Structure of Stieltjes classes of moment-equivalent probability laws

A Stieltjes class is a one-parameter family of moment-equivalent distribution functions constructed by modulation of a given indeterminate distribution function F, called the center of the class. Members of a Stieltjes class are mutually absolutely continuous, and conversely, any pair of moment-equivalent and mutually absolutely continuous distribution functions can be joined by a Stieltjes class. The center of a Stieltjes class is an equally weighted mixture of its extreme members, and this places restrictions on which distributions can belong to a Stieltjes class with a given center. The lognormal law provides interesting illustrations of the general ideas. In particular, it is possible for two moment equivalent infinitely divisible distributions to be joined by a Stieltjes class, and random scaling can be used to construct new Stieltjes classes from a given Stieltjes class.     

Discrete isoperimetric and Poincaré-type inequalities

We study some discrete isoperimetric and Poincaré-type inequalities for product probability measures μ ⁿ on the discrete cube {0, 1} ⁿ and on the lattice Z ⁿ . In particular we prove sharp lower estimates for the product measures of boundaries of arbitrary sets in the discrete cube. More generally, we characterize those probability distributions μ on Z which satisfy these inequalities on Z ⁿ . The class of these distributions can be described by a certain class of monotone transforms of the two-sided exponential measure. A similar characterization of distributions on R which satisfy Poincaré inequalities on the class of convex functions is proved in terms of variances of suprema of linear processes. Received: 30 April 1997 / Revised version: 5 June 1998     

Admissible and minimax multiparameter estimation in exponential families

Consider p independent distributions each belonging to the one parameter exponential family with distribution functions absolutely continuous with respect to Lebesgue measure. For estimating the natural parameter vector with p ≥ p₀ (p₀ is typically 2 or 3), a general class of estimators dominating the minimum variance unbiased estimator (MVUE) or an estimator which is a known constant multiple of the MVUE is produced under different weighted squared error losses. Included as special cases are some results of Hudson [13] and Berger [5]. Also, for a subfamily of the general exponential family, a class of estimators dominating the MVUE of the mean vector or an estimator which is a known constant multiple of the MVUE is produced. The major tool is to obtain a general solution to a basic differential inequality.     

On the Paper "Asymptotics for the Moments of Singular Distributions"

In their 1993 paper, W. Goh and J. Wimp derive interesting asymptotics for the moments c_n(α) ≡ c_n = ∫¹₀tⁿdα(t), N = 0, 1, 2, ..., of some singular distributions α (with support [0, 1]), which contain oscillatory terms. They suspect, that this is a general feature of singular distributions and that this behavior provides a striking contrast with what happens for absolutely continuous distributions. In the present note, however, we give an example of an absolutely continuous measure with asymptotics of moments containing oscillatory terms, and an example of a singular measure having very regular asymptotic behavior of its moments. Finally, we give a short proof of the fact that the drop-off rate of the moments is exactly the local measure dimension about 1 (if it exists).     

Large sample properties of the mle and mcle for the natural parameter of a truncated exponential family

Summary Consider a truncated exponential family of absolutely continuous distributions with natural parameter θ and truncation parameter γ. Strong consistency and asymptotic normality are shown to hold for the maximum likelihood and maximum conditional likelihood estimates of θ with γ unknown. Moreover, these two estimates are also shown to have the same limiting distribution, coinciding with that of the maximum likelihood estimate for θ when γ is assumed to be known.     

