Dependence Structure of Spacings of Order Statistics

Let X_1:n ≤ X_2:n ≤···≤ X_n:n denote the order statistics of a sample of n independent random variables X₁, X₂,…, X_n, all identically distributed as some X. It is shown that if X has a log-convex [log-concave] density function, then the general spacing vector (X_k1:n, X_k2:n ? X_k1:n,…, X_kr:n ? X_kr?1:n) is MTP₂ [S-MRR₂] whenever 1 ≤ k₁ < k₂ <···< k_r ≤ n and 1 ≤ r ≤ n. Multivariate likelihood ratio ordering of such general spacing vectors corresponding to two random samples is also considered. These extend some of the results in the literature for usual spacing vectors.     

Concomitants of multivariate order statistics from multivariate elliptical distributions

ABSTRACT

In this article, we consider a (k + 1)n-dimensional elliptically contoured random vector (X^T₁, X₂^T, …, X^T_k, Z^T)^T = (X₁₁, …, X_1n, …, X_k1, …, X_kn, Z₁, …, Z_n)^T and derive the distribution of concomitant of multivariate order statistics arising from X₁, X₂, …, X_k. Specially, we derive a mixture representation for concomitant of bivariate order statistics. The joint distribution of the concomitant of bivariate order statistics is also obtained. Finally, the usefulness of our result is illustrated by a real-life data.     

Bounds on the expected maximum

Let (X₁,X₂, …,X_n) be jointly distributed random variables. Define X_n:n = max(X₁,X₂, …,X_n).Bounds on E(X_n:n), obtained by putting constraints on the distributions and/or dependence structure of the X_i's, are surveyed.     

Some reliability properties of order statistics

Let X_i:j denote the i^th order statistic of a random sample of size j from a continuous life distribution. We show that if X_k:n, is IFR, IFRA, NBU, or DMRL, so are X_k+1:n, X_k+1:n?1 and X_k+1:n+1. Further we show that, in the first three cases, X_k+1:n+2 also shares the corresponding property if k ≤ (n+3)/2. We also present dual results for DFR, DFRA and NWU classes.     

Modified kolmogorov-smirnov tests of goodness of fit

The Kolmogorov-Smirnov (K–S) one-sided and two-sided tests of goodness of fit based on the test statistics D⁺ _n D^? _n and D_n are equivalent to tests based on taking the cumulative probability of the i–th order statistic of a sample of size n to be (i–.5)/n. Modified test statistics C⁺ _n, C^? _n and C_n are obtained by taking the cumulative probability to be i/(n+l). More generally, the cumula-tive probability may be taken to be (i?δ)/(n+l?2δ), as suggested by Blom (1958), where 0 less than or equal δ less than or equal .5. Critical values of the test statis-tics can be found by interpolating inversely in tables of the proba-bility integrals obtained by setting a=l/(n+l?2δ) in an expression given by Pyke (1959). Critical values for the D's (corresponding to δ=.5) have been tabulated to 5DP by Miller (1956) for n=1(1)100. The authors have made analogous tabulations for the C's (corresponding to δ=0) [previously tabulated by Durbin (1969) for n=1(1)60(2)100] and for the test statistics E⁺ _n, E^? _n and E_n corresponding to δ f.3. They have also made a Monte Carlo comparison of the power of the modified tests with that of the K–S test for several hypothetical distributions. In a number of cases, the power of the modified tests is greater than that of the K–S test, especially when the standard deviation is greater under the alternative than under the null hypo-thesis.     

Small samples confidencce intervals on common mean of two normal distributions variances

Let X₁,X₂,…,X_m be distributed normally with mean μ and variance σ² _X; Let Y₁,Y₂,…,Y_n be distributed normally with mean μ and variance σ² _Y; let X₁,X₂,…,X_m,Y₁,Y₂,…,Y_n be jointly independent. There have been several papers written concerning point estimation of μ for this problem, but very little is available in the literature concerning confidence intervals on the common mean μ. In this paper a method is proposed that results in a confidence interval with confidence coefficient essentially equal to a prescribed value 1 - α. The method is evaluated and compnred with other methods through the expected length of the confidence interval.     

Prediction Intervals for Future Order Statistics from a Generalized Modified Weibull

Viewing the future order statistics as latent variables at each Gibbs sampling iteration, several Bayesian approaches to predict future order statistics based on type-II censored order statistics, X₍₁₎, X₍₂₎, …, X_(r), of a size n( > r) random sample from a four-parameter generalized modified Weibull (GMW) distribution, are studied. Four parameters of the GMW distribution are first estimated via simulation study. Then various Bayesian approaches, which include the plug-in method, the Monte Carlo method, the Gibbs sampling scheme, and the MCMC procedure, are proposed to develop the prediction intervals of unobserved order statistics. Finally, four type-II censored samples are utilized to investigate the predictions.     

On characterization of the exponential distribution by spacings

Let X be a non-negative random variable with cumulative probability distribution function F. Suppse X₁, X₂, ..., X_n be a random sample of size n from F and X_i,n is the i-th smallest order statistics. We define the standardized spacings D_r,n=(n-r) (X_r+1,n-X_r,n), 1≤r≤n, with D_O,n=nX_1,n and D_n,n=0. Characterizations of the exponential distribution are given by considering the expectation and hazard rates of D_r,n.     

A zero-one law for large order statistics

Fix r ≥ 1, and let {M_nr} be the rth largest of {X₁,X₂,…X_n}, where X₁,X₂,… is a sequence of i.i.d. random variables with distribution function F. It is proved that P[M_nr ≤ u_n i.o.] = 0 or 1 according as the series Σ^∞_n=3Fⁿ(u_n)(log log n)^r/n converges or diverges, for any real sequence {u_n} such that n{1 -F(u_n)} is nondecreasing and divergent. This generalizes a result of Bamdorff-Nielsen (1961) in the case r = 1.     

