Regression of polynomial statistics on the sample mean and natural exponential families

A polynomial Q = Q(X ₁, …, X _n ) of degree m in independent identically distributed random variables with distribution function F is an unbiased estimator of a functional q(α ₁(F), …, α _m (F)), where q(u ₁, …, u _m ) is a polynomial in u ₁, …, u _m and α _j (F) is the jth moment of F (assuming the necessary moment of F exists). It is shown that the relation E(Q | X ₁ + … + X n) = 0 holds if and only if q(α ₁(θ), …, α _m (θ)) ≡ 0, where α _j (θ) is the jth moment of the natural exponential family generated by F. This result, based on the fact that X ₁ + … + X n is a complete sufficient statistic for a parameter θ in a sample from a natural exponential family of distributions F _θ(x) = ∫_−∞ ^x e ^θu−k(θ) dF(u), explains why the distributions appearing as solutions of regression problems are the same as solutions of problems for natural exponential families though, at the first glance, the latter seem unrelated to the former.     

Curved exponential families of stochastic processes and their envelope families

Exponential families of stochastic processes are usually curved. The full exponential families generated by the finite sample exponential families are called the envelope families to emphasize that their interpretation as stochastic process models is not straightforward. A general result on how to calculate the envelope families is given, and the interpretation of these families as stochastic process models is considered. For Markov processes rather explicit answers are given. Three examples are considered some in detail: Gaussian autoregressions, the pure birth process and the Ornstein-Uhlenbeck process. Finally, a goodness-of-fit test for censored data is discussed.     

Some characterizations of families of distributions,including logistic and exponential ones,by properties of order statistics

New characterizations of distributions based on properties of the maximal order statistics are obtained. The families of distributions that are characterized by some properties of maxima include exponential and logistic distributions as partial cases. Bibliography: 4 titles.     

Strong laws for exponential order statistics and spacings

We obtain upper and lower almost sure asymptotic bounds for spacings η _{K, N} − η _{K + 1, N} , where η _{1, N} > η _{2, N} > ⋯ > η _{N, N} are the order statistics of independent exponentially distributed random variables η ₁, η ₂, ..., η _N with mean 1; here N → ∞, and K = 1, 2, ... is fixed. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 477–491, October–December, 2006.     

On families of additive exponential sums

In this paper we compute geometric monodromy groups of additive exponential sums over BbbAⁿ. Our approach builds on work of N. Katz, and involves p-adic analysis of explicit sums and computation of the Galois group of an equation over a function field in characteristic 2. The paper also provides a brief historical outline of the problem and lists previously known results.     

Saddlepoint approximations for curved exponential families

An approximation of the density of the maximum likelihood estimator in curved exponential families is derived using a saddlepoint expansion. The approximation is particularly simple in nonlinear regression. An example is considered.     

k-U统计量的指数收敛速度

研究了k-U统计量的收敛速度,在一组适当的正则条件下,获得了k-U统计量的指数收敛速度,推广了U-统计量的指数收敛速度的相应结果.     

Lévy homeomorphic parametrization and exponential families

Summary Let P={P : } be an exponential family of probability distributions with the canonical parameter and consider the one to one mapping : P . It is shown that, under mild regularity assumptions, and ^–1 are continuous with respect to the Lévy metric in P and Euclidean metric in .     

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.     

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.     

A characterization of certain discrete exponential families

For independent random variables X and Y, define % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaabofaruWrL9MCNLwyaGqbciaa-bcacqGHHjIUcaWFGaGaa8hw% aiaa-TcacaWFzbaaaa!4551!\[{\rm{S}} \equiv X + Y\]. When the conditional expectations % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadweacaGGBbqefCuzVj3zPfgaiuGajaaqcaWFNbGccaGGOaGa% amiwaiaacMcacaGG8bGaam4uaiaac2facqGHHjIUcaWGHbGaaiikai% aadofacaGGPaaaaa!4BC4!\[E[g(X)|S] \equiv a(S)\]and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadweacaGGBbGaamiAaiaacIcacaWGybGaaiykaiaacYhacaWG% tbGaaiyxaiabggMi6kaadkgacaGGOaGaam4uaiaacMcaaaa!4894!\[E[h(X)|S] \equiv b(S)\]are given, then under certain assumptions, the density function of X has the form of u(x)k()e^ax, where u(x) is uniquely determined by the functions a(·) and b(·).     

Invariant finite-dimensional exponential families on homogeneous spaces

A general form is obtained for the finite-dimensional exponential family invariant with respect to a locally compact group G when it is defined on the measurable quotient spade (G/,A,) of this group with respect to a subgroup . Conditions for the existence of such families are derived. Examples are given of exponential families on a compact homogeneous space, and the general form of families in R_n invariant with respect to GL(n) is obtained.Translated from Matematicheskie Zametki, Vol. 7, No. 6, pp. 707–715, June, 1970.The author wishes to thank V. N. Tutubalin for his great interest in this work.     

Linear hash families and forbidden configurations

The classical orthogonal arrays over the finite field underlie a powerful construction of perfect hash families. By forbidding certain sets of configurations from arising in these orthogonal arrays, this construction yields previously unknown perfect, separating, and distributing hash families. When the strength s of the orthogonal array, the strength t of the hash family, and the number of its rows are all specified, the forbidden sets of configurations can be determined explicitly. Each forbidden set leads to a set of equations that must simultaneously hold. Hence computational techniques can be used to determine sufficient conditions for a perfect, separating, and distributing hash family to exist. In this paper the forbidden configurations, resulting equations, and existence results are determined when (s, t) ∈ {(2, 5), (2, 6), (3, 4), (4, 3)}. Applications to the existence of covering arrays of strength at most six are presented.      

The diagnonal multivariate natural exponential families and their classification

A natural exponential family (NEF)F in ? ⁿ ,n>1, is said to be diagonal if there existn functions,a ₁,...,a _n , on some intervals of ?, such that the covariance matrixV _F (m) ofF has diagonal (a ₁(m ₁),...,a _n (m _n )), for allm=(m ₁,...,m _n ) in the mean domain ofF. The familyF is also said to be irreducible if it is not the product of two independent NEFs in ? ^k and ? ^n-k , for somek=1,...,n?1. This paper shows that there are only six types of irreducible diagonal NEFs in ? ⁿ , that we call normal, Poisson, multinomial, negative multinomial, gamma, and hybrid. These types, with the exception of the latter two, correspond to distributions well established in the literature. This study is motivated by the following question: IfF is an NEF in ? ⁿ , under what conditions is its projectionp(F) in ? ^k , underp(x ₁,...,x _n )∶=(x ₁,...,x _k ),k=1,...,n?1, still an NEF in ? ^k ? The answer turns out to be rather predictable. It is the case if, and only if, the principalk×k submatrix ofV _F (m ₁,...,m _n ) does not depend on (m _k+1,...,m _n ).