Spectral structure of Anderson type Hamiltonians

We study self adjoint operators of the form?H _ω = H ₀ + ∑λ_ω(n) <δ _n ,·>δ _n ,?where the δ _n ’s are a family of orthonormal vectors and the λ_ω(n)’s are independently distributed random variables with absolutely continuous probability distributions. We prove a general structural theorem saying that for each pair (n,m), if the cyclic subspaces corresponding to the vectors δ _n and δ _m are not completely orthogonal, then the restrictions of H _ω to these subspaces are unitarily equivalent (with probability one). This has some consequences for the spectral theory of such operators. In particular, we show that “well behaved” absolutely continuous spectrum of Anderson type Hamiltonians must be pure, and use this to prove the purity of absolutely continuous spectrum in some concrete cases. Oblatum 27-V-1999 & 6-I-2000?Published online: 8 May 2000     

Simultaneous estimation of parameters in exponential families

Summary The paper considers estimation of the natural parameter vector or the mean vector from independent distributions each belonging to the one-parameter discrete or absolutely continuous exponential family. The usual estimators (maximum likelihood, minimum variance unbiased or best invariant) are improved simultaneously under various weighted squared error losses. Research supported by the NSF Grant Number MCS-8202116.     

Trimmed estimates in simultaneous estimation of parameters in exponential families

Let X₁,…, X_p be p (≥ 3) independent random variables, where each X_i has a distribution belonging to the one-parameter exponential family of distributions. The problem is to estimate the unknown parameters simultaneously in the presence of extreme observations. C. Stein (Ann. Statist.9 (1981), 1135–1151) proposed a method of estimating the mean vector of a multinormal distribution, based on order statistics corresponding to the |X_i|'s, which permitted improvement over the usual maximum likelihood estimator, for long-tailed empirical distribution functions. In this paper, the ideas of Stein are extended to the general discrete and absolutely continuous exponential families of distributions. Adaptive versions of the estimators are also discussed.     

Equipartition of a continuous mass distribution

Here are three samples of results. (1) Let m be a finite (absolutely) continuous mass distribution in ℝ², and let ℓ = {ℓ₁, ..., ℓ₅ ⊂ ℝ²} be a quintuple of rays with common origin such that any two adjacent angles between them make a sum of at most π. Then an affine image of ℓ subdivides m into five parts with any prescribed ratios. (2) For each finite continuous mass distribution m in ℝⁿ, there exist n mutually orthogonal hyperplanes any two of which quarter m. (3) Let m and m′ be two finite continuous mass distributions in ℝRⁿ with common center of symmetry O. Then there exist n hyperplanes through O any two of which quarter both m and m′. Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 92–106.     

Limit theorems for a nonhomogeneous Ornstein-Uhlenbeck process

We describe a construction of a summation scheme with replacements of random variables. As the limit, we obtain a time nonhomogeneous generalization of the Ornstein-Uhlenbeck process. This generalization is described by a transform of Lamperti type in which an arbitrary continuous monotone function is used instead of the exponential one. Bibliography: 16 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 339, 2006, pp. 111–134.     

Reliability and expectation bounds for coherent systems with exchangeable components

Sharp upper and lower bounds are obtained for the reliability functions and the expectations of lifetimes of coherent systems based on dependent exchangeable absolutely continuous components with a given marginal distribution function, by use of the concept of Samaniego's signature. We first show that the distribution of any coherent system based on exchangeable components with absolutely continuous joint distribution is a convex combination of distributions of order statistics (equivalent to the k-out-of-n systems) with the weights identical with the values of the Samaniego signature of the system. This extends the Samaniego representation valid for the case of independent and identically distributed components. Combining the representation with optimal bounds on linear combinations of distribution functions of order statistics from dependent identically distributed samples, we derive the corresponding reliability and expectation bounds, dependent on the signature of the system and marginal distribution of dependent components. We also present the sequences of exchangeable absolutely continuous joint distributions of components which attain the bounds in limit. As an application, we obtain the reliability bounds for all the coherent systems with three and four exchangeable components, expressed in terms of the parent marginal reliability function and specify the respective expectation bounds for exchangeable exponential components, comparing them with the lifetime expectations of systems with independent and identically distributed exponential components.     

