Computer Intensive Methods in Statistics

The problem of edge-detection in images will be formulated as a statistical changepoint problem using a Bayesian approach. It will be shown that the Gibbs sampler provides an effective procedure for the required Bayesian calculations. The use of the method for “quick and dirty” image segmentation will be illustrated.

Algorithms for efficient computer generation of matric-variate t random drawings are constructed which make use of two results in distribution theory. First, the definition of a matric-variate t distributed random matrix as the product of a matric-variate normal distributed random matrix and the square root of an inverted-Wishart distributed random matrix. Second, a decomposition of the Wishart and inverted Wishart matrix into triangular matrices. The different steps of the algorithm for matric-variate t drawings and the decomposition of the (inverted-) Wishart are explained. For illustrative purposes, the posterior density of the structural parameters of a simple market model is evaluated. These structural parameters are nonlinear functions of matric-variate t variables.

We evaluate two tests of residual autocorrelation in the linear regression model in a Bayesian framework. Each test checks if an approximate highest posterior density region of the parameters of the autoregressive process of the error contains the null hypothesis. The approximation consists in computing the posterior density of the coefficients of the AR process using augmented regressions. The first test uses the initial regression augmented with its lagged Bayesian residuals and can be done with tables of the Fisher distribution. The second test augments the initial regression with lagged dependent and explanatory variables, and requires numerical integration. The tests are evaluated through a small Monte-Carlo experiment, which indicates that the first test (easier to compute) is more powerful than the second one.

In the construction of multiparameter significance tests intensive computing may be involved at two levels: numerical computing in the construction of critical regions with particular optimality properties, symbolic computing to obtain a better understanding of a statistical model e.g. by giving the geometry of the model. Each level is explained in the construction of two tests for a simple null hypothesis in two lifetime models.

The g-LMMPUMα test maximizes the local mean power with respect to a metric g under all unbiased tests. This test is obtained for the two-parameter gamma family. The construction of the critical region involves the solution of a system of non-linear equations, two-dimensional numerical quadrature, numerical Fourier inversion and interpolation. The NAG Fortran library is a basic tool.

The purpose of this note is to report on pedagogical experiments centered around the use of a dedicated program in the learning of mathematical statistics and the practice of data analysis. The authors developed a set of lectures notes with a strong emphasis on: Graphical analyses, random number generation, simulation techniques, resampling methods, dynamic illustration of regression diagnostics, robust methods ... . Most of these concepts can be presented to undergraduate students but no appropriate textbook existed. Also, such a pedagogical experiment could not be conceived without the use of a computer program for use as a companion to the lectures. No satisfactory solution could be found from the existing commercial or public softwares. We discuss in detail some of the most salient features of the experiment and we describe the tools which the authors developed in the process.

This work presents a method to find rock conductivities in a zone from electromagnetic measurements on the surface of the earth. It uses a stochastic algorithm to find a Bayesian estimator of the conductivities. The algorithm is tested on a synthetic model made up of an heteregeneous thin sheet inbedded in a stratified substratum.

During the last few years, Markov Random Field (Mrf) models have already been successfully applied in some applications in image remote sensing in a context of conditional maximum likelihood estimation. Here, in the same context, we propose some original uses of Mrf, especially in image segmentation, noise filtering and discriminant analysis. For instance, we propose a Mrf model on the spectral signatures space, a strongly unified approach to classification and noise filtering as well as a particular model of noise.

We study the problem of estimating the edges in noisy images by linewise procedures. We show that the straightforward estimation method (naïve linewise procedure) does not attain the asymptotically minimax rate of accuracy as the number of observations tends to ∞. We propose the modified linewise procedure which has the asymptotically minimax rate.

This paper is concerned with nonparametric estimation of a regression function. The behaviour of kernel estimates depends on a smoothing parameter (i.e. the bandwidth). Bandwidth choice turns out to be of particular importance as well for practical use as to insure good asymptotic properties of the estimate. Various techniques have been proposed in the past ten last years to select optimal values of this parameter. This paper presents a survey on theoretical results concerned with bandwidth selection.

The usefulness of bootstrap in statistical analysis of regression models is demonstrated. Surveying earlier results, four specific problems are considered:

This paper investigates the problem of the choice of dimension in Principal Component Analysis (PCA). PCA is introduced as a model; a loss function assessing the stability of the fit is considered. The choice of dimension then amounts to the minimisation of an expected loss which has to be estimated. This is achieved by resampling methods. Different bootstrap and jackknife estimates are presented. The behaviour of these estimates are investigated on artificial data and on real data. The resulting choices are confronted with those given by naïve rules.