Exchangeable Variables Research Articles

Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models

Haploid population models with non-overlapping generations and fixed population size N are considered. It is assumed that the family sizes ν1,…,νN within a generation are exchangeable random variables. Rates of convergence for the finite-dimensional distributions of a properly time-scaled ancestral coalescent process are established and expressed in terms of the transition probabilities of the ancestral process, i.e., in terms of the joint factorial moments of the offspring variables ν1,…,νN.The Kingman coalescent appears in the limit as the population size N tends to infinity if and only if triple mergers are asymptotically negligible in comparison with binary mergers. In this case, a simple upper bound for the rate of convergence of the finite-dimensional distributions is derived. It depends (up to some constants) only on the three factorial moments E((ν1)2), E((ν1)3) and E((ν1)2(ν2)2), where (x)k := x(x-1)…(x-k+1). Examples are the Wright-Fisher model, where the rate of convergence is of order N-1, and the Moran model, with a convergence rate of order N-2.

A note on the monotonicity of the critical values of a step-up test

Abstract The monotonicity of the critical values is essential for the implementation of a step-up multiple test procedure. For a step-up test controlling the familywise error rate exactly at α, while a theoretical justification of the existence of monotone critical values is available in the literature for the case of i.i.d. test statistics, it is still an open problem for the general exchangeable case. Recently, Finner and Roters (1998 Ann. Statist. 26 505–524) gave an example of three exchangeable random variables casting doubt about this existence in general. We provide a positive result in this paper by proving that the critical values of a step-up test involving an equicorrelated trivariate standard normal distribution is indeed monotone when the common correlation is positive and at most (zα/22−z1/42)/(zα/22+z1/42), where zα is the upper 100α percent point of N(0,1). Our example raises the hope of establishing the desired monotonicity property in the case of test statistics with totally positive multivariate distributions in the sense of Karlin and Rinott.

A central limit theorem for exchangeable random variables on a locally compact abelian group

A central limit theorem is given for uniformly infinitesimal triangular arrays of random variables in which the random variables in each row are exchangeable and take values in a locally compact second countable abelian group. The limiting distribution in the theorem is Gaussian.

Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models

Haploid population models with non-overlapping generations and fixed population size N are considered. It is assumed that the family sizes 1, - - - , N within a generation are exchangeable random variables. Rates of convergence for the finite-dimensional distributions of a properly time-scaled ancestral coalescent process are established and expressed in terms of the transition probabilities of the ancestral process, i.e., in terms of the joint factorial moments of the offspring variables v1, - - - , N. The Kingman coalescent appears in the limit as the population size N tends to infinity if and only if triple mergers are asymptotically negligible in comparison with binary mergers. In this case, a simple upper bound for the rate of convergence of the finitedimensional distributions is derived. It depends (up to some constants) only on the three factorial moments E((v1)2), E((v1)3) and E((v1)2 (v2)2), where (x)k := x(x --1) ... (x k + 1). Examples are the Wright-Fisher model, where the rate of convergence is of order N-1, and the Moran model, with a convergence rate of order N-2.

Some results of stable convergence for exchangeable random variables in Hilbert spaces

Some results of stable convergence for exchangeable random variables in Hilbert spaces

A maximal inequality for real numbers with application to exchangeable random variables

A maximal inequality for real numbers with application to exchangeable random variables

Reverse submartingale property arising from exchangeable random variables

In this work, for an exchangeable sequence of random variables {Xi, i≥1}, and two nondecreasing sequences of positive integers {hn, n≥2} and {kn, n≥2}, where hn+kn≤n, ∀n≥2, we prove that {Rn,hn,kn/n, n≥2} forms a reverse submartingale sequence, where \(\), and X1,n≤X2,n≤…≤Xn,n are the order statistics based on {X1,…,Xn}.

Some consequences of exchangeability in random-matrix theory.

Properties of infinite sequences of exchangeable random variables result directly in explicit expressions for calculating asymptotic densities of eigenvalues rho(infinity)(lambda) of any ensemble of random matrices H whose distribution depends only on tr(H+H), where H+ is the Hermitian conjugate of H. For real symmetric matrices and for Hermitian matrices, the densities rho(infinity)(lambda) are constructed by summing up Wigner semicircles with varying radii and weights as confirmed by Monte Carlo simulations. Extensions to more general matrix ensembles are also considered.

