The mixing advantage is less than 2

Corresponding to n independent non-negative random variables X ₁,...,X _n , are values M ₁,...,M _n , where each M _i is the expected value of the maximum of n independent copies of X _i . We obtain an upper bound for the expected value of the maximum of X ₁,...,X _n in terms of M ₁,...,M _n . This inequality is sharp in the sense that the random variables can be chosen so that the bound is approached arbitrarily closely. We also present related comparison results.      

Three characterizations of non-binary correlation-immune and resilient functions

A functionf(X ₁,X ₂, ...,X _n ) is said to betth-order correlation-immune if the random variableZ=f(X ₁,X ₂,...,X _n ) is independent of every set oft random variables chosen from the independent equiprobable random variablesX ₁,X ₂,...,X _n . Additionally, if all possible outputs are equally likely, thenf is called at-resilient function. In this paper, we provide three different characterizations oft th-order correlation immune functions and resilient functions where the random variable is overGF (q). The first is in terms of the structure of a certain associated matrix. The second characterization involves Fourier transforms. The third characterization establishes the equivalence of resilient functions and large sets of orthogonal arrays.     

Bounds for the Concentration of Asymptotically Normal Statistics

Let X₁,..., X _n be independent, not necessarily identically distributed random variables. An optimal bound is derived for the concentration function of an arbitrary real-valued statistic T = T (X ₁,...,X _n ) for which ET² < . Applications are given for Wilcoxon"s rank-sum statistic, U-statistics, Student"s statistic, the two-sample Student statistic and linear regression.     

Berry–Esseen Bound for a Sample Sum from a Finite Set of Independent Random Variables

Let {X ₁,...,X _N} be a set of N independent random variables, and let S _n be a sum of n random variables chosen without replacement from the set {X ₁,...,X _N} with equal probabilities. In this paper we give an estimate of the remainder term for the normal approximation of S _n under mild conditions.     

Functions of elementary random variables and their application to system reliability

Summary LetX ₁,...,X _n be elementary random variables, i.e. random variables taking only finitely many values in . Then for an arbitray functionf(X ₁,...,X _n ) inX ₁,...,X _n a unique polynomial representation with the aid of Lagrange polynomials is given. This easily yields the moments as well as the distribution off(X ₁,...,X _n ) by terms of finitely many moments of the variablesX ₁,...,X _n . For n=1 a necessary and sufficient condition results thatm numbers are the firstm moments of a random variable takingm+1 different values. As an application of random functionsf(X ₁,...,X _n ) the reliability of technical systems with three states is treated.

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Zusammenfassung X ₁, ...,X _n seien elementare Zufallsvariable, d. h., Zufallsvariable, die nur endlich viele reelle Werte annehmen. Mit Hilfe von Lagrangepolynomen wird für eine beliebige Funktionf(X₁,...,X _n ) eine eindeutige polynomiale Darstellung angegeben. Daraus ergeben sich leicht die Momente wie auch die Verteilung von f(X₁,...,X _n ), ausgedrückt durch die Momente der VariablenX ₁,...,X _n . Fürn=1 erhält man eine notwendige und hinreichende Bedingung, daßm Zahlen die erstenm Momente einer Zufallsvariablen sind, diem+1 verschiedene Werte annimmt. Als Anwendung wird die Zuverlässigkeit eines technischen Systems mit drei Zuständen behandelt.

     

Extended large deviations

LetX ₁,X ₂,..., be a sequence ofi.i.d. random variables with a moment generating function finite in a neighborhood of 0. Further, for each integern1, letS _n denote the sum of the firstn terms in this sequence. We study the extended large deviation of such sums, meaning,P{S _n >n _n }, where _n is any sequence converging to infinity. We also derive functional extended large deviation theorems and then apply them to obtain functional versions of the Erdös-Rényi strong law of large numbers.Research partially supported by an NSF Grant.     

