Minimal auxiliary Markov chains through sequential elimination of states

When using an auxiliary Markov chain to compute the distribution of a pattern statistic, the computational complexity is directly related to the number of Markov chain states. Theory related to minimal deterministic finite automata have been applied to large state spaces to reduce the number of Markov chain states so that only a minimal set remains. In this paper, a characterization of equivalent states is given so that extraneous states are deleted during the process of forming the state space, improving computational efficiency. The theory extends the applicability of Markov chain based methods for computing the distribution of pattern statistics.     

Symmetry of functions and exchangeability of random variables

We present a new approach for measuring the degree of exchangeability of two continuous, identically distributed random variables or, equivalently, the degree of symmetry of their corresponding copula. While the opposite of exchangeability does not exist in probability theory, the contrary of symmetry is quite obvious from an analytical point of view. Therefore, leaving the framework of probability theory, we introduce a natural measure of symmetry for bivariate functions in an arbitrary normed function space. Restricted to the set of copulas this yields a general concept for measures of (non-)exchangeability of random variables. The fact that copulas are never antisymmetric leads to the notion of maximal degree of antisymmetry of copulas. We illustrate our approach by various norms on function spaces, most notably the Sobolev norm for copulas.     

Parallel discretization of the Markov chain approximation for the autoregressive moving average chart

Abstract

In the Markov chain model of an autoregressive moving average chart, the post-transition states of nonzero transition probabilities are distributed along one-dimensional lines of a constant gradient over the state space. By considering this characteristic, we propose discretizing the state space parallel to the gradient of these one-dimensional lines. We demonstrate that our method substantially reduces the computational cost of the Markov chain approximation for the average run length in two- and three-dimensional state spaces. Also, we investigate the effect of these one-dimensional lines on the computational cost. Lastly, we generalize our method to state spaces larger than three dimensions.     

Quotient spaces and statistical models

The purpose of this paper is to draw attention to the widespread occurrence of quotient spaces in statistical work. Quotient spaces are intrinsic to probability distributions, residuals and interaction in linear models, covariance functions and variograms of stochastic processes, etc. The theme is that explicit recognition of the quotient space can offer surprising conceptual simplification. The advantages of working directly with the quotient space are hard to describe in general. As the examples demonstrate, the answer lies partly in directness of approach.     

NEW LARGE SAMPLE AND BOOTSTRAP METHODS ON SHAPE SPACES IN HIGH LEVEL ANALYSIS OF NATURAL IMAGES

This paper, dedicated to the 80th birthday of Professor C. R. Rao, deals with asymptotic distributions of Fréchet sample means and Fréchet total sample variance that are used in particular for data on projective shape spaces or on 3D shape spaces. One considers the intrinsic means associated with Riemannian metrics that are locally flat in a geodesically convex neighborhood around the support of a probability measure on a shape space or on a projective shape space. Such methods are needed to derive tests concerning variability of planar projective shapes in natural images or large sample and bootstrap confidence intervals for 3D mean shape coordinates of an ordered set of landmarks from laser images.     

Estimation for distributions with monotone likelihood ratio:case of vector valued parameter

An extension of a result about the estimation in Karlin and Rubin is given for the following case:The sample space, the parameter space and the decision space are subsets of a multi-dimensional Euclidean space, there is defined a suitable partial ordering in each of spaces, and a probability distribution has monotone likelihood ratio with respect to the partial orderings (see Ishii, 1976). In the special case when the loss function is quadratic a simple proof of a result in Karlin and Rubin is given. Stein's estimators are discussed as examples.     

Bayes Hilbert Spaces

A Bayes linear space is a linear space of equivalence classes of proportional σ‐finite measures, including probability measures. Measures are identified with their density functions. Addition is given by Bayes' rule and substraction by Radon–Nikodym derivatives. The present contribution shows the subspace of square‐log‐integrable densities to be a Hilbert space, which can include probability and infinite measures, measures on the whole real line or discrete measures. It extends the ideas from the Hilbert space of densities on a finite support towards Hilbert spaces on general measure spaces. It is also a generalisation of the Euclidean structure of the simplex, the sample space of random compositions. In this framework, basic notions of mathematical statistics get a simple algebraic interpretation. A key tool is the centred‐log‐ratio transformation, a generalization of that used in compositional data analysis, which maps the Hilbert space of measures into a subspace of square‐integrable functions. As a consequence of this structure, distances between densities, orthonormal bases, and Fourier series representing measures become available. As an application, Fourier series of normal distributions and distances between them are derived, and an example related to grain size distributions is presented. The geometry of the sample space of random compositions, known as Aitchison geometry of the simplex, is obtained as a particular case of the Hilbert space when the measures have discrete and finite support.     

Measurements of separation among probability densities or random variables

Distance between two probability densities or two random variables is a well established concept in statistics. The present paper considers generalizations of distances to separation measurements for three or more elements in a function space. Geometric intuition and examples from hypothesis testing suggest lower and upper bounds for such measurements in terms of pairwise distances; but also in L_p spaces some useful non-pairwise separation measurements always lie within these bounds. Examples of such separation measurements are the Bayes probability of correct classification among several arbitrary distributions, and the expected range among several random variables.     

The table look-up rule

The table look-up rule problem can be described by the question: what is a good way for the table to represent the decision regions in the N-dimensional measurement space. This paper describes a quickly implementable table look-up rule based on Ashby’s representation of sets in his constraint analysis. A decision region for category c in the N-dimensional measurement space is considered to be the intersection of the inverse projections of the decision regions determined for category c by Bayes rules in smaller dimensional projection spaces. Error bounds for this composite decision rule are derived: any entry in the confusion matrix for the composite decision rule is bounded above by the minimum of that entry taken over all the confusion matrices of the Bayes decision rules in the smaller dimensional projection spaces.

