Characterizations of generalized mixtures of geometric and exponential distributions based on upper record values

Let X _{U (1)} < X _{U (2)} < … < X _{U ( n )} < … be the sequence of the upper record values from a population with common distribution function F. In this paper, we first give a theorem to characterize the generalized mixtures of geometric distribution by the relation between E[(X _{U ( n +1)}–X _{U ( n )})²|X _{U ( n )} = x] and the function of the failure rate of the distribution, for any positive integer n. Secondly, we also use the same relation to characterize the generalized mixtures of exponential distribution. The characterizing relations were motivated by the work of Balakrishnan and Balasubramanian (1995). Received: March 31, 1999; revised version: November 22, 1999     

Prediction based on linear combinations of order statistics and bivariate concomitants in the case of multivariate elliptical distributions

In this paper, by considering a (3n+1) -dimensional random vector (X₀, X^T, Y^T, Z^T)^T having a multivariate elliptical distribution, we derive the exact joint distribution of (X₀, a^TX_(n), b^TY_[n], c^TZ_[n])^T, where a, b, c∈?ⁿ, X_(n)=(X₍₁₎, …, X_(n))^T, X₍₁₎<···<X_(n), is the vector of order statistics arising from X, and Y_[n]=(Y_[1], …, Y_[n])^T and Z_[n]=(Z_[1], …, Z_[n])^T denote the vectors of concomitants corresponding to X_(n) ((Y_[r], Z_[r])^T, for r=1, …, n, is the vector of bivariate concomitants corresponding to X_(r)). We then present an alternate approach for the derivation of the exact joint distribution of (X₀, X_(r), Y_[r], Z_[r])^T, for r=1, …, n. We show that these joint distributions can be expressed as mixtures of four-variate unified skew-elliptical distributions and these mixture forms facilitate the prediction of X_(r), say, based on the concomitants Y_[r] and Z_[r]. Finally, we illustrate the usefulness of our results by a real data.     

ON THE NUMBER OF RECORDS NEAR THE MAXIMUM

Recent work has considered properties of the number of observations X_j, independently drawn from a discrete law, which equal the sample maximum X_(n) The natural analogue for continuous laws is the number K_n(a) of observations in the interval (X_(n)–a, X_(n)], where a > 0. This paper derives general expressions for the law, first moment, and probability generating function of K_n(a), mentioning examples where evaluations can be given. It seeks limit laws for n→ and finds a central limit result when a is fixed and the population law has a finite right extremity. Whenever the population law is attracted to an extremal law, a limit theorem can be found by letting a depend on n in an appropriate manner; thus the limit law is geometric when the extremal law is the Gumbel type. With these results, the paper obtains limit laws for ‘top end’ spacings X_(n) - X_(n-j) with j fixed.     

Randomly weighted sums and their maxima with heavy-tailed increments and dependence structure

Consider the randomly weighted sums S_m(θ) = ∑^m_{i = 1}θ_iX_i, 1 ? m ? n, and their maxima M_n(θ) = max?_{1 ? m ? n}S_m(θ), where X_i, 1 ? i ? n, are real-valued and dependent according to a wide type of dependence structure, and θ_i, 1 ? i ? n, are non negative and arbitrarily dependent, but independent of X_i, 1 ? i ? n. Under some mild conditions on the right tails of the weights θ_i, 1 ? i ? n, we establish some asymptotic equivalence formulas for the tail probabilities of S_n(θ) and M_n(θ) in the case where X_i, 1 ? i ? n, are dominatedly varying, long-tailed and subexponential distributions, respectively.     

Asymptotic Properties of Random Weighted Empirical Distribution Function

This article studies the asymptotic properties of the random weighted empirical distribution function of independent random variables. Suppose X₁, X₂, ???, X_n is a sequence of independent random variables, and this sequence is not required to be identically distributed. Denote the empirical distribution function of the sequence by F_n(x). Based on the random weighting method and F_n(x), the random weighted empirical distribution function H_n(x) is constructed and the asymptotic properties of H_n are discussed. Under weak conditions, the Glivenko–Cantelli theorem and the central limit theorem for the random weighted empirical distribution function are obtained. The obtained results have also been applied to study the distribution functions of random errors of multiple sensors.     

