Stochastic properties of INID progressive Type-II censored order statistics

Let X_1:n≤X_2:n≤?≤X_n:n denote the order statistics of random variables X₁,X₂,…,X_n which are independent but not necessarily identically distributed (INID), and let K₁,K₂ be two integer-valued random variables, independent of {X₁,…,X_n}, such that 1≤K₁≤K₂≤n. It is shown that if K₁ has a log-concave probability function and SI(K₂|K₁) then RTI(X_K2:n|X_K1:n), and if K₂ has a log-concave probability function and SI(K₁|K₂) then LTD(X_K1:n|X_K2:n), where SI, RTI and LTD are three notions of bivariate positive dependence. Based on these, we obtain that RTI and LTD whenever 1≤i<j≤m, where are progressive Type-II censored order statistics from INID random variables {X₁,…,X_n}. Furthermore, one result concerning the likelihood ratio ordering of the progressive Type-II censored order statistics is also given.     

Likelihood ratio orderings of spacings of heterogeneous exponential random variables

Let X₁,X₂,…,X_n be independent exponential random variables such that X_i has failure rate λ for i=1,…,p and X_j has failure rate λ^* for j=p+1,…,n, where p≥1 and q=n-p≥1. Denote by D_i:n(p,q)=X_i:n-X_i-1:n the ith spacing of the order statistics , where X_0:n≡0. It is shown that D_i:n(p,q)?_lrD_i+1:n(p,q) for i=1,…,n-1, and that if λ?λ^* then , and for i=1,…,n, where ?_lr denotes the likelihood ratio order. The main results are used to establish the dispersive orderings between spacings.     

The mixing advantage for bounded random variables

Corresponding to n independent non-negative random variables X₁,…,X_n concentrated on a bounded interval set are values M₁,…,M_n, where each M_i is the expected value of the maximum of n independent copies of X_i. We obtain a sharp upper bound for the expected value of the maximum of X₁,…,X_n in terms of M₁,…,M_n. This inequality is sharp. A similar result is demonstrated for minima.     

Limit Theorems for Randomly Selected Ratios of Order Statistics from a Pareto Distribution

Abstract

Consider independent and identically distributed random variables {X _nk , 1 ≤ k ≤ m, n ≥ 1} from the Pareto distribution. We randomly select a pair of order statistics from each row, X _n(i) and X _n(j), where 1 ≤ i < j ≤ m. Then we test to see whether or not Strong and Weak Laws of Large Numbers with nonzero limits for weighted sums of the random variables X _n(j)/X _n(i) exist where we place a prior distribution on the selection of each of these possible pairs of order statistics.     

Exact laws for sums of ratios of order statistics from the Pareto distribution

Consider independent and identically distributed random variables {X _nk, 1 ≤ k ≤ m, n ≤ 1} from the Pareto distribution. We select two order statistics from each row, X _n(i) ≤ X _n(j), for 1 ≤ i < j ≤ = m. Then we test to see whether or not Laws of Large Numbers with nonzero limits exist for weighted sums of the random variables R _ij = X _n(j)/X _n(i).     

Critically indecomposable graphs

An undirected graph G=(V,E) with a specific subset X⊂V is called X-critical if G and G(X), induced subgraph on X, are indecomposable but G(V−{w}) is decomposable for every w∈V−X. This is a generalization of critically indecomposable graphs studied by Schmerl and Trotter [J.H. Schmerl, W.T. Trotter, Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures, Discrete Mathematics 113 (1993) 191-205] and Bonizzoni [P. Bonizzoni, Primitive 2-structures with the (n−2)-property, Theoretical Computer Science 132 (1994) 151-178], who deal with the case where X is empty.We present several structural results for this class of graphs and show that in every X-critical graph the vertices of V−X can be partitioned into pairs (a₁,b₁),(a₂,b₂),…,(a_m,b_m) such that G(V−{a_j1,b_j1,…,a_jk,b_jk}) is also an X-critical graph for arbitrary set of indices {j₁,…,j_k}. These vertex pairs are called commutative elimination sequence. If G is an arbitrary indecomposable graph with an indecomposable induced subgraph G(X), then the above result establishes the existence of an indecomposability preserving sequence of vertex pairs (x₁,y₁),…,(x_t,y_t) such that x_i,y_i∈V−X. As an application of the commutative elimination sequence of an X-critical graph we present algorithms to extend a 3-coloring (similarly, 1-factor) of G(X) to entire G.     