MEM pixel correlated solutions for generalized moment and interpolation problems

In generalized moment problems (signed) measures are searched to fit given observations, or continuous functions are searched to fit given constraints. Known convex methods for solving such problems, and their stochastic interpretations via maximum entropy on the mean (MEM) and in a Bayesian sense are reviewed, with some improvements on previous results. Then the MEM and Bayesian approaches are extended to default models with a dependence structure, yielding new families of solutions. One family involves a transfer kernel, and allows using prior information such as modality, convexity, or Sobolev norms. Another family of solutions with possibly nonconvex criteria, is arrived at using default models with exchangeable random variables. The main technical tools are convex analysis and large deviations theory.

Multiple comparisons with a control in repeated measures incomplete block designs using R-estimators

New multiple comparison with a control (MCC) procedures are developed in repeated measures incomplete block design settings based on R-estimates. It is assumed that the errors within each subject are exchangeable random variables. The R-estimators of the treatment effects are obtained by minimizing a sum of Jaeckel (1972)-type dispersion functions. Based on the R-estimators, Dunnett-type multiple comparison procedures are developed for comparing test-treatments with a control-treatment. Under exchangeable errors, it is demonstrated that for Cox-type designs, the new procedures are more efficient than the existing nonparametric procedures. The new MCC procedures are applied to a data set in a clinical trial which consists of patients with reversible obstructive pulmonary disease.

On convergence of series of random variables with applications to martingale convergence and to convergence of series with orthogonal summands

The rate of convergence for an almost surely convergent series of random variables {Xn, n ≥ 1} is studied in this paper. More specifically, if Sn converges almost surely to a random variable S then the tail series is a well-defined sequence of random variable with Tn → 0 almost surely. The main result provides conditions for each of to hold for given numerical sequences . As special cases, new results are obtained when {Xn, n ≥ 1} is a martingale difference sequence and when the Xn 's are of the form , where {an, n ≥ 1} is a sequence of constants and {Yn, n ≥ 1} is an orthogonal system of exchangeable random variables. An application to the Polya urn scheme is considered. The current work generalizes and simplifies some of the recent results of Nam and Rosalsky [15]

On the Central Limit Problem for Partially Exchangeable Random Variables with Values in a Hilbert Space

This paper deals with a central limit problem for columnwise exchangeable arrays of a Hilbert space valued random variables satisfying the condition of conditional uniform asymptotic negligibility. It is proved that rowwise sums may converge in distribution only to an exchangeable sequence of random variables whose distribution is a mixture of infinitely divisible distributions. Conditions for convergence to a specific distribution and, in particular, to a mixture of Gaussian distributions, are also given.

A maximal inequality for partial sums of finite exchangeable sequences of random variables

Let X1, X2,... , X2. be a finite exchangeable sequence of Banach space valued random variables, i.e., a sequence such that all joint distributions are invariant under permutations of the variables. We prove that there is an absolute constant c such that if S j-j=1 X_, then P( sup JISjll > A) A/c), 1 0. This generalizes an inequality of Montgomery-Smith and Latala for independent and identically distributed random variables. Our maximal inequality is apparently new even if X1,X2,... is an infinite exchangeable sequence of random variables. As a corollary of our result, we obtain a comparison inequality for tail probabilities of sums of arbitrary random variables over random subsets of the indices. Montgomery-Smith [8] and Latala [7] have independently proved that if X1,-... Xn are independent and identically distributed Banach space valued random variables, then (1) P( sup ZX, > AZ A/C I 0 and 1 < k < n, where c is an absolute constant. It is obvious that this cannot hold for arbitrary independent random variables; as MontgomerySmith [8] notes, we need only let k -= n = 2, X1 1 and X2 -1 to see this. Levy's inequality says that (1) also holds for arbitrary independent symmetric random variables Xi (not necessarily identically distributed). For positive random variables, (1) is trivial, of course. A natural and much-studied extension of the concept of independent and identically distributed random variables is that of exchangeable random variables. We say that a finite sequence X1, ... , Xn of (not necessarily independent) random variables is exchangeable if the n-tuples (XI, ... , Xn) and (X(I), ... , Xr(n)) both have the same distribution whenever ir is a permutation of [n] If{,... ,n}. Evidently an exchangeable sequence of independent random variables is precisely a sequence of independent and identically distributed random variables. Received by the editors August 2, 1996 and, in revised form, December 2, 1996. 1991 Mathematics Subject Classification. Primary 60E15.

