Stochastic Comparisons and Dependence among Concomitants of Order Statistics

Let (X_i, Y_i) i=1, 2, …, n be n independent and identically distributed random variables from some continuous bivariate distribution. If X_(r) denotes the rth ordered X-variate then the Y-variate, Y_[r], paired with X_(r) is called the concomitant of the rth order statistic. In this paper we obtain new general results on stochastic comparisons and dependence among concomitants of order statistics under different types of dependence between the parent random variables X and Y. The results obtained apply to any distribution with monotone dependence between X and Y. In particular, when X and Y are likelihood ratio dependent, it is shown that the successive concomitants of order statistics are increasing according to likelihood ratio ordering and they are TP₂ dependent in pairs. If we assume that the conditional hazard rate of Y given X=x is decreasing in x, then the concomitants are increasing according to hazard rate ordering and are dependent according to the right corner set increasing property. Finally, it is proved that if Y is stochastically increasing in X, then the concomitants of order statistics are stochastically increasing and are associated. Analogous results are obtained when the variables X and Y are negatively dependent. We also prove that if the hazard rate of the conditional distribution of Y given X=x is decreasing in x and y, then the concomitants have DFR (decreasing failure rate) distributions and are ordered according to dispersive ordering.     

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.     

A Kernel Estimator of a Conditional Quantile

Let (X₁, Y₁), (X₂, Y₂), …, be two-dimensional random vectors which are independent and distributed as (X, Y). For 0<p<1, letξ(px) be the conditionalpth quantile ofYgivenX=x; that is,ξ(px)=inf{y : P(YyX=x)p}. We consider the problem of estimatingξ(px) from the data (X₁, Y₁), (X₂, Y₂), …, (X_n, Y_n). In this paper, a new kernel estimator ofξ(px) is proposed. The asymptotic normality and a law of the iterated logarithm are obtained.     

Dependence Properties of Multivariate Mixture Distributions and Their Applications

Consider a multivariate mixture model where the random variables X ₁, ..., X _n given (₁, ..., _n ), are conditionally independent. Conditions are obtained under which different kinds of positive dependence hold among X _i 's. The results obtained are applied to a variety of problems including the concomitants of order statistics and of record values; and to frailty models.     

First-order optimality conditions in set-valued optimization

A a set-valued optimization problem min _C F(x), x ∈X ₀, is considered, where X ₀ ⊂ X, X and Y are normed spaces, F: X ₀ ⊂ Y is a set-valued function and C ⊂ Y is a closed cone. The solutions of the set-valued problem are defined as pairs (x ⁰,y ⁰), y ⁰ ∈F(x ⁰), and are called minimizers. The notions of w-minimizers (weakly efficient points), p-minimizers (properly efficient points) and i-minimizers (isolated minimizers) are introduced and characterized through the so called oriented distance. The relation between p-minimizers and i-minimizers under Lipschitz type conditions is investigated. The main purpose of the paper is to derive in terms of the Dini directional derivative first order necessary conditions and sufficient conditions a pair (x ⁰, y ⁰) to be a w-minimizer, and similarly to be a i-minimizer. The i-minimizers seem to be a new concept in set-valued optimization. For the case of w-minimizers some comparison with existing results is done.     

Vertex Set Partitions Preserving Conservativeness

Let G be an undirected graph and ={X₁, …, X_n} be a partition of V(G). Denote by G/ the graph which has vertex set {X₁, …, X_n}, edge set E, and is obtained from G by identifying vertices in each class X_i of the partition . Given a conservative graph (G, w), we study vertex set partitions preserving conservativeness, i.e., those for which (G/ , w) is also a conservative graph. We characterize the conservative graphs (G/ , w), where is a terminal partition of V(G) (a partition preserving conservativeness which is not a refinement of any other partition of this kind). We prove that many conservative graphs admit terminal partitions with some additional properties. The results obtained are then used in new unified short proofs for a co-NP characterization of Seymour graphs by A. A. Ageev, A. V. Kostochka, and Z. Szigeti (1997, J. Graph Theory34, 357–364), a theorem of E. Korach and M. Penn (1992, Math. Programming55, 183–191), a theorem of E. Korach (1994, J. Combin. Theory Ser. B62, 1–10), and a theorem of A. V. Kostochka (1994, in “Discrete Analysis and Operations Research. Mathematics and its Applications (A. D. Korshunov, Ed.), Vol. 355, pp. 109–123, Kluwer Academic, Dordrecht).     

