Search results for "Applied Mathematics"
showing 10 items of 4379 documents
Horvitz-Thompson estimators for functional data: asymptotic confidence bands and optimal allocation for stratified sampling
2009
When dealing with very large datasets of functional data, survey sampling approaches are useful in order to obtain estimators of simple functional quantities, without being obliged to store all the data. We propose here a Horvitz--Thompson estimator of the mean trajectory. In the context of a superpopulation framework, we prove under mild regularity conditions that we obtain uniformly consistent estimators of the mean function and of its variance function. With additional assumptions on the sampling design we state a functional Central Limit Theorem and deduce asymptotic confidence bands. Stratified sampling is studied in detail, and we also obtain a functional version of the usual optimal …
On 1-Laplacian Elliptic Equations Modeling Magnetic Resonance Image Rician Denoising
2015
Modeling magnitude Magnetic Resonance Images (MRI) rician denoising in a Bayesian or generalized Tikhonov framework using Total Variation (TV) leads naturally to the consideration of nonlinear elliptic equations. These involve the so called $1$-Laplacian operator and special care is needed to properly formulate the problem. The rician statistics of the data are introduced through a singular equation with a reaction term defined in terms of modified first order Bessel functions. An existence theory is provided here together with other qualitative properties of the solutions. Remarkably, each positive global minimum of the associated functional is one of such solutions. Moreover, we directly …
Latin hypercube sampling with inequality constraints
2010
International audience; In some studies requiring predictive and CPU-time consuming numerical models, the sampling design of the model input variables has to be chosen with caution. For this purpose, Latin hypercube sampling has a long history and has shown its robustness capabilities. In this paper we propose and discuss a new algorithm to build a Latin hypercube sample (LHS) taking into account inequality constraints between the sampled variables. This technique, called constrained Latin hypercube sampling (cLHS), consists in doing permutations on an initial LHS to honor the desired monotonic constraints. The relevance of this approach is shown on a real example concerning the numerical w…
A computationally fast alternative to cross-validation in penalized Gaussian graphical models
2015
We study the problem of selection of regularization parameter in penalized Gaussian graphical models. When the goal is to obtain the model with good predicting power, cross validation is the gold standard. We present a new estimator of Kullback-Leibler loss in Gaussian Graphical model which provides a computationally fast alternative to cross-validation. The estimator is obtained by approximating leave-one-out-cross validation. Our approach is demonstrated on simulated data sets for various types of graphs. The proposed formula exhibits superior performance, especially in the typical small sample size scenario, compared to other available alternatives to cross validation, such as Akaike's i…
Multivariate nonparametric estimation of the Pickands dependence function using Bernstein polynomials
2017
Abstract Many applications in risk analysis require the estimation of the dependence among multivariate maxima, especially in environmental sciences. Such dependence can be described by the Pickands dependence function of the underlying extreme-value copula. Here, a nonparametric estimator is constructed as the sample equivalent of a multivariate extension of the madogram. Shape constraints on the family of Pickands dependence functions are taken into account by means of a representation in terms of Bernstein polynomials. The large-sample theory of the estimator is developed and its finite-sample performance is evaluated with a simulation study. The approach is illustrated with a dataset of…
Importance sampling correction versus standard averages of reversible MCMCs in terms of the asymptotic variance
2017
We establish an ordering criterion for the asymptotic variances of two consistent Markov chain Monte Carlo (MCMC) estimators: an importance sampling (IS) estimator, based on an approximate reversible chain and subsequent IS weighting, and a standard MCMC estimator, based on an exact reversible chain. Essentially, we relax the criterion of the Peskun type covariance ordering by considering two different invariant probabilities, and obtain, in place of a strict ordering of asymptotic variances, a bound of the asymptotic variance of IS by that of the direct MCMC. Simple examples show that IS can have arbitrarily better or worse asymptotic variance than Metropolis-Hastings and delayed-acceptanc…
Explicit, identical maximum likelihood estimates for some cyclic Gaussian and cyclic Ising models
2017
Cyclic models are a subclass of graphical Markov models with simple, undirected probability graphs that are chordless cycles. In general, all currently known distributions require iterative procedures to obtain maximum likelihood estimates in such cyclic models. For exponential families, the relevant conditional independence constraint for a variable pair is given all remaining variables, and it is captured by vanishing canonical parameters involving this pair. For Gaussian models, the canonical parameter is a concentration, that is, an off-diagonal element in the inverse covariance matrix, while for Ising models, it is a conditional log-linear, two-factor interaction. We give conditions un…
Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality
2017
We provide a full quantitative version of the Gaussian isoperimetric inequality: the difference between the Gaussian perimeter of a given set and a half-space with the same mass controls the gap between the norms of the corresponding barycenters. In particular, it controls the Gaussian measure of the symmetric difference between the set and the half-space oriented so to have the barycenter in the same direction of the set. Our estimate is independent of the dimension, sharp on the decay rate with respect to the gap and with optimal dependence on the mass.
On the function of modified nucleosides in the RNA world.
1998
Presumably ribosome and transfer RNA (tRNA) evolved from a pre-existing function in the RNA stage of life and were secondarily adapted for protein synthesis. Various possible initial functions of the primitive ribosome (protoribosome) have been suggested. The initial function of the primitive ribosome and primitive genetic translation would have been quite similar. It is possible that, initially, both functions coexisted in the protoribosome. Given that the three-dimensional structure of ribosomal RNAs shows only minor variations throughout time, it is, then, most likely that present ribosomes can still recall (remember) the most important parts of the mechanism of their initial function. A…
Improvements and Modifications of Tarone's Multiple Test Procedure for Discrete Data
1998
Tarone (1990, Biometrics 46, 515-522) proposed a multiple test procedure for discrete test statistics improving the usual Bonferroni procedure. However, Tarone's procedure is not monotone depending on the predetermined multiple level a. Roth (1998, Journal of Statistical Planning and Inference, in press) developed a monotone version of Tarone's procedure. We present a similar procedure that is both monotone and an improvement of Tarone's proposal. Based on this extension, we derive a step-down procedure that is a corresponding improvement of Holm's (1979, Scandinavian Journal of Statistics 6, 65-70) sequentially rejective procedure. It is shown how adjusted p-values can be computed for the …