On simulated Gaussian Data, probability of error with the table look-up rule is comparable to the optimum Bayes rule.     

On two definitions of identification

Summary. The paper analyzes the connection between the identification of a statistical experiment defined by the injectivity of the sampling probability indexation and by the minimal sufficiency of the c-field characterizing the parametrization. If the parameter space is a BLACKWELL space and if the observation generates a separable ?7-field, these two concepts are proved to be equivalent     

On the efficiency of affine minimax rules in estimating a bounded multivariate normal mean

Problems involving bounded parameter spaces, for example T-minimax and minimax esyimation of bounded parameters, have received much attention in recent years. The existing literature is rich. In this paper we consider T-minimax estimation of a multivariate bounded normal mean by affine rules, and discuss the loss of efficiency due to the use of such rules instead of optimal, unrestricted rules. We also investigate the behavior of 'probability restricted' affine rules, i.e., rules that have a guaranteed large probability of being in the bounded parameter space of the problem.     

Least-square regularized regression with non-iid sampling

We study the least-square regression learning algorithm generated by regularization schemes in reproducing kernel Hilbert spaces. A non-iid setting is considered: the sequence of probability measures for sampling is not identical and the sampling may be dependent. When the sequence of marginal distributions for sampling converges exponentially fast in the dual of a Hölder space and the sampling process satisfies a polynomial strong mixing condition, we derive learning rates for the learning algorithm.     

Perfect simulation of conditionally specified models

We discuss how the ideas of producing perfect simulations based on coupling from the past for finite state space models naturally extend to multivariate distributions with infinite or uncountable state spaces such as auto-gamma, auto-Poisson and autonegative binomial models, using Gibbs sampling in combination with sandwiching methods originally introduced for perfect simulation of point processes.     

Stability of Bayesian decisions

Stability of a Bayes decision is analysed with respect to small changes of the probability measure on the space of the states of nature. The problem leads to the study of continuity aspects of the infimum of the Bayes functional. These are approached through the epigraphical convergence of integral functionals as the minimal setting for convergence of infima.     

Stabilized Sieve Sampling: A Point-Estimator Analysis

A modification of sieve sampling is proposed that returns a constant sample size. It is a scheme that selects line items with probability proportional to size (PPS) and nearly without replacement. An unbiased estimator of the total error amount is presented and its variance derived. Conditions under which the scheme is more efficient than sieve sampling and than PPS with replacement sampling are given.     

Achieving limiting distributions for Markov chains using back buttons

As a simple model for browsing the World Wide Web, we consider Markov chains with the option of moving back to the previous state. We develop an algorithm which uses back buttons to achieve essentially any limiting distribution on the state space. This corresponds to spending the desired total fraction of time at each web page. On finite state spaces, our algorithm always succeeds. On infinite state spaces the situation is more complicated, and is related to both the tail behaviour of the distributions, and the properties of convolution equations.     

First hitting probabilities for semi-Markov chains and estimation

We first consider a stochastic system described by an absorbing semi-Markov chain (SMC) with finite state space, and we introduce the absorption probability to a class of recurrent states. Afterwards, we study the first hitting probability to a subset of states for an irreducible SMC. In the latter case, a non-parametric estimator for the first hitting probability is proposed and the asymptotic properties of strong consistency and asymptotic normality are proven. Finally, a numerical application on a five-state system is presented to illustrate the performance of this estimator.     

The computation of standard normal distribution integral in any required precision based on reliability method

Based on reliability theory, the value of the standard normal distribution integral can be obtained by calculating the probability of the failure domain of the linear performance function. After the sample space is divided into some sub-sample spaces, a number of sub-failure domains are obtained. In the paper, the methods of computing the probabilities of sub-failure domains are discussed. All the formulae and the steps of computing the standard normal distribution integral which meet any required precision are given in the paper. Examples show that it is easy for the method to compute the standard normal distribution integral.     

Convergence of compensators of simple point processes

Processes are viewed as random variables with values in the space of càdlàg functions endowed with the J₁ topology. For a sequence of point processes with their minimal filtration, the convergence of their compensators is studied. It is shown that convergence in probability of the processes and of the corresponding conditional distributions of their jump times implies, with some additional hypothesis, the convergence in probability of the compensators. The result is then applied to convergence to a quasi-left-continuous point process with independent increments.     

Sieve bootstrapping the memory parameter in long-range dependent stationary functional time series

We consider a sieve bootstrap procedure to quantify the estimation uncertainty of long-memory parameters in stationary functional time series. We use a semiparametric local Whittle estimator to estimate the long-memory parameter. In the local Whittle estimator, discrete Fourier transform and periodogram are constructed from the first set of principal component scores via a functional principal component analysis. The sieve bootstrap procedure uses a general vector autoregressive representation of the estimated principal component scores. It generates bootstrap replicates that adequately mimic the dependence structure of the underlying stationary process. We first compute the estimated first set of principal component scores for each bootstrap replicate and then apply the semiparametric local Whittle estimator to estimate the memory parameter. By taking quantiles of the estimated memory parameters from these bootstrap replicates, we can nonparametrically construct confidence intervals of the long-memory parameter. As measured by coverage probability differences between the empirical and nominal coverage probabilities at three levels of significance, we demonstrate the advantage of using the sieve bootstrap compared to the asymptotic confidence intervals based on normality.