Anna Dembińska

Linearity of regression for non-adjacent record values

Abstract Let X1,X2,… be a sequence of iid random variables having a continuous distribution; by R1,R2,… denote the corresponding record values. All the distributions allowing linearity of regressions either E(Rm+k|Rm) or E(Rm|Rm+k) are identified.

How Many Observations Fall in a Neighborhood of an Order Statistic

Asymptotic behavior of the number of independent identically distributed observations in a left or right neighborhood of k n th order statistic from the sample of size n, for k n /n → α ∈ [0, 1], is studied. It appears that the limiting laws are of the Poisson type.

A Characterization of Geometric Distribution Through kth Weak Records

ABSTRACT Let be the sequence of kth weak records from a distribution F having support on non negative integers. We show that is a constant almost surely (a.s.) iff the underlying distribution is geometric. We also prove that for n ≥ 1 and k ≥ 2 the property (a.s.) does not hold in the geometric case.

Asymptotic properties of numbers of observations near sample quantiles

In this paper, we show that the proportions of observations falling in the left and right vicinity of the k n th order statistic converge in probability to some population quantities. We then prove that suitably normalized versions of these variables are jointly asymptotically normal under some conditions. A generalization of this result to the case of proportions of observations in the vicinity of two or more central order statistics is established next. Some concluding remarks and a potential statistical application of these results are finally made.

On the asymptotic independence of numbers of observations near order statistics

In this paper, we consider the numbers of observations in two-sided neighbourhoods of the kth and (n−r)th order statistics from a sample of size n and show that they are asymptotically independent as n→∞. We also establish a result that generalizes all the existing results regarding the asymptotic independence of numbers of observations in the left and right neighbourhoods of order statistics. Finally, we consider the limiting joint behaviour of numbers of observations in the neighbourhoods of s central order statistics and establish that they are asymptotically independent.

A Review on Characterizations of Discrete Distributions Based on Records and kth Records

This is a survey of characterizations of discrete distributions via properties of record values. Characterization results based on records and weak records are presented. Then the concepts of kth records, strong kth records, and weak kth records are recalled. Finally, characterizations of the geometric parent involving these three types of kth records are discussed.

On the asymptotics of numbers of observations in random regions determined by order statistics

In this paper, we consider random variables counting numbers of observations that fall into regions determined by extreme order statistics and Borel sets. We study multivariate asymptotic behavior of these random variables and express their joint limiting law in terms of independent multinomial and negative multinomial laws. First, we give our results for samples with deterministic size; next we explain how to generalize them to the case of randomly indexed samples.

Characterizations of geometric distribution through progressively Type-II right-censored order statistics

In this paper, we consider characterizations of geometric distribution based on some properties of progressively Type-II right-censored order statistics. Specifically, we establish characterizations through conditional expectation, identical distribution, and independence of functions of progressively Type-II right-censored order statistics. Moreover, extensions of these results to generalized order statistics are also sketched. These generalize the corresponding results known for the case of ordinary order statistics.

Asymptotic normality of numbers of observations in random regions determined by order statistics

In this paper, we study the joint limiting behaviour of numbers of observations that fall into regions determined by order statistics and Borel sets. We show that suitably centred and normed versions of these numbers are asymptotically multivariate normal under some conditions. We consider two cases: one where the population distribution function is discontinuous and the other where it is continuous and the order statistics are extreme. Finally, we compare results obtained for the two cases with their analogues for absolutely continuous distribution function and central-order statistics.

Linearity of regression for adjacent order statistics-discrete case

Let Xx,Xz be a random sample from a discrete distribution on the integers with the corresponding order statistics X\a < -^2:2In [5] Nagaraja considered the problem of characterizing distributions with the property E(Xi:2\Xj:2, X2-.2 — Xi-2 > m) = ajXj:2 + bj where m is a nonnegative integer, a, b -some constants and i,j € {1,2}, i ^ j. However some of the main results of that paper are not complete or incorrect. In this note, developing Nagarajas ideas, the family of distributions with the linearity of regression property is completely characterized.