On Mittag-Leffler functions and related distributions

The distribution F () = 1 – E (–), 0 < 1; 0 , where E (x) is the Mittag-Leffler function is studied here with respect to its Laplace transform. Its infinite divisibility and geometric infinite divisibility are proved, along with many other properties. Its relation with stable distribution is established. The Mittag-Leffler process is defined and some of its properties are deduced.     

Admissible minimax estimators of a mean vector of scale mixtures of multivariate normal distributions

The problem of estimating a mean vector of scale mixtures of multivariate normal distributions with the quadratic loss function is considered. For a certain class of these distributions, which includes at least multivariate-t distributions, admissible minimax estimators are given.     

On characterizations of distributions by mean absolute deviation and variance bounds

In this paper we present a bound for the mean absolute deviation of an arbitrary real-valued function of a discrete random variable. Using this bound we characterize a mixture of two Waring (hence geometric) distributions by linearity of a function involved in the bound. A double Lomax distribution is characterized by linearity of the same function involved in the analogous bound for a continuous distribution. Finally, we characterize the Pearson system of distributions and the generalized hypergeometric distributions by a quadratic function involved in a similar bound for the variance of a function of a random variable.     

Characterizations of mixtures of discrete distributions by a regression point

Given that the conditional distribution p_s(y|x) of Y, given X = x is an x-fold convolution of a nonnegative integer-valued r.v. ξ for every s= P[ξ = 0] > 0, the distribution of X, hence also of Y, is characterized by the regression point m(0) = E[X|Y = 0]. An infinite variety of generalized distributions (of Y) can be characterized by arbitrarily varying the distribution of X.     

Identifiability of Modified Power Series Mixtures via Posterior Means

Problems of specification of discrete bivariate statistical models by a modified power series conditional distribution and a regression function are studied. An identifiability result for a wide class of such mixtures with infinite support is obtained. Also the finite support case within a more specific model is considered. Applications for Poisson, (truncated) geometric, and binomial mixtures are given. From the viewpoint of Bayesian analysis unique determination of the prior by a Bayes estimate of the mean for modified power series mixtures is investigated.     

On a new system of discrete distributions and characterizations of several discrete distributions by equality of distributions

This paper deals with a new system of discrete distributions. It also gives several characterizations of the Waring (and hence the Yule) distribution (and its truncated versions), the super-Poisson, the discrete uniform and other discrete distributions by using this system and other such systems existing in the literature, and linear regression. Continuous analogues of the above results are also briefly discussed.     

Identifiability of the multinormal and other distributions under competing risks model

Let X₁, X₂ ,…, X_p be p random variables with joint distribution function F(x₁ ,…, x_p). Let Z = min(X₁, X₂ ,…, X_p) and I = i if Z = X_i. In this paper the problem of identifying the distribution function F(x₁ ,…, x_p), given the distribution Z or that of the identified minimum (Z, I), has been considered when F is a multivariate normal distribution. For the case p = 2, the problem is completely solved. If p = 3 and the distribution of (Z, I) is given, we get a partial solution allowing us to identify the independent case. These results seem to be highly nontrivial and depend upon Liouville's result that the (univariate) normal distribution function is a nonelementary function. Some other examples are given including the bivariate exponential distribution of Marshall and Olkin, Gumbel, and the absolutely continuous bivariate exponential extension of Block and Basu.     

On generalized stable and related laws

A probabilistic characterization is given for Schneider?s generalized stable laws. They belong to the larger family of laws which are invariant under length-biasing followed by a random beta scaling. Questions of infinite divisibility and self-decomposability are pursued. Moment determinacy of reciprocal generalized stable laws are investigated.     

Trace operators,Feynman distributions,and multiparameter white noise

Within the framework of white noise analysis on the probability space = ^* R ^d R ^M , the recent work by Johnson and Kallianpur on the Hu-Meyer formula, traces, and natural extensions is generalized to the multiparameter case:d>1. Besides providing a more general setting for these topics, the paper gives an alternative definition for the traces, a distributional version of the natural extension, and a generalized Kallianpur-Feynman distribution. The development illustrates how traces and natural extensions are intimately related to Wick products and the change of covariance formula from quantum field theory, as well as to the projective tensor product of Hilbert spaces from functional analysis.     

The block sphericity test—exact and near‐exact distributions for the likelihood ratio statistic

Using a suitable decomposition of the null hypothesis of the sphericity test for several blocks of variables, into a sequence of conditionally independent null hypotheses, we show that it is possible to obtain the expressions for the likelihood ratio test statistic, for its hth null moment, and for the characteristic function of its logarithm. The exact distribution of the logarithm of the likelihood ratio test statistic is obtained in the form of a sum of a generalized integer gamma distribution with the sum of a given number of independent logbeta distributions, taking the form of a single generalized integer gamma distribution when each set of variables has two variables. The development of near‐exact distributions arises, from the previous decomposition of the null hypothesis and from the consequent‐induced factorization of the characteristic function, as a natural and practical way to approximate the exact distribution of the test statistic. A measure based on the exact and approximating characteristic functions, which gives an upper bound on the distance between the corresponding distribution functions, is used to assess the quality of the near‐exact distributions proposed and to compare them with an asymptotic approximation on the basis of Box's method. Copyright © 2011 John Wiley & Sons, Ltd.     