Limiting distributions of Kolmogorov-Lévy-type statistics under the alternative

Let X_i, 1 ≤ i ≤ n, be independent identically distributed random variables with a common distribution function F, and let G be a smooth distribution function. We derive the limit distribution of □{ρ_α(F_n, G) - _α(F, G)}, where F_n is the empirical distribution function based on X₁,…,X_n and _α is a Kolmogorov-Lévy-type metric between distribution functions. For α ≤ 0 and two distribution functions F and G the metric p_α is given by p_α(F, G) = inf {? ≤ 0: G(x - α?) - ? F(x) ≤ G(x + α?) + ? for all x ?}.     

Inequalities for the partial sums of elliptical order statistics related to genetic selection

Let X₁,…,X_n be exchangeable normal variables with a common correlation p, and let X₍₁₎ > … > X_(n) denote their order statistics. The random variable σⁿ_i=_n—_k+1x_i, called the selection differential by geneticists, is of particular interest in genetic selection and related areas. In this paper we give results concerning a conjecture of Tong (1982) on the distribution of this random variable as a function of ρ. The same technique used can be applied to yield more general results for linear combinations of order statistics from elliptical distributions.     

Bayesian Estimation of Change Point in Inverse Weibull Sequence

A sequence of independent lifetimes X ₁,…, X _m , X _m+1,…, X _n were observed from inverse Weibull distribution with mean stress θ₁ and reliability R ₁(t ₀) at time t ₀ but later it was found that there was a change in the system at some point of time m and it is reflected in the sequence after X _m by change in mean stress θ₁ and in reliability R ₂(t ₀) at time t ₀. The Bayes estimators of m, R ₁(t ₀) and R ₂(t ₀) are derived when a poor and a more detailed prior information is introduced into the inferential procedure. The effects of correct and wrong prior information on the Bayes estimators are studied.     

Characterizations based on regression assumptions of order statistics

Let X₍₁₎≤X₍₂₎≤···≤X_(n) be the order statistics from independent and identically distributed random variables {X_i, 1≤i≤n} with a common absolutely continuous distribution function. In this work, first a new characterization of distributions based on order statistics is presented. Next, we review some conditional expectation properties of order statistics, which can be used to establish some equivalent forms for conditional expectations for sum of random variables based on order statistics. Using these equivalent forms, some known results can be extended immediately.     

Bayes Estimation of Change Point in Non Standard Mixture of Left-Truncated Exponential With Unknown Proportions

A sequence of independent lifetimes X ₁, X ₂,…, X _m , X _m+1,…, X _n were observed from the mixture of a degenerate and left-truncated exponential (LTE) distribution, with reliability R _1τ at time τ and minimum life length η with unknown proportion p ₁ and θ₁ but later it was found that there was a change in the system at some point of time m and it is reflected in the sequence after X _m by change in reliability R _2τ at time τ and unknown proportion p ₂ and θ₂. This distribution occurs in many practical situations, for instance; life of a unit may have a LTE distribution but some of the units fail instantaneously. Apart from mixture distributions, the phenomenon of change point is also observed in several situations in life testing and reliability estimation problems. It may happen that at some point of time instability in the sequence of failure times is observed. The problem of study is: When and where this change has started occurring. This is called change point inference problem. The estimators of m, R ₁(t ₀), R ₂(t ₀), p ₁, and p ₂ are derived under asymmetric loss functions namely Linex loss & general entropy loss functions. Both the non informative and informative prior are considered. The effects of prior consideration on Bayes estimates of change point are also studied.     

An extension of almost sure central limit theorem for self-normalized products of sums for mixing sequences

ABSTRACT

Let X, X₁, X₂, … be a sequence of strictly stationary φ-mixing random variables with EX = μ > 0. In this paper, we show that a self-normalized version of almost sure central limit theorem (ASCLT) holds under the assumptions that the mixing coefficients satisfy ∑^∞_{n = 1}φ^1/2(2ⁿ) < ∞ and the weight sequence {d_k} satisfies a mild growth condition similar to Kolmogorov’s condition for the LIL. This shows that logarithmic averages, used traditionally in ASCLT for products of sums, can be replaced by other averages, leading to considerably sharper results.     

Limit results for ordered uniform spacings

Let Δ _k:n  = X _k,n  − X _k-1,n (k = 1, 2, . . . , n + 1) be the spacings based on uniform order statistics, provided X _0,n  = 0 and X _n+1,n  = 1. Obtained from uniform spacings, ordered uniform spacings 0 = Δ_0,n  < Δ_1,n  < . . . < Δ _n+1,n , are discussed in the present paper. Distributional and limit results for them are in the focus of our attention.     