On a conjecture of Kakosyan,Klebanov and Melamed

Suppose X₁, X₂, ..., X_m is a random sample of size m from a population with probability density function f(x), x>0 and let X_1,m<...m,m be the corresponding order statistics. We assume m as an integer valued random variable with P(m=k)=p(1?p)^k?1, k=1, 2, ... and 0 and n X_1,n for fixed n characterizes the exponential distribution. In this paper we prove that under the assumption of monotone hazard rate the identical distribution of and (n?r+1) (X_r,n?X_r?1,n) for some fixed r and n with 1≤r≤n, n≥2, X_0,n=0, characterizes the exponential distribution. Under the assumption of monotone hazard rate the conjecture of Kakosyan, Klebanov and Melamed follows from the above result with r=1.     

Improved Ordering Results for Fail-Safe Systems with Exponential Components

Let X_{2: n} and Y_{2: m} be the second order statistics from n independent exponential variables with hazards λ₁, …, λ_n, and an independent exponential sample of size m with hazard change to λ, respectively. When m ? n, we obtain necessary and sufficient conditions for comparing X_{2: n} and Y_{2: m} in mean residual life, dispersive, hazard rate, and likelihood ratio orderings based on some inequalities between λ_i’s and λ. The established results show how one can compare an (n ? 1)-out-of-n system consisting of heterogeneous components with exponential lifetimes with any (m ? 1)-out-of-m system consisting of homogeneous components with exponential lifetimes.     

Some competitors of the mood test for the two-sample scale problem

Let X_l,…,X_n (Y_l,…,Y_m) be a random sample from an absolutely continuous distribution with distribution function F(G).A class of distribution-free tests based on U-statistics is proposed for testing the equality of F and G against the alternative that X's are more dispersed then Y's. Let 2 ? C ? n and 2 ? d ? m be two fixed integers. Let ?_c,d(X_il,…,X_ic ; Y_jl,…,X_jd)=1(-1)when max as well as min of {X_il,…,X_ic ; Y_jl,…,Y_jd } are some X_i's (Y_j's)and zero oterwise. Let S_c,d be the U-statistic corresponding to ?_c,d.In case of equal sample sizes, S22 is equivalent to Mood's Statistic.Large values of S_c,d are significant and these tests are quite efficient     

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).     

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.     

Prediction intervals with a Dirichlet-process prior distribution

Let X₁, …,X_n, and Y₁, … Y_n be consecutive samples from a distribution function F which itself is randomly chosen according to the Ferguson (1973) Dirichlet-process prior distribution on the space of distribution functions. Typically, prediction intervals employ the observations X₁,…, X_n in the first sample in order to predict a specified function of the future sample Y₁, …, Y_n. Here one- and two-sided prediction intervals for at least q of N future observations are developed for the situation in which, in addition to the previous sample, there is prior information available. The information is specified via the parameter α of the Dirichlet process prior distribution.     

Nonparametric estimation of survival functions for incomplete observations when the life time distribution is proportionally related ot the censoring time distribution

Let X₁, X₂…,X_n be a random sample from [ILM0001] and let Y₁, …,Y_n be a random sample from [ILM0002]. Then instead of observing a complete sample X₁,…X_n, we can only observe the pairs Z_i. = min(X_i.,Y_i) and [ILM0003] In this paper, we consider estimation of survival function [ILM0004] when [ILM0005], where β is an unknown positive real number.

     

Estimation of the derivative of a density and related functions

Let Y₁,…,Y _n, (Y₁ <Y₂<…<Y _n) be the order statistics of a random sample from a distribution F with density f on the realline. This paper gives a class of estimators of the derivativef'(x) of the density f at points x for which f has

a continuoussecond derivative. These estimators are based on spacings inthe order statistics Y_j+kn -y _j j = 1,…,n-k_n,k_n<n.     

On the asymptotic expectation and variance of the number of boundary crossings

Let X₁,X₂,… be independent and identically distributed nonnegative random variables with mean μ, and let S_n = X₁ + … + X_n. For each λ > 0 and each n ≥ 1, let A_n be the interval [λn^Y, ∞), where γ > 1 is a constant. The number of times that S_n is in A_n is denoted by N. As λ tends to zero, the asymtotic behavior of N is studied. Specifically under suitable conditions, the expectation of N is shown to be (μλ^?1)^β + o(λ^?β/2 where β = 1/(γ-1) and the variance of N is shown to be (μλ^?1)^β(βμ¹)²σ² + o(λ^?β) where σ² is the variance of X_n.     

Evaluation of a method for setting confidence intervals on the common mean of two normal populations

Let X₁, , X₂, …, X be distributed N(µ, σ² _x), let Y₁, Y₂, …, Y"_n be distributed N(µ, σ² _y), and let X , X , … X_m, Y₁, Y₂, …, Y_n be mutually independent. In this paper a method for setting confidence intervals on the common mean µ is proposed and evaluated.     

Zur abschätzung von lösungen linearer integralgleichungen mit hilfe der monte-carlo-methode

Let X₁, X₂, … be i.i.d.r.v. and write (X₁+…X_n?A_n)/B_n?F_n, where B_n >0.A_nER₁, n≥1. It is known that solely one–sided asymptotic assumptions imposed on F_n imply F_n0. In the present note we show that stronger one–sided assumptions lead even to the existence of EX₁ ³ so that the BERRY-ESSEEN inequalities hold true.