Convergence in the Wasserstein Metric for Markov Chain Monte Carlo Algorithms with Applications to Image Restoration

Abstract

In this paper, we show how the time for convergence to stationarity of a Markov chain can be assessed using the Wasserstein metric, rather than the usual choice of total variation distance. The Wasserstein metric may be more easily applied in some applications, particularly those on continuous state spaces. Bounds on convergence time are established by considering the number of iterations required to approximately couple two realizations of the Markov chain to within ε tolerance. The particular application considered is the use of the Gibbs sampler in the Bayesian restoration of a degraded image, with pixels that are a continuous grey-scale and with pixels that can only take two colours. On finite state spaces, a bound in the Wasserstein metric can be used to find a bound in total variation distance. We use this relationship to get a precise O(N log N) bound on the convergence time of the stochastic Ising model that holds for appropriate values of its parameter as well as other binary image models. Our method employing convergence in the Wasserstein metric can also be applied to perfect sampling algorithms involving coupling from the past to obtain estimates of their running times.     

Bootstrap confidence intervals for the coefficient of quartile variation

ABSTRACT

In non-normal populations, it is more convenient to use the coefficient of quartile variation rather than the coefficient of variation. This study compares the percentile and t-bootstrap confidence intervals with Bonett's confidence interval for the quartile variation. We show that empirical coverage of the bootstrap confidence intervals is closer to the nominal coverage (0.95) for small sample sizes (n = 5, 6, 7, 8, 9, 10 and 15) for most distributions studied. Bootstrap confidence intervals also have smaller average width. Thus, we propose using bootstrap confidence intervals for the coefficient of quartile variation when the sample size is small.     

A Robust Score Test for Testing Several Coefficients of Variation with Unknown Underlying Distributions

A parametric robust test is proposed for comparing several coefficients of variation. This test is derived by properly correcting the normal likelihood function according to the technique suggested by Royall and Tsou. The proposed test statistic is asymptotically valid for general random variables, as long as their underlying distributions have finite fourth moments.

Simulation studies and real data analyses are provided to demonstrate the effectiveness of the novel robust procedure.     

Influence Functions for the Coefficient of Variation,Its Inverse,and CV Comparisons

Expressions are found for the influence function of the coefficient of variation, CV, and its reciprocal, the signal to noise ratio. These functions are free of units, which permits the comparison of the values of the CVs of continuous positive distributions to a perturbation by a small amount of probability at x. For a CV ≤0.5, the influence function response will be negative, of modest size, for values of x near E(X). For such values of a CV and of x, the influence function for 1/CV will be positive and its values will be substantial. These results imply similar behavior by the sample coefficient of variation or its reciprocal, which is supported by simulation studies in the literature. Values of the CV ≥1 are associated with large negative responses of their influence functions. The distributions producing such responses often have densities that decrease from positive infinite to zero on the positive axis with a long tail to the right. An influence function for the difference of two coefficients of variation is also obtained.     

Quadratic Programming and Penalized Regression

Quadratic programming is a versatile tool for calculating estimates in penalized regression. It can be used to produce estimates based on L ₁ roughness penalties, as in total variation denoising. In particular, it can calculate estimates when the roughness penalty is the total variation of a derivative of the estimate. Combining two roughness penalties, the total variation and total variation of the third derivative, results in an estimate with continuous second derivative but controls the number of spurious local extreme values. A multiresolution criterion may be included in a quadratic program to achieve local smoothing without having to specify smoothing parameters.     

An Improved Class of Estimators for the General Population Parameters Using Auxiliary Information

This article suggests an improved class of estimators for estimating the general population parameter using information on an auxiliary variable. The properties of the suggested class of estimators have been studied under large sample approximation. The general results are then applied to estimate the population coefficient of variation of study variable using auxiliary information. An empirical study is given in support of the theoretical results.     

On random walks on affine group

Diaconis' presumption that the number of steps required to get close to uniform for a random walk on the affine group A _pis c(p)p ²with c(p) →ã is verified. We also discuss the random number generation associated with the random walk on the affine group. The number of steps to force the generated number to become random is improved. A modified version of Diacohis-Shahshahani's upper bound lemma is given and applied