Posterior robustness in simultaneous estimation problem with exchangeable contaminated priors

Let X 1, …, X p be independent random variables with densities ƒ i(x i∣θ i), i = 1, …, p . Suppose that θ 1, …, θ p are exchangeable random variables. In this paper we consider the Bayesian robustness or posterior sensitivity analysis with respect to the class of ε-contaminated prior distributions Γ AM = {(1 − ε)π 0 + εq: q ∈ Q AM } where L AM consists of all the exchangeable distributions on ( θ 1, …, θ p ) and for each distribution in L AM which can be expressed as a mixture of product of i.i.d. distributions. Most of the previous studies on posterior sensitivity analysis focus on a one-dimensional parameter space, and the posterior quantities considered in those analyses are the posterior probability of a subset of the parameter space, the posterior mean, and the posterior variance. The loss function is usually not taken into consideration, or if it is, only the zero-one or squared error loss is considered. It is worthwhile to consider other loss functions and other interesting posterior quantities, which are the goals of the paper. Under the weighted squared error loss L(θ,δ = ∑ i = 1 p w i(θ i)(0 i − δ i) 2 , we study the posterior sensitivity of the Bayes action, the posterior expected loss and some interested posterior quantities with respect to the class Γ AM. By using the de Finetti mixture representation of exchangeable random variables, the problem of finding the range of each component of the Bayes action over the class Γ AM can be reduced to a much smaller class of distributions with support at at most p points. The range of other posterior quantities can also be obtained in a similar way. One example for simultaneous estimation of independent Poisson means is given. In this example we compute the ranges of Bayes action and discuss their posterior robustness issues.

The role of the covariance matrix in the least-squares estimation for a common mean

For n > 1 let X = ( X 1,…, X n )′ have a mean vector θ 1 and covariance matrix σ 2 Σ, where 1 = (1,…,1)′, Σ is a known positive definite matrix, and σ 2 > 0 is either known or unknown. This model has been found useful when the observations X 1,…, X n from a population with mean θ are not independent. We show how the variance of θ , the least-squares estimator of θ, depends on the covariance structure of Σ. More specifically, we give expressions for Var( θ ), obtain its lower and upper bounds (which involve only the smallest and the largest eigenvalues of Σ), and show how the dependence of X 1,…, X n plays a role in Var θ . Examples of applications are given for M-matrices, for exchangeable random variables, for a class of covariance matrices with a block-correlation structure, and for twin data.

ON THE FOURTH MOMENT OF THE MAXIMUM OF PARTIAL SUMS OF $n$-EXCHANGEABLE RANDOM VARIABLES WITH APPLICATIONS

&#x0D; &#x0D; &#x0D; A general formula is obtained for the fourth moment of the maximum of partial sums of $n$-exchangeable random variables derived from a result of Spitzer. The formula is applied in particular to obtain the fourth moment of the maximum of adjusted partial sums of normal summands. This is of direct relevance to reservoir design and the analysis of the structure of stochastic processes and time series. &#x0D; &#x0D; &#x0D;

Ultra Logconcave Sequences and Negative Dependence

We prove that the convolution of two ultra-logconcave sequences is ultra-log-concave. This was conjectured recently by Pemantle and implies that a natural negative dependence property is preserved under the operation of “joining” families of exchangeable Bernoulli random variables.

A Glivenko-Cantelli theorem for exchangeable random variables

For an exchangeable sequence of random variables, almost surely, the difference between the empirical and the predictive distribution functions converges to zero uniformly.

On Extensions of an Inequality of Kolmogorov

The inequality of Kolmogorov (Sankhya, 1963) has been extended to a sequence of independent sub-Gaussian and other random variables. All the earlier results in the literature on this problem concerned only on the very special case of Bernoulli variables. We use martingale inequalities to establish a key result. A similar inequality is also proved for U-statistics based on exchangeable random variables.

On pairyise independent and independent exchangeable random variables

Let be an infinite sequence of exchangeable random variables. In this note, we prove that if are pairwise independent, then are mutually independent. A counter example is also provided to show that this result may not be true for a finite sequence of exchangeable random variables.