Continuous mappings on subspaces of products with the κ-box topology

Much of General Topology addresses this issue: Given a function fC(Y,Z) with YY^′ and ZZ^′, find , or at least , such that ; sometimes Z=Z^′ is demanded. In this spirit the authors prove several quite general theorems in the context Y^′=(X_I)_κ=∏_iIX_i in the κ-box topology (that is, with basic open sets of the form ∏_iIU_i with U_i open in X_i and with U_i≠X_i for <κ-many iI). A representative sample result, extending to the κ-box topology some results of Comfort and Negrepontis, of Noble and Ulmer, and of Hušek, is this. Theorem Let ωκα (that means: κ<α, and [β<α and λ<κ]β^λ<α) with α regular, be a set of non-empty spaces with each d(X_i)<α, π[Y]=X_J for each non-empty JI such that |J|<α, and the diagonal in Z be the intersection of <α-many regular-closed subsets of Z×Z. Then (a) Y is pseudo-(α,α)-compact, (b) for every fC(Y,Z) there is J[I]^<α such that f(x)=f(y) whenever x_J=y_J, and (c) every such f extends to .     

相关序的进一步研究

Let (X1,X2,…,Xn) and (Y1,Y2,…Yn) be real random vectors with the same marginal distributions,if (X1,X2,…,Xn)≤c(Y1,Y2,…Yn), it is showed in this paper that ∑i=1^n Xi≤cx∑i=1^n Yi and max1≤k≤n∑i=1^k Xi≤icx max1≤k≤n∑i=1^k Yi hold. Based on this fact,a more general comparison theorem is obtained.     

Estimation of the Location of the Maximum of a Regression Function Using Extreme Order Statistics

In this paper, we consider the problem of approximating the location,x₀∈C, of a maximum of a regresion function,θ(x), under certain weak assumptions onθ. HereCis a bounded interval inR. A specific algorithm considered in this paper is as follows. Taking a random sampleX₁, …, X_nfrom a distribution overC, we have (X_i, Y_i), whereY_iis the outcome of noisy measurement ofθ(X_i). Arrange theY_i's in nondecreasing order and take the average of ther X_i's which are associated with therlargest order statistics ofY_i. This average,x₀, will then be used as an estimate ofx₀. The utility of such an algorithm with fixed r is evaluated in this paper. To be specific, the convergence rates ofx₀tox₀are derived. Those rates will depend on the right tail of the noise distribution and the shape ofθ(·) nearx₀.     

Nonparametric Empirical Bayes Estimation of the Matrix Parameter of the Wishart Distribution

We consider independent pairs (X₁, Σ₁), (X₂, Σ₂), …, (X_n, Σ_n), where eachΣ_iis distributed according to some unknown density functiong(Σ) and, givenΣ_i=Σ,X_ihas conditional density functionq(xΣ) of the Wishart type. In each pair the first component is observable but the second is not. After the (n+1)th observationX_n+1is obtained, the objective is to estimateΣ_n+1corresponding toX_n+1. This estimator is called the empirical Bayes (EB) estimator ofΣ. An EB estimator ofΣis constructed without any parametric assumptions ong(Σ). Its posterior mean square risk is examined, and the estimator is demonstrated to be pointwise asymptotically optimal.     

Copositive approximation of periodic functions

Let f be a real continuous 2π-periodic function changing its sign in the fixed distinct points y _i ∈ Y:= {y _i } _i∈ℤ such that for x ∈ [y _i , y _i−1], f(x) ≧ 0 if i is odd and f(x) ≦ 0 if i is even. Then for each n ≧ N(Y) we construct a trigonometric polynomial P _n of order ≦ n, changing its sign at the same points y _i ∈ Y as f, and

On multipartite posets

A poset P=(X,) is m-partite if X has a partition X=X₁X_m such that (1) each X_i forms an antichain in P, and (2) xy implies xX_i and yX_j where i<j. In this article we derive a tight asymptotic upper bound on the order dimension of m-partite posets in terms of m and their bipartite sub-posets in a constructive and elementary way.     

Ordering distributions by scaled order statistics

Motivated by applications in reliability theory, we define a preordering (X ₁, ...,X _n) (Y ₁ ...,Y _n) of nonnegative random vectors by requiring thek-th order statistic ofa ₁ X ₁,..., a _n X _n to be stochastically smaller than thek-th order statistic ofa ₁ Y ₁, ...,a _n Y _n for all choices ofa _i >0,i=1, 2, ...,n. We identify a class of functionsM _{k, n} such that if and only ifE(X)E(Y) for allM _k,n. Some preservation results related to the ordering are obtained. Some applications of the results in reliability theory are given.Supported by the Air Force Office of Scientific Research, U.S.A.F., under Grant AFOSR-84-0205.     

One-Point Discontinuities of Separately Continuous Functions on the Product of Two Compact Spaces

We investigate the existence of a separately continuous function f: X × Y → ℝ with a one-point set of discontinuity points in the case where the topological spaces X and Y satisfy conditions of compactness type. In particular, it is shown that, for compact spaces X and Y and nonisolated points x ₀ ∈ X and y ₀ ∈ Y, a separately continuous function f: X × Y → ℝ with the set of discontinuity points {(x ₀, y ₀)} exists if and only if there exist sequences of nonempty functionally open sets in X and Y that converge to x ₀ and y ₀, respectively.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 1, pp. 94–101, January, 2005.     