Finding upper-triangular representations for phase-type distributions with 3 distinct real poles

In this paper we seek particular representations for absolutely continuous Phase-type distributions with 3 distinct real poles. First, we define subsets of these Phase-type distributions given the 3 distinct poles. One subset contains distributions that have upper triangular PH representation of order n, but do not have a triangular one of PH order n−1. This is done by using the invariant polytope approach. For any distribution in our subsets we give an invariant polytope containing the corresponding distribution by finding the vertices of the polytope. Second, we propose a method that actually constructs the generator matrix of the required PH representation from the invariant polytope. Consequently, our method constructs an upper triangular PH representation that has minimal order among the upper triangular PH representations given the probability density function of a PH distribution.     

Estimating a bounded parameter for symmetric distributions

For the problem of estimating under squared error loss the parameter of a symmetric distribution which is subject to an interval constraint, we develop general theory which provides improvements on various types of inadmissible procedures, such as maximum likelihood procedures. The applications and further developments given include: (i) symmetric location families such as the exponential power family including double-exponential and normal, Student and Cauchy, a Logistic type family, and scale mixture of normals in cases where the variance is lower bounded; (ii) symmetric exponential families such as those related to a Binomial(n,p) model with bounded |p−1/2| and to a Beta(α + θ, α −θ) model; and (iii) symmetric location distributions truncated to an interval (−c,c). Finally, several of the dominance results are studied with respect to model departures yielding robustness results, and specific findings are given for scale mixture of normals and truncated distributions. Research supported by NSERC of Canada.     

Double shrinkage estimation of ratio of scale parameters

The problems of estimating ratio of scale parameters of two distributions with unknown location parameters are treated from a decision-theoretic point of view. The paper provides the procedures improving on the usual ratio estimator under strictly convex loss functions and the general distributions having monotone likelihood ratio properties. In particular,double shrinkage improved estimators which utilize both of estimators of two location parameters are presented. Under order restrictions on the scale parameters, various improvements for estimation of the ratio and the scale parameters are also considered. These results are applied to normal, lognormal, exponential and pareto distributions. Finally, a multivariate extension is given for ratio of covariance matrices.     

Characterization of types of asymptotic discernibility of families of hypotheses

We give a characterization of the types of asymptotic discernibility of families of hypotheses in the case of hypothetical measures that are not, in general, mutually absolutely continuous. The case when the logarithm of the likelihood ratio admits an asymptotic expansion of the type of an expansion with local asymptotic normality is examined in detail. Examples are studied.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 64–71, 1987.     

On Distributional Properties of Perpetuities

We study probability distributions of convergent random series of a special structure, called perpetuities. By giving a new argument, we prove that such distributions are of pure type: degenerate, absolutely continuous, or continuously singular. We further provide necessary and sufficient criteria for the finiteness of p-moments, p>0, as well as exponential moments. In particular, a formula for the abscissa of convergence of the moment generating function is provided. The results are illustrated with a number of examples at the end of the article.      

Multivariate distributions having Weibull properties

Random variables X₁ ,…, X_n are said to have a joint distribution with Weibull minimums after arbitrary scaling if min_i(a_iX_i) has a one dimensional Weibull distribution for arbitrary constants a_i > 0, i = 1,…, n. Some properties of this class are demonstrated, and some examples are given which show the existence of a number of distributions belonging to the class. One of the properties is found to be useful for computing component reliability importance. The class is seen to contain an absolutely continuous Weibull distribution which can be generated from independent uniform and gamma distributions.     

Abelian theorems for a class of probability distributions inR d and their application

A class of multidimensional absolutely continuous distributions is considered. Each of them has a moment-generating function that is finite in a bounded set S and, therefore, generates a family of so-called conjugate or associated distributions. At the focus of our attention are the limiting distributions for this family that appear as the conjugating parameter tends to the boundary of S. As in the one-dimensional case, each such limiting distribution can be obtained as a consequence of an Abelian theorem. Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part II.     

Second order asymptotic comparison of estimators of a common parameter in the double exponential case

Summary The problem to estimate a common parameter for the pooled sample from the double exponential distributions is discussed in the presence of nuisance parameters. The maximum likelihood estimator, a weighted median, a weighted mean and others are asymptotically compared up to the second order, i.e. the ordern ^−1/2 with the asymptotic expansions of their distributions. University of Electro-communications