Characterizations of geometric distribution and discrete IFR (DFR) distributions using order statistics

Let X be a discrete random variable the set of possible values (finite or infinite) of which can be arranged as an increasing sequence of real numbers a₁<a₂<a₃<…. In particular, a_i could be equal to i for all i. Let X_1n≦X_2n≦?≦X_nn denote the order statistics in a random sample of size n drawn from the distribution of X, where n is a fixed integer ≧2. Then, we show that for some arbitrary fixed k(2≦k≦n), independence of the event {X_kn=X_1n} and X_1n is equivalent to X being either degenerate or geometric. We also show that the montonicity in i of P{X_kn = X_1n | X_1n = a_i} is equivalent to X having the IFR (DFR) property. Let a_i = i and $\text{G(i) = P(X≧i), i = 1, 2, …}$. We prove that the independence of {X_2n ? X_1n ∈B} and X_1n for all i is equivalent to X being geometric, where B = {m} (B = {m,m+1,…}), provided G(i) = q^i?1, 1≦i≦m+2 (1≦i≦m+1), where 0<q<1.     

Some multivariate singular unified skew-normal distributions and their application

Abstract

We introduce here the truncated version of the unified skew-normal (SUN) distributions. By considering a special truncations for both univariate and multivariate cases, we derive the joint distribution of consecutive order statistics X_{(r, ..., r + k)} = (X_(r), ..., X_{(r + K)})^T from an exchangeable n-dimensional normal random vector X. Further we show that the conditional distributions of X_{(r + j, ..., r + k)} given X_{(r, ..., r + j ? 1)}, X_{(r, ..., r + k)} given (X_(r) > t)?and X_{(r, ..., r + k)} given (X_{(r + k)} < t) are special types of singular SUN distributions. We use these results to determine some measures in the reliability theory such as the mean past life (MPL) function and mean residual life (MRL) function.     

Limiting behavior of randomly weighted averages of symmetric heavy-tailed random variables

In this paper we consider a sequence of independent continuous symmetric random variables X₁, X₂, …, with heavy-tailed distributions. Then we focus on limiting behavior of randomly weighted averages S_n = R⁽ⁿ⁾₁X₁ + ??? + R⁽ⁿ⁾_nX_n, where the random weights R⁽ⁿ⁾₁, …, R_n⁽ⁿ⁾ which are independent of X₁, X₂, …, X_n, are the cuts of (0, 1) by the n ? 1 order statistics from a uniform distribution. Indeed we prove that c_nS_n converges in distribution to a symmetric α-stable random variable with c_n = n^{1 ? 1/α}/Γ^1/α(α + 1).     

On characterizing distributions via regression of functions of generalized order statistics

Let X(1,n,m₁,k),X(2,n,m₂,k),…,X(n,n,m,k) be n generalized order statistics from a continuous distribution F which is strictly increasing over (a,b),−∞≤a<b≤∞, the support of F. Let g be an absolutely continuous and monotonically increasing function in (a,b) with finite g(a⁺),g(b⁻) and E(g(X)). Then for some positive integer s,1<s≤n, we give characterization of distributions by means of

     

On moments of the supremum of normed weighted averages

Let {X, X_n; n ≥ 1} be a sequence of real-valued iid random variables, 0 < r < 2 and p > 0. Let D = { A = (a_nk; 1 ≤ k ≤ n, n ≥ 1); a_nk, ? R and sup_{n, k} |a_n,k| < ∞}. Set S_n( A ) = ∑ⁿ_k=1a_{n, k}X_k for A ? D and n ≥ 1. This paper is devoted to determining conditions whereby E{sup_{n ≥ 1}, |S_n( A )|/n^1/r}^p < ∞ or E{sup_{n ≥ 2} |S_n( A )|/2n log n)^1/2}^p < ∞ for every A ? D. This generalizes some earlier results, including those of Burkholder (1962), Choi and Sung (1987), Davis (1971), Gut (1979), Klass (1974), Siegmund (1969) and Teicher (1971).     

Strict stationarity of ar(p) processes generated by nonlinear random functions with additive perturbations

Let {X_n} be a generalized autoregressive process of order ρ defined by X_n=Φ_n(X_n-ρ,…,X_n-1)-η_m, where {φ_n} is a sequence of i.i.d. random maps taking values on H, and {η_n} is a sequence of i.i.d. random variables. Let H be a collection of Borel measurable functions on R_P to R. By considering the associated Markov process, we obtain sufficient conditions for stationarity, (geometric) ergodicity of {X_n}.