Inequalities for sums of independent geometrical random variables

Summary We give a survey of known results regarding Schur-convexity of probability distribution functions. Then we prove that the functionF(p ₁,...,p_n;t)=P(X₁+...+X_n≤t) is Schur-concave with respect to (p ₁,...,p_n) for every realt, whereX _i are independent geometric random variables with parametersp _i. A generalization to negative binomial random variables is also presented.     

Linear combination of concomitants of order statistics with application to testing and estimation

Summary Let (X ₁,Y ₁), (X ₂,Y ₂),…, (X _n,Y _n) be i.i.d. as (X, Y). TheY-variate paired with therth orderedX-variateX _rn is denoted byY _rn and terms the concomitant of therth order statistic. Statistics of the form are considered. The asymptotic normality ofT _n is established. The asymptotic results are used to test univariate and bivariate normality, to test independence and linearity ofX andY, and to estimate regression coefficient based on complete and censored samples.     

Limit theorems for trimmed sums

This paper studies the heavily trimmed sums (*) _{[ns] + 1} ^[nt] X _j ⁽ⁿ⁾ , where {X _j ⁽ⁿ⁾ } _{j = 1} ⁿ are the order statistics from independent random variables {X ₁,...,X _n } having a common distributionF. The main theorem gives the limiting process of (*) as a process oft. More smoothly trimmed sums like _{j = 1} ^[nt] J(j/n)X _j ⁽ⁿ⁾ are also discussed.     

Ratio comparisons of supremum and stop rule expectations

Summary Suppose X ₁,X ₂,...,X_n are independent non-negative random variables with finite positive expectations. Let T _n denote the stop rules for X ₁,...,X _n. The main result of this paper is that E(max{X ₁,...,X _n }) <2 sup{EX _t :tT _n }. The proof given is constructive, and sharpens the corresponding weak inequalities of Krengel and Sucheston and of Garling.Partially supported by AFOSR Grant F49620-79-C-0123     

An optimal Berry-Esseen type inequality for expectations of smooth functions

We provide an optimal Berry-Esseen type inequality for Zolotarev’s ideal ζ₃-metric measuring the difference between expectations of sufficiently smooth functions, like |·|³, of a sum of independent random variables X ₁,..., X _n with finite third-order moments and a sum of independent symmetric two-point random variables, isoscedastic to the X _i . In the homoscedastic case of equal variances, and in particular, in case of identically distributed X ₁,..., X _n the approximating law is a standardized symmetric binomial one. As a corollary, we improve an already optimal estimate of the accuracy of the normal approximation due to Tyurin (2009).     

On products of unbounded operators

A symmetric operator X^ is attached to each operator X that leaves the domain of a given positive operator A invariant and makes the product AX symmetric. Some spectral properties of X^ are derived from those of X and, as a consequence, various conditions ensuring positivity of products of the form AX ₁ ... X _n are proved. The question of ^-complete positivity of the mapping p → Ap(X ₁,...,X _n) defined on complex polynomials in n variables is investigated. It is shown that the set ω is related to the McIntosh-Pryde joint spectrum of (X ₁,...,X _n) in case all the operators A, X ₁,...,X _n are bounded. Examples illustrating the theme of the paper are included. This revised version was published online in June 2006 with corrections to the Cover Date.     

The exact density function of the ratio of two dependent linear combinations of chi-square variables

A computable expression is derived for the raw moments of the random variableZ=N/D whereN= ₁ ⁿ m _iX_i+ _n ^+1s m _iX_i,D= _n ^+1s l _iX_i+ _s ^+1r n _iX_i, and theX _i's are independently distributed central chi-square variables. The first four moments are required for approximating the distribution ofZ by means of Pearson curves. The exact density function ofZ is obtained in terms of sums of generalized hypergeometric functions by taking the inverse Mellin transform of theh-th moment of the ratioN/D whereh is a complex number. The casen=1,s=2 andr=3 is discussed in detail and a general technique which applies to any ratio having the structure ofZ is also described. A theoretical example shows that the inverse Mellin transform technique yields the exact density function of a ratio whose density can be obtained by means of the transformation of variables technique. In the second example, the exact density function of a ratio of dependent quardratic forms is evaluated at various points and then compared with simulated values.     