On a continuous analogue of the stochastic difference equation Xn=ρXn-1+Bn

Let B₁, B₂, ... be a sequence of independent, identically distributed random variables, letX₀ be a random variable that is independent ofB_n forn?1, let ρ be a constant such that 0<ρ<1 and letX₁,X₂, ... be another sequence of random variables that are defined recursively by the relationshipsX_n=ρX_n-1+B_n. It can be shown that the sequence of random variablesX₁,X₂, ... converges in law to a random variableX if and only ifE[log⁺¦B₁¦]<∞. In this paper we let {B(t):0≦t<∞} be a stochastic process with independent, homogeneous increments and define another stochastic process {X(t):0?t<∞} that stands in the same relationship to the stochastic process {B(t):0?t<∞} as the sequence of random variablesX₁,X₂,...stands toB₁,B₂,.... It is shown thatX(t) converges in law to a random variableX ast →+∞ if and only ifE[log⁺¦B(1)¦]<∞ in which caseX has a distribution function of class L. Several other related results are obtained. The main analytical tool used to obtain these results is a theorem of Lukacs concerning characteristic functions of certain stochastic integrals.     

The uniform convergence and precise asymptotics of generalized stochastic order statistics

Let X(i,n,m,k), i=1,…,n, be generalized order statistics based on F. For fixed r∈N, and a suitable counting process N(t), t>0, we mainly discuss the precise asymptotic of the generalized stochastic order statistics X(N(n)−r+1,N(n),m,k). It not only makes the results of Yan, Wang and Cheng [J.G. Yan, Y.B. Wang, F.Y. Cheng, Precise asymptotics for order statistics of a non-random sample and a random sample, J. Systems Sci. Math. Sci. 26 (2) (2006) 237-244] as the special case of our result, and presents many groups of weighted functions and boundary functions, but also permits a unified approach to several models of ordered random variables.     

A random intersection digraph: Indegree and outdegree distributions

Let S(1),…,S(n),T(1),…,T(n) be random subsets of the set [m]={1,…,m}. We consider the random digraph D on the vertex set [n] defined as follows: the arc i→j is present in D whenever S(i)∩T(j)≠0?. Assuming that the pairs of sets (S(i),T(i)), 1≤i≤n, are independent and identically distributed, we study the in- and outdegree distributions of a typical vertex of D as n,m→∞.     

Order of magnitude bounds for expectations of Δ2-functions of generalized random bilinear forms

Let Φ be a symmetric function, nondecreasing on [0,∞) and satisfying a Δ₂ growth condition, (X ₁,Y ₁), (X ₂,Y ₂),…,(X _n ,Y _n ) be arbitrary independent random vectors such that for any given i either Y _i =X _i or Y _i is independent of all the other variates. The purpose of this paper is to develop an approximation of

Simultaneous rational approximation to the successive powers of a real number

Let n be an integer ≥ 1 and let θ be a real number which is not an algebraic number of degree ≤ [n/2]. We show that there exist ? > 0 and arbitrary large real numbers X such that the system of linear inequalities |x₀| ≤ X and |x₀θ^j − x_j| ≤ ?X^−1/[n/2] for 1 < j < n, has only the zero solution in rational integers x₀,…, x_n. This result refines a similar statement due to H. Davenport and W. M. Schmidt, where the upper integer part [n/2] is replaced everywhere by the integer part [n/2]. As a corollary, we improve slightly the exponent of approximation to 0 by algebraic integers of degree n + 1 over Q obtained by these authors.     

