An identity on the maximum of a set of random variables

It is shown that if X₁, X₂, …, X_n are symmetric random variables and max(X₁, …, X_n)⁺ = max(0, X₁, …, X_n), then E[max(X₁,…,X_n)⁺]=[max(X₁,X₁,+X₂,+X₁,+X₃,…X₁,+X_n)⁺], and in the case of independent identically distributed symmetric random variables, E[max(X₁, X₂)⁺] = E[(X₁)⁺] + (1/2)E[(X₁ + X₂)⁺], so that for independent standard normal random variables, E[max(X₁, X₂)⁺] = (1/√2π)[1 + (1/√2)].     

On confidence sequences for the mean vector of a multivariate normal distribution

Let X₁, X₂,… be idd random vectors with a multivariate normal distribution N(μ, Σ). A sequence of subsets {R_n(a₁, a₂,…, a_n), n ≥ m} of the space of μ is said to be a (1 − α)-level sequence of confidence sets for μ if P(μ R_n(X₁, X₂,…, X_n) for every n ≥ m) ≥ 1 − α. In this note we use the ideas of Robbins Ann. Math. Statist. 41 (1970) to construct confidence sequences for the mean vector μ when Σ is either known or unknown. The constructed sequence R_n(X₁, X₂, …, X_n) depends on Mahalanobis' or Hotelling's according as Σ is known or unknown. Confidence sequences for the vector-valued parameter in the general linear model are also given.     

p-Norm bounds on the expectation of the maximum of a possibly dependent sample

Let X₁, X₂,…, X_n be identically distributed possibly dependent random variables with finite pth absolute moment assumed without loss of generality to be equal to 1. Denote the order statistics by X_1:n, X_2:n,…, X_n:n. Bounds are derived for E(X_n:n) when it is assumed that the X_i's are (i) arbitrarily dependent and (ii) independent. The effect of assuming a symmetric common distribution for the X_i's is discussed. Analogous bounds are described for the expected range of the sample. Bounds on expectations of general linear combinations of order statistics are described in the independent case.     

Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability

The basic result of the paper states: Let F₁, …, F_n, F₁^′,…, F_n^′ have proportional hazard functions with λ₁ ,…, λ_n , λ₁^′ ,…, λ_n^′ as the constants of proportionality. Let X₍₁₎ ≤ … ≤ X_(n) (X₍₁₎^′ ≤ … ≤ X_(n)^′) be the order statistics in a sample of size n from the heterogeneous populations {F₁ ,…, F_n}({F₁^′ ,…, F_n^′}). Then (λ₁ ,…, λ_n) majorizes (λ₁^′ ,…, λ_n^′) implies that (X₍₁₎ ,…, X_(n)) is stochastically larger than (X₍₁₎^′ ,…, X_(n)^′). Earlier results stochastically comparing individual order statistics are shown to be special cases. Applications of the main result are made in the study of the robustness of standard estimates of the failure rate of the exponential distribution, when observations actually come from a set of heterogeneous exponential distributions. Further applications are made to the comparisons of linear combinations of Weibull random variables and of binomial random variables.     

Some New Results on Stochastic Comparisons of Spacings from Heterogeneous Exponential Distributions

Some new results are obtained on stochastic orderings between random vectors of spacings from heterogeneous exponential distributions and homogeneous ones. LetD₁, …, D_nbe the normalized spacings associated with independent exponential random variablesX₁, …,X_n, whereX_ihas hazard rateλ_i,i=1, 2, …, n. LetD*₁, …, D*_nbe the normalized spacings of a random sampleY₁, …, Y_nof sizenfrom an exponential distribution with hazard rateλ=∑ⁿ_i=1 λ_i/n. It is shown that for anyn2, the random vector (D₁, …, D_n) is greater than the random vector (D*₁, …, D*_n) in the sense of multivariate likelihood ratio ordering. It also follows from the results proved in this paper that for anyjbetween 2 andn, the survival function ofX_j:n−X_1:nis Schur convex.     

Study of some measures of dependence between order statistics and systems

Let be a random vector, and denote by X_1:n,X_2:n,…,X_n:n the corresponding order statistics. When X₁,X₂,…,X_n represent the lifetimes of n components in a system, the order statistic X_n−k+1:n represents the lifetime of a k-out-of-n system (i.e., a system which works when at least k components work). In this paper, we obtain some expressions for the Pearson’s correlation coefficient between X_i:n and X_j:n. We pay special attention to the case n=2, that is, to measure the dependence between the first and second failure in a two-component parallel system. We also obtain the Spearman’s rho and Kendall’s tau coefficients when the variables X₁,X₂,…,X_n are independent and identically distributed or when they jointly have an exchangeable distribution.     