Hoeffding's inequalities: A counterexample

A sequence : ₀ satisfiesHoeffding's inequality of order n if wheneverX ₁,...,X _n are independent nonnegative integer-valued elementary random variables and are independent identically distributed nonnegative integer-valued elementary random variables, the common distribution of which is the average of those ofX ₁,...,X _n. We show that for each integerm greater than 2 there exists a sequence satisfying Hoeffding's inequality of every order greater thanm but not that of orderm. This answers a question raised by Berg, Christensen, and Ressel.     

On isometric stability of complemented subspaces of L p

We show that the Rudin-Plotkin isometry extension theorem inL _pimplies that whenX andY are isometric subspaces ofL _pandp is not an even integer, 1≤p<∞, thenX is complemented inL _pif and only ifY is; moreover, the constants of complementation ofX andY are equal. We provide examples demonstrating that this fact fails whenp is an even integer larger than 2.     

On the homology of certain spaces looped beyond their connectivity

Very little is known aboutH _*(Ωⁿ X) whenn is larger than the connectivity ofX. In this paper we calculate this whenX=Ω^∞ S ^∞ andn=1 or 2, and whenX=JU(q) or JSO(3) andn is arbitrary. Some information is also given whenX is a sphere. The authors were partially supported by the NSF.     

Divergence rates for the number of rare numbers

Suppose thatX ₁,X ₂, ... is a sequence of i.i.d. random variables taking value inZ ⁺. Consider the random sequenceA(X)(X ₁,X ₂,...). LetY _n be the number of integers which appear exactly once in the firstn terms ofA(X). We investigate the limit behavior ofY _n /E[Y _n ] and establish conditions under which we have almost sure convergence to 1. We also find conditions under which we dtermine the rate of growth ofE[Y _n ]. These results extend earlier work by the author.     

Limiting distributions of homogeneous functions of sample spacings for distributions approaching the uniform distribution

Summary X ₁,...,X _n are independent random variables, identically distributed over the unit interval, with common probability density function 1 + r(x)/n for all sufficiently large n, where is a positive constant, and |r(x)| <D. V ₁, ..., V _n+1 are the sample spacings generated by X ₁,..., X _n . It is shown that in many cases, the asymptotic joint distribution of homogeneous functions of V ₁,..., V _n+1 can be found directly from the asymptotic joint distribution of homogeneous functions of independent exponential random variables.Research supported by NSF Grant GP 3783.     

The bivariate gamma exponential distribution with application to drought data

The exponential and the gamma distributions have been the traditional models for drought duration and drought intensity data, respectively. However, it is often assumed that the drought duration and drought intensity are independent, which is not true in practice. In this paper, an application of the bivariate gamma exponential distribution is provided to drought data from Nebraska. The exact distributions ofR =X +Y,P =XY andW =X/(X +Y) and the corresponding moment properties are derived whenX andY follow this bivariate distribution.     

Strongly consistent estimators of k-th order regression curves and rates of convergence

Summary Let X and Y be two jointly distributed real valued random variables, and let the conditional distribution of X given Y be either in a Lebesgue exponential family or in a discrete exponential family. Let r_k be the k-th order regression curve of Y on X. Let X _n=(X ₁,..., X_n) be a random sample of size n on X. For a subset S of the real line R, statistics based on X_n are exhibited and sufficient conditions are given under which is close to O(n ^–1/2) with probability one. To obtain this result, with uf (u known and f unknown) denoting the unconditional (on y) density of X, the problem of estimating r _k (·) is reduced to the one of estimating f ^(k) (·)/f(·) if the density is wrt the Lebesgue measure on R and f ^(k) is the k-th order derivative of f; and to the one of estimating f(·+k)/f(·) if the density is wrt the counting measure on a countable subset of R.