On the dependence between the extreme order statistics in the proportional hazards model

Let X₁,…,X_n be a random sample from an absolutely continuous distribution with non-negative support, and let Y₁,…,Y_n be mutually independent lifetimes with proportional hazard rates. Let also X₍₁₎<?<X_(n) and Y₍₁₎<?<Y_(n) be their associated order statistics. It is shown that the pair (X₍₁₎,X_(n)) is then more dependent than the pair (Y₍₁₎,Y_(n)), in the sense of the right-tail increasing ordering of Avérous and Dortet-Bernadet [LTD and RTI dependence orderings, Canad. J. Statist. 28 (2000) 151-157]. Elementary consequences of this fact are highlighted.     

Stochastic comparisons of order statistics from gamma distributions

Let (X₁,X₂,…,X_n) and (Y₁,Y₂,…,Y_n) be gamma random vectors with common shape parameter α(0<α?1) and scale parameters (λ₁,λ₂,…,λ_n), (μ₁,μ₂,…,μ_n), respectively. Let X₍₎=(X₍₁₎,X₍₂₎,…,X_(n)), Y₍₎=(Y₍₁₎,Y₍₂₎,…,Y_(n)) be the order statistics of (X₁,X₂,…,X_n) and (Y₁,Y₂,…,Y_n). Then (λ₁,λ₂,…,λ_n) majorizes (μ₁,μ₂,…,μ_n) implies that X₍₎ is stochastically larger than Y₍₎. However if the common shape parameter α>1, we can only compare the the first- and last-order statistics. Some earlier results on stochastically comparing proportional hazard functions are shown to be special cases of our results.     

Convergence of Point Processes with Weakly Dependent Points

For each n≥1, let {X _j,n }_1≤j≤n be a sequence of strictly stationary random variables. In this article, we give some asymptotic weak dependence conditions for the convergence in distribution of the point process $N_{n}=\sum_{j=1}^{n}\delta_{X_{j,n}}For each n≥1, let {X _j,n }_1≤j≤n be a sequence of strictly stationary random variables. In this article, we give some asymptotic weak dependence conditions for the convergence in distribution of the point process N_n=?_j=1ⁿd_Xj,nN_{n}=\sum_{j=1}^{n}\delta_{X_{j,n}} to an infinitely divisible point process. From the point process convergence we obtain the convergence in distribution of the partial sum sequence S _n =∑ _j=1 ⁿ X _j,n to an infinitely divisible random variable whose Lévy measure is related to the canonical measure of the limiting point process. As examples, we discuss the case of triangular arrays which possess known (row-wise) dependence structures, like the strong mixing property, the association, or the dependence structure of a stochastic volatility model.     

Bitableau bases for Garsia-Haiman modules of hollow type

Garsia-Haiman modules C[Xn,Yn]/Iγ are quotient rings in the variables Xn={x₁,x₂,…,xn} and Yn={y₁,y₂,…,yn} that generalize the quotient ring C[Xn]/I, where I is the ideal generated by the elementary symmetric polynomials ej(Xn) for 1?j?n. A bitableau basis for the Garsia-Haiman modules of hollow type is constructed. Applications of this basis to representation theory and other related polynomial spaces are considered.     