Multivariate Liouville distributions

A random vector (X₁, …, X_n), with positive components, has a Liouville distribution if its joint probability density function is of the formf(x₁ + … + x_n)x₁^a₁.1 … x_n^a_n.1 with thea_i all positive. Examples of these are the Dirichlet and inverted Dirichlet distributions. In this paper, a comprehensive treatment of the Liouville distributions is provided. The results pertain to stochastic representations, transformation properties, complete neutrality, marginal and conditional distributions, regression functions, and total positivity and reverse rule properties. Further, these topics are utilized in various characterizations of the Dirichlet and inverted Dirichlet distributions. Matrix analogs of the Liouville distributions are also treated, and many of the results obtained in the vector setting are extended appropriately.     

On a characterization of the exponential distribution by spacings

Summary LetX be a non-negative random variable with probability distribution functionF. SupposeX _i,n (i=1,…,n) is theith smallest order statistics in a random sample of sizen fromF. A necessary and sufficient condition forF to be exponential is given which involves the identical distribution of the random variables (n−i)(X _i+1,n−X_i,n) and (n−j)(X _j+1,n−X_j,n) for somei, j andn, (1≦i<j<n). The work was partly completed when the author was at the Dept. of Statistics, University of Brasilia, Brazil.     

Domains of attraction of nonnormal operator-stable laws

A sequence of independent, identically distributed random vectors X₁, X₂, X₃,… is said to belong to the domain of attraction of a random vector Y is there exist linear operators A_n and constant vectors b_n such that A_n(X₁,…, X_n)+b_n converges in distribution to Y. We present a simple, necessary, and sufficient condition for the existence of such A_n, B_n in the case where Y has no normal component.     

On the Maximal Multiplicity of Parts in a Random Integer Partition

We study the asymptotic behavior of the maximal multiplicity μ_n = μ_n(λ) of the parts in a partition λ of the positive integer n, assuming that λ is chosen uniformly at random from the set of all such partitions. We prove that πμ_n/(6n)^1/2 converges weakly to max _jX_j/j as n→∞, where X₁, X₂, … are independent and exponentially distributed random variables with common mean equal to 1.2000 Mathematics Subject Classification: Primary—05A17; Secondary—11P82, 60C05, 60F05     

L1-optimal estimates for a regression type function in R

Let X₁, X₂, …, X_n be random vectors that take values in a compact set in R^d, d ≥ 1. Let Y₁, Y₂, …, Y_n be random variables (“the responses”) which conditionally on X₁ = x₁, …, X_n = x_n are independent with densities f(y | x_i, θ(x_i)), i = 1, …, n. Assuming that θ lives in a sup-norm compact space Θ_q,d of real valued functions, an optimal L₁-consistent estimator of θ is constructed via empirical measures. The rate of convergence of the estimator to the true parameter θ depends on Kolmogorov's entropy of Θ_q,d.     

The asymptotic distributions of sums of record values for distributions with lognormal-type tails

Let X1,X2,... be a sequence of i.i.d. random variables and let X(1),X(2),... be the associatedrecord value sequence. We focus on the asymptotic distributions of sums of records, Tn=∑nk=1X(k), forX1 ∈ LN(γ). In this case, we find that 2 is a strange point for parameter γ. When γ＞ 2, Tn is asymptoticallynormal, while for 2 ＞γ＞ 1, we prove that Tn cannot converge in distribution to any non-degenerate lawthrough common centralizing and normalizing and log Tn is asymptotically normal.     

On the approximate behavior of the posterior distribution for an extreme multivariate observation

The behavior of the posterior for a large observation is considered. Two basic situations are discussed; location vectors and natural parameters.Let X = (X₁, X₂, …, X_n) be an observation from a multivariate exponential distribution with that natural parameter Θ = (Θ₁, Θ₂, …, Θ_n). Let θ_x^* be the posterior mode. Sufficient conditions are presented for the distribution of Θ − θ_x^* given X = x to converge to a multivariate normal with mean vector 0 as |x| tends to infinity. These same conditions imply that E(Θ | X = x) − θ_x^* converges to the zero vector as |x| tends to infinity.The posterior for an observation X = (X₁, X₂, …, X_n is considered for a location vector Θ = (Θ₁, Θ₂, …, Θ_n) as x gets large along a path, γ, in Rⁿ. Sufficient conditions are given for the distribution of γ(t) − Θ given X = γ(t) to converge in law as t → ∞. Slightly stronger conditions ensure that γ(t) − E(Θ | X = γ(t)) converges to the mean of the limiting distribution.These basic results about the posterior mean are extended to cover other estimators. Loss functions which are convex functions of absolute error are considered. Let δ be a Bayes estimator for a loss function of this type. Generally, if the distribution of Θ − E(Θ | X = γ(t)) given X = γ(t) converges in law to a symmetric distribution as t → ∞, it is shown that δ(γ(t)) − E(Θ | X = γ(t)) → 0 as t → ∞.     

