, which is assumed to belong in the space H^1_0. We argued that, although it would be natural to look for an unbiased estimator which belongs to the same space as the target function, its risk will always be infinite. We also distinguished between the spaces W^{\alpha,2}, where the observation itself turns out to be an efficient estimator and the spaces W^{\alpha,p}, p\in(2,\infty), where it is not. In conclusion, Malliavin Calculus provides a useful instrument for giving alternative expressions of the score function without involving the likelihood function directly. Consequently we obtain the Fisher Information, the Cramer-Rao lower bound and study the asymptotic behaviour of the model. On the other side it enables us to apply Stein's argument in the context of functional estimation

A two-sample comparison of mean survival times of uncured sub-populations

Comparing the survival times among two groups is a common problem in time-to-event analysis, for example if one would like to understand whether one medical treatment is superior to another. In the standard survival analysis setting, there has been a lot of discussion on how to quantify such difference and what can be an intuitive, easily interpretable, summary measure. In the presence of subjects that are immune to the event of interest (`cured'), we illustrate that it is not appropriate to just compare the overall survival functions. Instead, it is more informative to compare the cure fractions and the survival of the uncured sub-populations separately from each other. Our research is mainly driven by the question: if the cure fraction is similar for two available treatments, how else can we determine which is preferable? To this end, we estimate the mean survival times in the uncured fractions of both treatment groups ($MST_u$) and develop permutation tests for inference. In the first out of two connected papers, we focus on nonparametric approaches. The methods are illustrated with medical data of leukemia patients. In Part II we adjust the mean survival time of the uncured for potential confounders, which is crucial in observational settings. For each group, we employ the widely used logistic-Cox mixture cure model and estimate the $MST_u$ conditionally on a given covariate value. An asymptotic and a permutation-based approach have been developed for making inference on the difference of conditional $MST_u$'s between two groups. Contrarily to available results in the literature, in the simulation study we do not observe a clear advantage of the permutation method over the asymptotic one to justify its increased computational cost. The methods are illustrated through a practical application to breast cancer data

A two-sample comparison of mean survival times of uncured subpopulations

Comparing the survival times among two groups is a common problem in time-to-event analysis, for example if one would like to understand whether one medical treatment is superior to another. In the standard survival analysis setting, there has been a lot of discussion on how to quantify such difference and what can be an intuitive, easily interpretable, summary estimand. In the presence of subjects that are immune to the event of interest (‘cured’), we illustrate that it is not appropriate to just compare the overall survival functions. Instead, it is more informative to compare the cure fractions and the survival of the uncured subpopulations separately from each other. Our research is mainly driven by the question: if the cure fraction is similar for two available treatments, how else can we determine which is preferable? To this end, we estimate the mean survival times in the uncured fractions of both treatment groups and develop both permutation and asymptotic tests for inference. We first propose a nonparametric approach which is then extended to account for covariates by means of the semi-parametric logistic-Cox mixture cure model. The methods are illustrated through practical applications to breast cancer and leukemia data.</p