Search results for "Theorem"
showing 10 items of 1250 documents
Confidence bands for Horvitz-Thompson estimators using sampled noisy functional data
2013
When collections of functional data are too large to be exhaustively observed, survey sampling techniques provide an effective way to estimate global quantities such as the population mean function. Assuming functional data are collected from a finite population according to a probabilistic sampling scheme, with the measurements being discrete in time and noisy, we propose to first smooth the sampled trajectories with local polynomials and then estimate the mean function with a Horvitz-Thompson estimator. Under mild conditions on the population size, observation times, regularity of the trajectories, sampling scheme, and smoothing bandwidth, we prove a Central Limit theorem in the space of …
Fractional Brownian motion and Martingale-differences
2004
Abstract We generalize a result of Sottinen (Finance Stochastics 5 (2001) 343) by proving an approximation theorem for the fractional Brownian motion, with H> 1 2 , using martingale-differences.
Bayesian analysis of a disability model for lung cancer survival
2016
Bayesian reasoning, survival analysis and multi-state models are used to assess survival times for Stage IV non-small-cell lung cancer patients and the evolution of the disease over time. Bayesian estimation is done using minimum informative priors for the Weibull regression survival model, leading to an automatic inferential procedure. Markov chain Monte Carlo methods have been used for approximating posterior distributions and the Bayesian information criterion has been considered for covariate selection. In particular, the posterior distribution of the transition probabilities, resulting from the multi-state model, constitutes a very interesting tool which could be useful to help oncolog…
A hierarchical Bayesian birth cohort analysis from incomplete registry data: evaluating the trends in the age of onset of insulin-dependent diabetes …
2005
Childhood diabetes is one of the major non-communicable diseases in children under 15 years of age. It requires a life-long insulin treatment and may lead to serious complications. Along with the worldwide increase in the incidence several countries have recently reported a decreasing trend in the age of onset of the disease. The aim of this study is to analyse long-term data on the incidence of the childhood diabetes in Finland from the birth cohorts perspective. The annual incidence data were available for the period 1965--1996 which translates into 1951--1996 birth cohorts. Hence the data consist of completely and partially observed cohorts. Bayesian modelling was employed in the analysi…
Recursive estimation of the conditional geometric median in Hilbert spaces
2012
International audience; A recursive estimator of the conditional geometric median in Hilbert spaces is studied. It is based on a stochastic gradient algorithm whose aim is to minimize a weighted L1 criterion and is consequently well adapted for robust online estimation. The weights are controlled by a kernel function and an associated bandwidth. Almost sure convergence and L2 rates of convergence are proved under general conditions on the conditional distribution as well as the sequence of descent steps of the algorithm and the sequence of bandwidths. Asymptotic normality is also proved for the averaged version of the algorithm with an optimal rate of convergence. A simulation study confirm…
Central Limit Theorem for Linear Eigenvalue Statistics for a Tensor Product Version of Sample Covariance Matrices
2017
For $$k,m,n\in {\mathbb {N}}$$ , we consider $$n^k\times n^k$$ random matrices of the form $$\begin{aligned} {\mathcal {M}}_{n,m,k}({\mathbf {y}})=\sum _{\alpha =1}^m\tau _\alpha {Y_\alpha }Y_\alpha ^T,\quad {Y}_\alpha ={\mathbf {y}}_\alpha ^{(1)}\otimes \cdots \otimes {\mathbf {y}}_\alpha ^{(k)}, \end{aligned}$$ where $$\tau _{\alpha }$$ , $$\alpha \in [m]$$ , are real numbers and $${\mathbf {y}}_\alpha ^{(j)}$$ , $$\alpha \in [m]$$ , $$j\in [k]$$ , are i.i.d. copies of a normalized isotropic random vector $${\mathbf {y}}\in {\mathbb {R}}^n$$ . For every fixed $$k\ge 1$$ , if the Normalized Counting Measures of $$\{\tau _{\alpha }\}_{\alpha }$$ converge weakly as $$m,n\rightarrow \infty $$…
A generalization of the Binomial distribution based on the dependence ratio
2014
We propose a generalization of the Binomial distribution, called DR-Binomial, which accommodates dependence among units through a model based on the dependence ratio (Ekholm et al., Biometrika, 82, 1995, 847). Properties of the DR-Binomial are discussed, and the constraints on its parameter space are studied in detail. Likelihood-based inference is presented, using both the joint and profile likelihoods; the usefulness of the DR-Binomial in applications is illustrated on a real dataset displaying negative unit-dependence, and hence under-dispersion compared with the Binomial. Although the DR-Binomial turns out to be a reparameterization of Altham's Additive-Binomial and Kupper–Haseman's Cor…
Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small-noise limit
2011
In the context of self-stabilizing processes, that is processes attracted by their own law, living in a potential landscape, we investigate different properties of the invariant measures. The interaction between the process and its law leads to nonlinear stochastic differential equations. In [S. Herrmann and J. Tugaut. Electron. J. Probab. 15 (2010) 2087–2116], the authors proved that, for linear interaction and under suitable conditions, there exists a unique symmetric limit measure associated to the set of invariant measures in the small-noise limit. The aim of this study is essentially to point out that this statement leads to the existence, as the noise intensity is small, of one unique…
Lattices and dual lattices in optimal experimental design for Fourier models
1998
Number-theoretic lattices, used in integration theory, are studied from the viewpoint of the design and analysis of experiments. For certain Fourier regression models lattices are optimal as experimental designs because they produce orthogonal information matrices. When the Fourier model is restricted, that is a special subset of the full factorial (cross-spectral) model is used, there is a difficult inversion problem to find generators for an optimal design for the given model. Asymptotic results are derived for certain models as the dimension of the space goes to infinity. These can be thought of as a complexity theory connecting designs and models or as special type of Nyquist sampling t…
On (n-l)-wise and joint independence and normality of n Random variables: an example
1981
An example is given of a vector of n random variables such that any (n-1)-dimensional subvector consists of n-1 independent standard normal variables. The whole vector however is neither independent nor normal.