420 research outputs found
Functional Cramér–Rao bounds and Stein estimators in Sobolev spaces, for Brownian motion and Cox processes
No full textWe 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 H01⊂Wα,2⊂L2. In both situations, Cramér–Rao lower bounds are obtained, entailing in particular that no unbiased estimators (not necessarily adapted) with finite risk in H01 exist. By Malliavin calculus techniques, we also study super-efficient Stein type estimators (in the Gaussian case)
Malliavin Calculus with Applications to Statistical Inference
No full textThe 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
Smoothed Isotonic Estimators of a Monotone Baseline Hazard in the Cox Model
No full textWe 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.Statistic
Coupled PDE for Ultrasound Despeckling Using ENI Classification
No full textAbstractSpeckle is a type of noise which is often present in ultrasound images. Speckle is formed due to constructive or destructive interference of ultrasound waves. Due to the granular pattern of speckle noise, it hides important details in ultrasound images. Many despeckling techniques are proposed in the literature, but most of them fail to reach a balance between the removal of speckle noise and preservation of the fine details in the image. In this work, an improved coupled PDE model is proposed which combines second order selective degenerate diffusion (SDD) model and fourth order PDE model based on the assumption that speckle in ultrasound image follows Gamma distribution. An edge noise interior (ENI) method is used to control the diffusion. With the help of ENI controlling function, the diffusion at edge pixels and noisy pixels are selectively accomplished with varying speed. Thus, the proposed model preserves the edges and fine texture details in the image. The model is tested on simulated images after corrupting the images with various levels of Gamma noise. Further, we have tested it on real ultrasound images also. The performance of the proposed model is compared with other similar techniques and the proposed method outperforms other state-of-the-art methods, both in terms of qualitative and quantitative measures
Petrolio e Mezzogiorno. Lo sviluppo del Sud nel cinema industriale: il caso Eni (1960-1965)
No full textThe Eni (Ente nazionale idrocarburi), the oil and gas state ownes enterprise, founded by Enrico Mattei in 1953, has used many advertising communication’s strategies to diffuse its activities throughout the national territory and abroad. Among the many instruments that the company has used (billboards advertising on the road, magazines, newspapers, house organ “Il gatto selvatico”, television spots “Carosello”) there are also the documentaries that are seen in Italian cinemas. The author explores the representation that Eni gave the south of Italy in the years of the “miracolo economico” (economic miracle) with its industrial documentaries. Eni show himself as one of the key factors for economic and social development of southern Italy and resolve the “Questione meridionale”. The study of documentaries shows that while a country is moving towards modernity, still has problems of poverty and backwardness that even industrial progress can be solved, despite the company advertises itself as the motor of a new history for the Italy
A two-sample comparison of mean survival times of uncured sub-populations
No full textComparing 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 () 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 conditionally on a
given covariate value. An asymptotic and a permutation-based approach have been
developed for making inference on the difference of conditional '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
Uniform moment bounds for the standard estimators in the Cox proportional hazard model
No full textInternational audienceWe consider the Cox regression model and show that the regression parameter estimator and the Breslow estimator for the cumulative hazard have uniformly bounded moments of any order if we restrict to a sequence of events with probability converging to one. These results are needed, for example, when studying global errors of shape restricted estimators of the baseline hazard functio
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