Point prediction of future order statistics from an exponential distribution

Two-sample point prediction is considered for a two-parameter exponential distribution. Several point predictors such as the best unbiased predictor, best invariant predictor and maximum likelihood predictor are obtained for future order statistics on the basis of observed record values in two cases: where the location parameter is known and unknown. These predictors are compared in the sense of their mean squared prediction errors. Finally, some numerical results are given to illustrate the proposed procedures.     

Extremes of Shepp statistics for Gaussian random walk

Let (ξ _i , i ≥ 1) be a sequence of independent standard normal random variables and let be the corresponding random walk. We study the renormalized Shepp statistic and determine asymptotic expressions for when u,N and T→ ∞ in a synchronized way. There are three types of relations between u and N that give different asymptotic behavior. For these three cases we establish the limiting Gumbel distribution of when T,N→ ∞ and present corresponding normalization sequences.      

Limsup results and a uniform LIL for partial sums of an LNQD sequence

In this paper, we establish some limsup results and a generalized uniform law of the iterated logarithm (LIL) for the increments of partial sums of a strictly stationary and linearly negative quadrant dependent (LNQD) sequence of random variables whose covariance coefficients decay polynomially.     

Strong laws for exponential order statistics and spacings

We obtain upper and lower almost sure asymptotic bounds for spacings η _{K, N} − η _{K + 1, N} , where η _{1, N} > η _{2, N} > ⋯ > η _{N, N} are the order statistics of independent exponentially distributed random variables η ₁, η ₂, ..., η _N with mean 1; here N → ∞, and K = 1, 2, ... is fixed. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 477–491, October–December, 2006.     

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.     

Random walk in mixed random environment without uniform ellipticity

We study a random walk in random environment on ?₊. The random environment is not homogeneous in law, but is a mixture of two kinds of site, one in asymptotically vanishing proportion. The two kinds of site are (i) points endowed with probabilities drawn from a symmetric distribution with heavy tails at 0 and 1, and (ii) “fast points” with a fixed systematic drift. Without these fast points, the model is related to the diffusion in heavy-tailed (“stable”) random potential studied by Schumacher and Singh; the fast points perturb that model. The two components compete to determine the behaviour of the random walk; we identify phase transitions in terms of the model parameters. We give conditions for recurrence and transience and prove almost sure bounds for the trajectories of the walk.     

Rates of uniform convergence of extreme order statistics

Summary Bounds for the convergence uniformly over all Borel sets of the largest order statistic as well as of the joint distribution of extremes are established which reveal in which way these rates are determined by the distance of the underlying density from the density of the corresponding generalized Pareto distribution. The results are highlighted by several examples among which there is a bound for the rate at which the joint distribution of thek largest order statistics from a normal distribution converges uniformly to its limit.     

Limit theorems for sums of independent random variables defined on a nonrecurrent random walk

Let {X_i} _i= -, {_i} _i=1 be two independent sequences of randoms variables, where the _i are identically distributed and assume integer values. LetIn the paper the question of the asymptotic behavior as n of the quantity is considered. It is shown that the distribution of W_n converges to the distribution of the normal law and that the estimate of the rate of convergence has the same order as the classical estimate of Berry-Esseen.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 85, pp. 17–29, 1979.The author is grateful to I. A. Ibragimov for his attention to the present work and for useful discussions.     

Sequential order statistics with an order statistics prior

In the model of sequential order statistics, prior distributions are considered for the model parameters, which, for example, describe increasing load put on remaining components. Gamma priors are examined as well as priors out of a class of extended truncated Erlang distributions (ETED), which is introduced along with some properties. The choice of independent priors in both set-ups leads to respective independent, conjugate posterior distributions for the model parameters of sequential order statistics. Since, in practical applications, the model parameters will often be increasingly ordered, a multivariate prior is applied being the joint distribution of common ETED-order statistics. Whatever baseline distribution of the sequential order statistics is chosen, the joint posterior distribution turns out to be a Weinman multivariate exponential distribution. Posterior moments are given explicitly, and HPD credible sets for the model parameters are stated.     

Uniform convergence of sums of order statistics to stable laws

Let X ₁, X ₂,... denote an i.i.d. sequence of real valued random variables which ly in the domain of attraction of a stable law Q with index 0<1. under=" a=" von=" mises=" condition=" we=" show=" that=" the=" sum=" of=" order=" statistics=">

Reducts of the random partial order

We determine, up to the equivalence of first-order interdefinability, all structures which are first-order definable in the random partial order. It turns out that there are precisely five such structures. We achieve this result by showing that there exist exactly five closed permutation groups which contain the automorphism group of the random partial order, and thus expose all symmetries of this structure.