Some remarks on the strong law of large numbers for Banach-space valued weakly integrable random variables

The purpose of this paper is to show the equivalence of almost sure convergence of S_n/n, n ≥ 1 and lim sup_n→∞S_n/n < ∞ a.e., where S_n = X₁ + X₂ + … + X_n and X₁, X₂,… are independent identically distributed random elements in a separable Banach space with EX₁ < ∞. This result disproves a result of Pop-Stojanovic [8].     

Checking the adequacy of the multivariate semiparametric location shift model

Let X,X₁,…,X_m,…, Y,Y₁,…,Y_n,… be independent d-dimensional random vectors, where the X_j are i.i.d. copies of X, and the Y_k are i.i.d. copies of Y. We study a class of consistent tests for the hypothesis that Y has the same distribution as X+μ for some unspecified . The test statistic L is a weighted integral of the squared modulus of the difference of the empirical characteristic functions of and Y₁,…,Y_n, where is an estimator of μ. An alternative representation of L is given in terms of an L²-distance between two nonparametric density estimators. The finite-sample and asymptotic null distribution of L is independent of μ. Carried out as a bootstrap or permutation procedure, the test is asymptotically of a given size, irrespective of the unknown underlying distribution. A large-scale simulation study shows that the permutation procedure performs better than the bootstrap.     

Characterizations of multivariate normality II. Through linear regressions

It is established that a vector (X′₁, X′₂, …, X′_k) has a multivariate normal distribution if (i) for each X_i the regression on the rest is linear, (ii) the conditional distribution of X₁ about the regression does not depend on the rest of the variables, and (iii) the conditional distribution of X₂ about the regression does not depend on the rest of the variables, provided that the regression coefficients satisfy some more conditions that those given by [4]J. Multivar. Anal. 6 81–94].     

Third-order asymptomic properties of a class of test statistics under a local alternative

Suppose that {X_i; I = 1, 2, …,} is a sequence of p-dimensional random vectors forming a stochastic process. Let p_n, θ(X_n), X_n ^np, be the probability density function of X_n = (X₁, …, X_n) depending on θ Θ, where Θ is an open set of ¹. We consider to test a simple hypothesis H : θ = θ₀ against the alternative A : θ ≠ θ₀. For this testing problem we introduce a class of tests , which contains the likelihood ratio, Wald, modified Wald, and Rao tests as special cases. Then we derive the third-order asymptotic expansion of the distribution of T under a sequence of local alternatives. Using this result we elucidate various third-order asymptotic properties of T (e.g., Bartlett's adjustments, third-order asymptotically most powerful properties). Our results are very general, and can be applied to the i.i.d. case, multivariate analysis, and time series analysis. Two concrete examples will be given. One is a Gaussian ARMA process (dependent case), and the other is a nonlinear regression model (non-identically distributed case).     

On the transition from a Markov chain to a continuous time process

Starting from a real-valued Markov chain X₀,X₁,…,X_n with stationary transition probabilities, a random element {Y(t);t[0, 1]} of the function space D[0, 1] is constructed by letting Y(k/n)=X_k, k= 0,1,…,n, and assuming Y (t) constant in between. Sample tightness criteria for sequences {Y(t);t[0,1]};_n of such random elements in D[0, 1] are then given in terms of the one-step transition probabilities of the underlying Markov chains. Applications are made to Galton-Watson branching processes.     

A uniform bound for the deviation of empirical distribution functions

If X₁, …, X_n are independent R^d-valued random vectors with common distribution function F, and if F_n is the empirical distribution function for X₁, …, X_n, then, among other things, it is shown that P{sup_x F_n(x) ε} 2e²(2n)^de^−2nε2 for all nε² ≥ d². The inequality remains valid if the X_i are not identically distributed and F(x) is replaced by Σ_iP{X_i ≤ x}/n.     

Asymptotic distribution of the increase of the largest canonical correlation when one of the vectors is augmented

Let r₁ > r₂ > … be the sample canonical correlations in a sample of size n from a multivariate normal population partitioned into two subvectors with population canonical correlations ₁ > ₂ > …. Let one of the subvectors be augmented by adding one or more variables to it. For the increase in the largest canonical correlation, Δr in the sample and Δ in the population, it is shown that √n(Δr − Δ) → ^DN(0, σ²) and a formula for σ² is derived.