On the range of heterogeneous samples

Let R_n be the range of a random sample X₁,…,X_n of exponential random variables with hazard rate λ. Let S_n be the range of another collection Y₁,…,Y_n of mutually independent exponential random variables with hazard rates λ₁,…,λ_n whose average is λ. Finally, let r and s denote the reversed hazard rates of R_n and S_n, respectively. It is shown here that the mapping t?s(t)/r(t) is increasing on (0,∞) and that as a result, R_n=X_(n)−X₍₁₎ is smaller than S_n=Y_(n)−Y₍₁₎ in the likelihood ratio ordering as well as in the dispersive ordering. As a further consequence of this fact, X_(n) is seen to be more stochastically increasing in X₍₁₎ than Y_(n) is in Y₍₁₎. In other words, the pair (X₍₁₎,X_(n)) is more dependent than the pair (Y₍₁₎,Y_(n)) in the monotone regression dependence ordering. The latter finding extends readily to the more general context where X₁,…,X_n form a random sample from a continuous distribution while Y₁,…,Y_n are mutually independent lifetimes with proportional hazard rates.     

Maximal Sidon sets and matroids

A subset X of an abelian group Γ, written additively, is a Sidon set of orderh if whenever {(a_i,m_i):i∈I} and {(b_j,n_j):j∈J} are multisets of size h with elements in X and ∑_i∈Im_ia_i=∑_j∈Jn_jb_j, then {(a_i,m_i):i∈I}={(b_j,n_j):j∈J}. The set X is a generalized Sidon set of order(h,k) if whenever two such multisets have the same sum, then their multiset intersection has size at least k. It is proved that if X is a generalized Sidon set of order (2h−1,h−1), then the maximal Sidon sets of order h contained in X have the same cardinality. Moreover, X is a matroid where the independent subsets of X are the Sidon sets of order h.     

Oberwolfach rectangular table negotiation problem

We completely solve certain case of a “two delegation negotiation” version of the Oberwolfach problem, which can be stated as follows. Let H(k,3) be a bipartite graph with bipartition X={x₁,x₂,…,x_k},Y={y₁,y₂,…,y_k} and edges x₁y₁,x₁y₂,x_ky_k−1,x_ky_k, and x_iy_i−1,x_iy_i,x_iy_i+1 for i=2,3,…,k−1. We completely characterize all complete bipartite graphs K_n,n that can be factorized into factors isomorphic to G=mH(k,3), where k is odd and mH(k,3) is the graph consisting of m disjoint copies of H(k,3).     

Limiting distribution of sums of nonnegative stationary random variables

Summary Let {X _n,j,−∞<j<∞∼,n≧1, be a sequence of stationary sequences on some probability space, with nonnegative random variables. Under appropriate mixing conditions, it is shown thatS _n=X_n,1+…+X _n,n has a limiting distribution of a general infinitely divisible form. The result is applied to sequences of functions {f _n(x)∼ defined on a stationary sequence {X _j∼, whereX _n.f=f_n(X_j). The results are illustrated by applications to Gaussian processes, Markov processes and some autoregressive processes of a general type. This paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under the sponsorship of the National Sciences Foundation, Grant MCS 82-01119.     

Multivariate weighted renewal functions

Let (X,Y),(X₁,Y₁),(X₂,Y₂),… denote independent positive random vectors with common distribution function F(x,y)=P(X?x,Y?y) with F(x,y)<1 for all x,y. Based on the X_i and the Y_j we construct the sum sequences and respectively. For a double sequence of weighting constants {b(n,m)} we associate a weighted renewal function G(x,y) defined as . The function G(x,y) can be expressed in terms of well-known renewal quantities. The main goal of this paper is to study asymptotic properties of G(x,y). In the one-dimensional case such results have been obtained among others by Omey and Teugels [Weighted renewal functions: a hierarchical approach, Adv. in Appl. Probab. 34 (2002) 394-415.] and Alsmeyer [Some relations between harmonic renewal measures and certain first passage times, Statist. Probab. Letters 12 (1991) 19-27; On generalized renewal measures and certain first passage times, Ann. Probab. 20 (1992) 1229-1247]. Here we prove a multivariate version of the elementary renewal theorem and moreover we obtain a rate of convergence result in this elementary renewal theorem. We close the paper with an application and some concluding remarks. For convenience we prove and formulate the results in the two-dimensional case only.