50 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)
Development and application of generalized MUSTA schemes
No full textNumerical methods for solving non-linear systems of hyperbolic conservation laws via finite volume methods or discontinuous Galerkin finite element methods require, as the building block, a monotone numerical flux. The simplest approach for providing a monotone numerical utilizes a symmetric stencil and does not explicitly make use of wave propagation information, giving rise to centred or symmetric schemes. A more refined approach utilizes wave propagation information through the exact or approximate solution of the Riemann problem, giving rise to Godunov methods. Conventional approximate Riemann solvers are usually complex and are not available for many systems of practical interest, such as for models for compressible multi-phase flows. It is thus desirable to construct a numerical flux that emulates the best flux available (upwind) with the simplicity and generality of symmetric schemes. Here we build upon MUSTA approach [2,3], which leads to schemes that have the simplicity and generality of symmetric schemes and the accuracy of upwind schemes. First we present a new flux that is an average of symmetric fluxes and which reproduces the Godunov upwind scheme for the model hyperbolic equation. For non-linear systems it is found that this flux gives superior results to those of the whole family of incomplete Riemann solvers that do not explicitly account for linearly degenerate fields. Then we incorporate this flux into the MUSTA multi-staging approach, as predictor and corrector. It is found that the resulting MUSTA schemes reproduce the Godunov upwind scheme for the model hyperbolic equation for any number of stages, including multiple space dimensions. They are linearly stable in two and three space dimensions and the stability region is identical to that of the Godunov upwind method. For non-linear systems the MUSTA scheme with one or two stages gives results that are indistinguishable from those of Riemann solvers, such as the exact Riemann solver or Roe approximate Riemann solver. Finally, we assess the schemes on carefully chosen test problems for the one-dimensional equations of magneto hydrodynamics and nonlinear elasticity. High-order examples are provided for the multidimensional Euler equations in the framework of finite-volume WENO schemes [1]. The results illustrate the accuracy and efficiency of new methods combined with the ease of coding
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
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
Musta Tritonus : Livemusiikkikuunnelman tehokeinot
No full textOpinnäytetyöprojekti Musta Tritonus -livemusiikkikuunnelma yhdistää valmiiksi äänitetyn kuunnelman elävään musiikkiin. Projektin tekijä vastasi käsikirjoituksesta, säveltämisestä,
äänityksestä ja ohjauksesta. Projektiin osallistui 24 muusikkoa, jotka koostuivat pääasiassa Jyväskylän ammattikorkeakoulun opiskelijoista. Käsikirjoitus perustuu Kainuun ammattiopistossa opinnäytetyössä käytettyyn Mustan Tritonuksen Seura -tarinaympäristöön.
Suomessa kuunnelma on ollut taidemuotona pienessä mutta sisukkaassa asemassa. Projektin tavoite oli elävöittää kuunnelmakulttuuria, luoda elokuvamainen elämys näkörajoitteisille sekä tarjota projektiin osallistuville uusia kokemuksia. Kaksi ja puoli vuotta valmisteltu projekti huipentui kahteen esitykseen Jyväskylän ammattikorkeakoulun Hannikaissalissa joulukuussa 2013.
Opinnäytetyön kirjallisessa osassa raportoitiin kuunnelman vaiheet juonisynopsiksesta valmiikseen esitykseen. Kuunnelman eri tehokeinoja analysoitiin käsikirjoitusoppaan ja musikaalianalyysien avulla. Kirjallisessa osassa pyrittiin luomaan standardi, jonka avulla livemusiikkikuunnelman
tehojeinoja voitaisiin tarkastella. Merkittävimmät tehokeinot olivat draaman kaari, esityksen rytmitys sekä kuulijan huomion vangitsevat henkilöhahmot. Projektin massiivisuudesta johtuen Mustaa Tritonusta tuskin tullaan esittämään uudelleen lähiaikoina. Lisää Mustan Tritonuksen Seuran seikkailuja tullaan kuitenkin julkaisemaan tekijän kotisivuilla.The thesis project called Black Tritone: A Live Music Radio Drama combines a pre-recorded radio drama with live music. The project was written, composed, recorded and mixed by the author of this thesis. The project was participated by 24 musicians of whom most were studying at JAMK University of Applied Sciences. The script was based on a story environment called The Order of the Black Tritone used in an earlier thesis of the author completed in the Kainuu Conservatory.
In Finland radio drama as an art form has had a small and, yet, a headstrong position. The aim of the project was to invigorate the radio drama culture, create cinematic experiences for the visually challenged and give the participants new experiences. The project took two and a half years to finish and it was premiered twice at the chamber music hall of JAMK University of Applied Sciences, called
Hannikaissali, in December 2013.
The written part of the thesis dealt with the process of producing a live music radio drama from a plot synopsis to the complete performance. The radio drama’s methods of enhancement were analyzed based on script guides and a musical analysis. The written part aimed to create a standard
based on which those methods of a live music radio drama could be viewed. The most important enhancement methods were the dramatic structure, the performance's pacing and the characters that capture the listeners' attention. Due to its massiveness the Black Tritone is unlikely to be reperformed
in the near future. The stories of the Order of the Black Tritone will still continue on the author's website
Smooth nonparametric estimation under monotonicity constraints
No full textIn 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.Statistic