Classes of orderings of measures and related correlation inequalities II. Multivariate reverse rule distributions

A function f(x) defined on = ₁ × ₂ × … × _n where each _i is totally ordered satisfying f(x y) f(x y) ≥ f(x) f(y), where the lattice operations and refer to the usual ordering on , is said to be multivariate totally positive of order 2 (MTP₂). A random vector Z = (Z₁, Z₂,…, Z_n) of n-real components is MTP₂ if its density is MTP₂. Classes of examples include independent random variables, absolute value multinormal whose covariance matrix Σ satisfies −DΣ⁻¹D with nonnegative off-diagonal elements for some diagonal matrix D, characteristic roots of random Wishart matrices, multivariate logistic, gamma and F distributions, and others. Composition and marginal operations preserve the MTP₂ properties. The MTP₂ property facilitate the characterization of bounds for confidence sets, the calculation of coverage probabilities, securing estimates of multivariate ranking, in establishing a hierarchy of correlation inequalities, and in studying monotone Markov processes. Extensions on the theory of MTP₂ kernels are presented and amplified by a wide variety of applications.     

Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions

A function f(x) defined on $\text{X}$ = $\text{X}$₁ × $\text{X}$₂ × … × $\text{X}$_n where each $\text{X}$_i is totally ordered satisfying f(x ∨ y) f(x ∧ y) ≥ f(x) f(y), where the lattice operations ∨ and ∧ refer to the usual ordering on $\text{X}$, is said to be multivariate totally positive of order 2 (MTP₂). A random vector Z = (Z₁, Z₂,…, Z_n) of n-real components is MTP₂ if its density is MTP₂. Classes of examples include independent random variables, absolute value multinormal whose covariance matrix Σ satisfies ?DΣ^?1D with nonnegative off-diagonal elements for some diagonal matrix D, characteristic roots of random Wishart matrices, multivariate logistic, gamma and F distributions, and others. Composition and marginal operations preserve the MTP₂ properties. The MTP₂ property facilitate the characterization of bounds for confidence sets, the calculation of coverage probabilities, securing estimates of multivariate ranking, in establishing a hierarchy of correlation inequalities, and in studying monotone Markov processes. Extensions on the theory of MTP₂ kernels are presented and amplified by a wide variety of applications.     

Asymptotic behavior of a class of stochastic semigroups in the Bernoulli scheme

A family of subalgebras describing the space of complex-valued 2×2 matrices is selected. In this space, a stochastic semigroupY _n =X _n X _–1 ...X ₁,n= , is considered, where {X_i, i= } are independent equally distributed random matrices taking two values. For a stochastic semigroupY _n , whose phase space belongs to one of the subalgebras, the index of exponential growth is determined explicitly.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 11, pp. 1580–1584, November, 1993.     

Comonotone Jackson's Inequality

Let 2s points y_i=−πy_2s<…<y₁<π be given. Using these points, we define the points y_i for all integer indices i by the equality y_i=y_i+2s+2π. We shall write fΔ⁽¹⁾(Y) if f is a 2π-periodic continuous function and f does not decrease on [y_i, y_i−1], if i is odd; and f does not increase on [y_i, y_i−1], if i is even. In this article the following Theorem 1—the comonotone analogue of Jackson's inequality—is proved. 1. If fΔ⁽¹⁾(Y), then for each nonnegative integer n there is a trigonometric polynomial τ_n(x) of order n such that τ_nΔ⁽¹⁾(Y), and |f(x)−π_n(x)|c(s) ω(f; 1/(n+1)), x , where ω(f; t) is the modulus of continuity of f, c(s)=const. Depending only on s.     

Order Determination for Multivariate Autoregressive Processes Using Resampling Methods

LetX₁, …, X_nbe observations from a multivariate AR(p) model with unknown orderp. A resampling procedure is proposed for estimating the orderp. The classical criteria, such as AIC and BIC, estimate the orderpas the minimizer of the function[formula]wherenis the sample size,kis the order of the fitted model, Σ²_kis an estimate of the white noise covariance matrix, andC_nis a sequence of specified constants (for AIC,C_n=2m²/n, for Hannan and Quinn's modification of BIC,C_n=2m²(ln ln n)/n, wheremis the dimension of the data vector). A resampling scheme is proposed to estimate an improved penalty factorC_n. Conditional on the data, this procedure produces a consistent estimate ofp. Simulation results support the effectiveness of this procedure when compared with some of the traditional order selection criteria. Comments are also made on the use of Yule–Walker as opposed to conditional least squares estimations for order selection.     

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].