Search results for "Statistics & Probability"
showing 10 items of 436 documents
Importance sampling type estimators based on approximate marginal Markov chain Monte Carlo
2020
We consider importance sampling (IS) type weighted estimators based on Markov chain Monte Carlo (MCMC) targeting an approximate marginal of the target distribution. In the context of Bayesian latent variable models, the MCMC typically operates on the hyperparameters, and the subsequent weighting may be based on IS or sequential Monte Carlo (SMC), but allows for multilevel techniques as well. The IS approach provides a natural alternative to delayed acceptance (DA) pseudo-marginal/particle MCMC, and has many advantages over DA, including a straightforward parallelisation and additional flexibility in MCMC implementation. We detail minimal conditions which ensure strong consistency of the sug…
The smallest singular value of a shifted $d$-regular random square matrix
2017
We derive a lower bound on the smallest singular value of a random d-regular matrix, that is, the adjacency matrix of a random d-regular directed graph. Specifically, let $$C_1<d< c n/\log ^2 n$$ and let $$\mathcal {M}_{n,d}$$ be the set of all $$n\times n$$ square matrices with 0 / 1 entries, such that each row and each column of every matrix in $$\mathcal {M}_{n,d}$$ has exactly d ones. Let M be a random matrix uniformly distributed on $$\mathcal {M}_{n,d}$$ . Then the smallest singular value $$s_{n} (M)$$ of M is greater than $$n^{-6}$$ with probability at least $$1-C_2\log ^2 d/\sqrt{d}$$ , where c, $$C_1$$ , and $$C_2$$ are absolute positive constants independent of any other parameter…
Multiple smoothing parameters selection in additive regression quantiles
2021
We propose an iterative algorithm to select the smoothing parameters in additive quantile regression, wherein the functional forms of the covariate effects are unspecified and expressed via B-spline bases with difference penalties on the spline coefficients. The proposed algorithm relies on viewing the penalized coefficients as random effects from the symmetric Laplace distribution, and it turns out to be very efficient and particularly attractive with multiple smooth terms. Through simulations we compare our proposal with some alternative approaches, including the traditional ones based on minimization of the Schwarz Information Criterion. A real-data analysis is presented to illustrate t…
Local bandwidth selection for kernel density estimation in a bifurcating Markov chain model
2020
International audience; We propose an adaptive estimator for the stationary distribution of a bifurcating Markov Chain onRd. Bifurcating Markov chains (BMC for short) are a class of stochastic processes indexed by regular binary trees. A kernel estimator is proposed whose bandwidths are selected by a method inspired by the works of Goldenshluger and Lepski [(2011), 'Bandwidth Selection in Kernel Density Estimation: Oracle Inequalities and Adaptive Minimax Optimality',The Annals of Statistics3: 1608-1632). Drawing inspiration from dimension jump methods for model selection, we also provide an algorithm to select the best constant in the penalty. Finally, we investigate the performance of the…
Posterior moments and quantiles for the normal location model with Laplace prior
2021
We derive explicit expressions for arbitrary moments and quantiles of the posterior distribution of the location parameter η in the normal location model with Laplace prior, and use the results to approximate the posterior distribution of sums of independent copies of η.
Local Asymptotic Normality for Shape and Periodicity in the Drift of a Time Inhomogeneous Diffusion
2017
We consider a one-dimensional diffusion whose drift contains a deterministic periodic signal with unknown periodicity $T$ and carrying some unknown $d$-dimensional shape parameter $\theta$. We prove Local Asymptotic Normality (LAN) jointly in $\theta$ and $T$ for the statistical experiment arising from continuous observation of this diffusion. The local scale turns out to be $n^{-1/2}$ for the shape parameter and $n^{-3/2}$ for the periodicity which generalizes known results about LAN when either $\theta$ or $T$ is assumed to be known.
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…
Ergodicity and limit theorems for degenerate diffusions with time periodic drift. Application to a stochastic Hodgkin−Huxley model
2016
We formulate simple criteria for positive Harris recurrence of strongly degenerate stochastic differential equations with smooth coefficients on a state space with certain boundary conditions. The drift depends on time and space and is periodic in the time argument. There is no time dependence in the diffusion coefficient. Control systems play a key role, and we prove a new localized version of the support theorem. Beyond existence of some Lyapunov function, we only need one attainable inner point of full weak Hoermander dimension. Our motivation comes from a stochastic Hodgkin−Huxley model for a spiking neuron including its dendritic input. This input carries some deterministic periodic si…
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…
Pairwise Markov properties for regression graphs
2016
With a sequence of regressions, one may generate joint probability distributions. One starts with a joint, marginal distribution of context variables having possibly a concentration graph structure and continues with an ordered sequence of conditional distributions, named regressions in joint responses. The involved random variables may be discrete, continuous or of both types. Such a generating process specifies for each response a conditioning set that contains just its regressor variables, and it leads to at least one valid ordering of all nodes in the corresponding regression graph that has three types of edge: one for undirected dependences among context variables, another for undirect…