Linearity of regression for non-adjacent order statistics

Let X ₁,X ₂,…,X _n be a random sample from a continuous distribution with the corresponding order statistics X _1:n≤X _2:n≤…≤X _n:n. All the distributions for which E(X _{k+r: n}|X _k:n)=a X _k:n+b are identified, which solves the problem stated in Ferguson (1967). Received February 1998     

Characterizations of distributions via linearity of regression of generalized order statistics

Let X (r, n, m, k), 1 r n, denote generalized order statistics based on an absolutely continuous distribution function F. We characterize all distribution functions F for which the following linearity of regression holds E(X(r+l,n,m,k) | X(r,n,m,k))=aX(r,n,m,k)+b.We show that only exponential, Pareto and power distributions satisfy this equation. Using this result one can obtain characterizations of exponential, Pareto and power distributions in terms of sequential order statistics, Pfeifers records and progressive type II censored order statistics. Received July 2001/Revised August 2002     

On characterizations of distributions by regression of adjacent generalized order statistics

Let , r ≥ 1, denote generalized order statistics, with arbitrary parameters , based on distribution function F. In this paper we characterize continuous distributions F by the regression of adjacent generalized order statistics, i.e. where are continuous and increasing functions and ψ is strictly increasing. Further we investigate in detail the case when ψ(x) = x and g is a linear function of the form g(x) = cx + d for some .     

On functionals of order statistics

Summary Let g_n be real functions,U _ni, 1in, the ordered sample ofn independentU(0,1) distributed random variables, andc _ni(), 1in, 01 be (known) real numbers,n=1, 2, ... The random quantity , 01, is studied. Based on a method proposed byShorack [1972] the main result is the weak convergence of to Gaussian processes, where , 01. The convergence is with respect to theSkorokhod [1956]-topologiesM ₂,M ₁ onD (I) and the -topology onC(I), depending on the conditions imposed on thec _ni().     

Generalized densities of order statistics

Let X ₁, ... , X _n be independent identically distributed random variables with distribution F . We derive expressions for generalized joint 'densities' of order statistics of X ₁, ... , X _n , for arbitrary distributions F , in terms of Radon–Nikodym derivatives with respect to product measures based on F . We then give formulae for conditional distributions of order statistics and use them to derive results concerning Markov properties of order statistics, formulae for distributions of trimmed sums, and other useful representations. Our approach leads to simple and natural expressions which appear not to have been given before.     

An identity for expectations of functions of order statistics

Summary We consider an identity for expectations of general functions of order statistics valid in a parametric class of probability distributions. Corresponding characterization results are indicated.     

An identity for the product moments of order statistics

A general identity for the product moments of successive order statistics is given, which is valid in a class of probability distributions including Weibull, Pareto, exponential and Burr distributions.     

Exponential inequalities for tail probabilities of generalized order statistics

We establish exponential bounds for the probability that a generalized order statistic exceeds a given threshold or falls below a given threshold. As a main tool we apply the Bernstein inequality for sums of independent random variables.Received May 2003     

Extreme order statistics on Galton-Watson trees

This paper studies the asymptotic behaviour of extreme order statistics of i.i.d. random scores ascribed to each individual in a Galton-Watson family tree. Of interest is the asymptotic behaviour of the order statistics within thenth generation, or up to and including thenth generation, and the index of the generation up to thenth which contains the largest observation.     

Moment inequalities for order statistics from ordered families of distributions

Summary In this paper some inequalities for the variance and covariance of convex monotone functions of order statistics from ordered families of distributions are presented. The considered order relations in the set of distributions are the stochastic ordering relation and the convex ordering relation. Stochastic comparisons of spacings and their sums are also given. As corollaries the results for IFR and DFR distributions are obtained.     

Moment inequalities for order statistics with applications to characterizations of distributions

General inequalities of Hölder type between moments of order statistics and moments of record values respectively are derived. Special choices of the involved sample sizes and ranks and discussions of when equality is attained in these inequalities yield several characterizations of well known distributions, such as the uniform, polynomial, Pareto, reflected Pareto, exponential, Weibull distribution and some others.     

Testing for Linearity

The problem of testing for linearity and the number of regimes in the context of self‐exciting threshold autoregressive (SETAR) models is reviewed. We describe least‐squares methods of estimation and inference. The primary complication is that the testing problem is non‐standard, due to the presence of parameters which are only defined under the alternative, so the asymptotic distribution of the test statistics is non‐standard. Simulation methods to calculate asymptotic and bootstrap distributions are presented. As the sampling distributions are quite sensitive to conditional heteroskedasticity in the error, careful modeling of the conditional variance is necessary for accurate inference on the conditional mean. We illustrate these methods with two applications — annual sunspot means and monthly U.S. industrial production. We find that annual sunspots and monthly industrial production are SETAR(2) processes.     

Sharp bounds on moments of generalized order statistics

Sharp lower and upper bounds on expected values of generalized order statistics are proven by the use of rearranged Moriguti's inequality. The method yields improvements of known quantile and moment bounds for expectations of order and record statistics based on independent identically distributed random variables. The bounds are attainable providing new characterizations of two-point distributions. Received: January 1999     

Marginal distributions of sequential and generalized order statistics

Expressions for marginal distribution functions of sequential order statistics and generalized order statistics are presented without any restrictions imposed on the model parameters. The results are related to the relevation transform, to the distribution of the product of Beta distributed random variables, and to Meijers G-functions. Some selected applications in the areas of moments, conditional distributions, recurrence relations, and reliability properties are shown. Key words:Order statistics; Generalized order statistics; Sequential order statistics; Record values; Distribution theory; Meijers G-function; Recurrence relations; Reliability properties.     

On the asymptotic Fisher information in order statistics

Metrika - We extend the result of Efron and Johnstone (1990), who expressed the Fisher information in terms of the hazard function, to express the Fisher information in order statistics as an...     

Predicting the sample mean by extreme order statistics

Summary This paper considers the prediction of the sample mean by extreme order statistics when the population distribution is known. The predictor and its mean square error are found. The problem is studied in details for the normal model.