A decomposition of the beta distribution,related order and asymptotic behavior

Let β _{v, w} be any beta variate with p.d.f.(Γ(v+w)/Γ(v)Γ(w))x ^v−1(1−x) ^w−1 and letU _{v, w} =-log β _{v, w} . ThenU _v,w =U ^CM +U ^PF , whereU ^CM andU ^PF are independent with completely monotone andPF _∞ densities, respectively. It is shown thatU _{v, w} is infinitely divisible and β _{v, w} correspondingly infinitely factorable. The asymptotic behavior ofU _{v, w} and β _{v, w} for largev, w is described. For different modes of increase ofv andw,U _{v, w} is asymptotically normal, gamma or extreme value distributed. The decomposition is employed to provide an algorithm for generating random β _{v, w} distributed numbers. Many of the results are based on insights provided by the classical theory of the Gamma function in the complex plane. This work was supported in part by the United States Air Force, Office of Scientific Research, under grant No. AFOSR-79-0043.     

Some properties of subexponential distributions

The nonnegative random variableX is said to have a subexponential distribution if we have (1-G(t))/(1-F(t))→2 ast→∞, whereF(t)=P{X≤t} andG(t) is the convolution ofF(t) with itself. Conditions on the distribution of independent nonnegative random variablesX andY such that max(X, Y) and min(X, Y) have a subexponential distribution are given. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 138–144, July, 1997. Translated by N. K. Kulman     

Explicit representation of roots on <Emphasis Type="Italic">p</Emphasis>-adic solenoids and non-uniqueness of embeddability into rational one-parameter subgroups

This note generalizes known results concerning the existence of roots and embedding one-parameter subgroups on p-adic solenoids. An explicit representation of the roots leads to the construction of two distinct rational embedding one-parameter subgroups. The results contribute to enlighten the group structure of solenoids and to point out difficulties arising in the context of the embedding problem in probability theory. As a consequence, the uniqueness of embedding of infinitely divisible probability measures on p-adic solenoids is solved under a certain natural condition.     

The distribution of the likelihood ratio for mixtures of densities from the one-parameter exponential family

We here consider testing the hypothesis ofhomogeneity against the alternative of a two-component mixture of densities. The paper focuses on the asymptotic null distribution of 2 log _n , where _n is the likelihood ratio statistic. The main result, obtained by simulation, is that its limiting distribution appears pivotal (in the sense of constant percentiles over the unknown parameter), but model specific (differs if the model is changed from Poisson to normal, say), and is not at all well approximated by the conventional ₍₂₎ ² -distribution obtained by counting parameters. In Section 3, the binomial with sample size parameter 2 is considered. Via a simple geometric characterization the case for which the likelihood ratio is 1 can easily be identified and the corresponding probability is found. Closed form expressions for the likelihood ratio _n are possible and the asymptotic distribution of 2 log _n is shown to be the mixture giving equal weights to the one point distribution with all its mass equal to zero and the ²-distribution with 1 degree of freedom. A similar result is reached in Section 4 for the Poisson with a small parameter value (0.1), although the geometric characterization is different. In Section 5 we consider the Poisson case in full generality. There is still a positive asymptotic probability that the likelihood ratio is 1. The upper precentiles of the null distribution of 2 log _n are found by simulation for various populations and shown to be nearly independent of the population parameter, and approximately equal to the (1–2)100 percentiles of ₍₁₎ ² . In Sections 6 and 7, we close with a study of two continuous densities, theexponential and thenormal with known variance. In these models the asymptotic distribution of 2 log _n is pivotal. Selected (1–) 100 percentiles are presented and shown to differ between the two models.     

Robust characterizations of k‐wise independence over product spaces and related testing results

A discrete distribution D over Σ₁ ×··· ×Σ_n is called (non‐uniform) k ‐wise independent if for any subset of k indices {i₁,…,i_k} and for any z₁∈Σ,…,z_k∈Σ, Pr_X～D[X···X = z₁···z_k] = Pr_X～D[X = z₁]···Pr_X～D[X = z_k]. We study the problem of testing (non‐uniform) k ‐wise independent distributions over product spaces. For the uniform case we show an upper bound on the distance between a distribution D from k ‐wise independent distributions in terms of the sum of Fourier coefficients of D at vectors of weight at most k. Such a bound was previously known only when the underlying domain is {0,1}ⁿ. For the non‐uniform case, we give a new characterization of distributions being k ‐wise independent and further show that such a characterization is robust based on our results for the uniform case. These results greatly generalize those of Alon et al. (STOC'07, pp. 496–505) on uniform k ‐wise independence over the Boolean cubes to non‐uniform k ‐wise independence over product spaces. Our results yield natural testing algorithms for k ‐wise independence with time and sample complexity sublinear in terms of the support size of the distribution when k is a constant. The main technical tools employed include discrete Fourier transform and the theory of linear systems of congruences.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Nonnull distributions of some statistics associated with testing for the equality of two covariance matrices

The nonnull distribution of some statistics, used for testing Σ₁ = Σ₂ are obtained as mixtures of incomplete beta functions as well as mixtures of incomplete gamma functions. The introduction of the convergence factors and certain recurrence relations are useful in the computation of the power of the tests as well as computation of exact percentage points for tests of significance.     

Positive solutions in the sense of distributions of singular boundary value problems

We obtain positive solutions in the sense of distributions of singular boundary value problems using perturbation and variational methods.

     

The multiplication of distributions in the study of delta shock wave for the nonlinear chromatography system

In the frame of $\alpha -$solutions defined in the setting of distributional products, the discontinuous solutions to the Riemann problem for a nonlinear chromatography system are constructed. All the discontinuous solutions are obtained within a convenient space of distributions including discontinuous functions and Dirac delta measures. The constructed $\alpha -$solutions are reasonable in comparison with the known results by using other techniques.