We introduce a new measure of inaccuracy based on extropy between distributions of the nth upper (lower) record value and parent random variable and discuss some properties of it. A characterization problem for the proposed extropy inaccuracy measure has been studied. It is also shown that the defined measure of inaccuracy is invariant under scale but not under location transformation. We characterize certain specific lifetime distribution functions. Nonparametric estimators based on the empirical and kernel methods for the proposed measures are also obtained. The performance of estimators is also discussed using a real dataset.

Due to the importance of generalized order statistics (GOS) in many branches of Statistics, a wide interest has been shown in investigating stochastic comparisons of GOS. In this article, we study the likelihood ratio ordering of $p$-spacings of GOS, establishing some flexible and applicable results. We also settle certain unresolved related problems by providing some useful lemmas. Since we do not impose restrictions on the model parameters (as previous studies did), our findings yield new results for comparison of various useful models of ordered random variables including order statistics, sequential order statistics, $k$-record values, Pfeifer's record values, and progressive Type-II censored order statistics with arbitrary censoring plans. Some results on preservation of logconvexity properties among spacings are provided as well.

We study the asymptotic properties of a minimal spanning tree formed by n points uniformly distributed in the unit square, where the minimality is amongst all rooted spanning trees with a direction of growth. We show that the number of branches from the root of this tree, the total length of these branches, and the length of the longest branch each converges weakly. This model is related to the study of record values in the theory of extreme-value statistics and this relation is used to obtain our results. The results also hold when the tree is formed from a Poisson point process of intensity n in the unit square.

Let n points be placed uniformly at random in a subset A of the plane. A point is said to be maximal in the configuration if no other point is larger in both coordinates. We show that, for large n and for many sets A, the number of maximal points is approximately normally distributed. The argument uses Stein's method, and is also applicable in higher dimensions.

We study a random record model where the observation Xi has continuous distribution function Fαi (αi > 0) and the number of available observations is random and independent of the observations. We obtain the joint distribution of the record values and inter-record times for our model. We investigate the distribution of the number of records when the number of observations has one of the common distributions and the α's increase geometrically or linearly. A particularly interesting case arises when the observations arrive at time points paced by a Poisson point process. For this model we obtain distributional results for the inter-arrival times of records for a large class of combinations of α structures and intensity functions.

Let Y0, Y1, Y2, ··· be an i.i.d. sequence of random variables with absolutely continuous distribution function F, and let {N(t), t ≧ 0} be a Poisson process with rate λ (t) and mean Λ(t), independent of the Yj's. We associate Y0 with the point t = 0, and Yj with the jth point of N(·), j ≧ 1. The first Yj (j ≧ 1) to exceed all previous ones is the first record value, and the time of its occurrence is the first record time; subsequent record values and times are defined analogously. For general Λ, we give the joint distribution of the values and times of the first n records to occur after a fixed time T, 0 ≦ T < ∞. Assuming that F satisfies Von Mises regularity conditions, and that λ (t)/Λ (t) → c ∈ (0, ∞) as t → ∞, we find the limiting joint p.d.f. of the values and times of the first n records after T, as T → ∞. In the course of this we correct a result of Gaver and Jacobs (1978). We also consider limiting marginal and conditional distributions. In addition, we extend a known result for the limit as the number of recordsK → ∞, and we compare the results for the limit as T → ∞ with those for the limit as K → ∞.

We consider a class of secretary problems in which the order of arrival of candidates is no longer uniformly distributed. By a suitable embedding in a time-transformed extremal process it is shown that the asymptotic winning probability is again 1/e as in the classical situation. Extensions of the problem to more than one choice are also considered.

By Ignatov's theorem the sets of values in an i.i.d. sequence that are the kth largest at their appearance (k = 1, 2, ···) are supports of i.i.d. Poisson processes. The present paper contains an elementary and short proof for the case where the underlying distribution function F is discrete, and then extends the result to general F.

A complete characterization of elementary pure birth processes is given by means of record counting processes from independent (non-identically) distributed random variables.

The limit distribution of the k th maximum from a random sample of size n when n → ∞ is identified as the distribution of the k th lower record value from one of three extreme value distributions. This fact is used in giving a different canonical representation and new proofs of the results of Hall (1978) for this limiting random variable. A characterization of the exponential distribution based on upper record values is given.

A non-homogeneous version of the classical record process is presented which allows two different characterizations of exponential distributions by independent non-stationary record increments. A connection with the interarrival times of the corresponding record counting process (which is pure birth) is also pointed out.

We define a vector-valued scheduled maxima sequence M by considering simultaneously the maxima of several i.i.d. sequences, with the number of observations considered from each sequence at any time determined by a random scheduling sequence J. It is shown that the max-min (vector) sequence derived from i.i.d. can be represented as a mixture of scheduled maxima sequences, giving results for this sequence and the range A functional limit theorem for the scheduled maxima sequence shows convergence to independent extremal processes. Embedding in a scheduled extremal process gives strong laws, central limit theorems, and laws of the iterated logarithm for the record time of the scheduled maxima sequence, and hence for the max-min sequence and the range.

The inverse of an extremal process {Y(t), t ≧ 0} is an additive process whose Lévy measure can be computed. This measure controls among other things the Poisson number of jumps of Y while Y is in the vertical window (c, d]. A simple transformation of the inverse of the extremal process governed by Λ (x) = exp{– e–x} is also extremal-Λ (x) and this fact enables one to relate behavior of Y-Λ at t = ∞ to behavior near t = 0. Some extensions of these ideas to sample sequences of maxima of i.i.d. random variables are carried out.

{Xn, n ≧ 1} are i.i.d. unbounded random variables with continuous d.f. F(x) =1 —e –R(x). Xj is a record value of this sequence if Xj >max {X1, …, Xj-1} The almost sure behavior of the sequence of record values {XLn} is studied. Sufficient conditions are given for lim supn→∞XLn/R–l(n)=ec, lim inf n → ∞XLn/R−1 (n) = e−c, a.s., 0 ≦ c ≦ ∞, and also for lim supn→∞ (XLn—R–1(n))/an =1, lim infn→∞ (XLn—R–1(n))/an = − 1, a.s., for suitably chosen constants an. The a.s. behavior of {XLn} is compared to that of the sequence {Mn}, where Mn = max {X1, …, Xn}. The method is to translate results for the case where the Xn's are exponential to the general case by means of an extended theory of regular variation.

First, asymptotic results for inter-record times when the CDF of the underlying IID process is not necessarily continuous are obtained, by a stochastic order argument, from known results for the continuous case. Then the asymptotic behaviour of the bivariate process of upper-record values and inter-record times is studied. Finally, assuming continuity of the underlying CDF, we derive the law of the process of total times spent in sets of states, viewing upper record values as states and inter-record times as times spent in a state, the process so viewed being a discrete time continuous state Markov jump process.

The possible relevance of this result to single lane road traffic flow is indicated.

A limit theorem for inter-record times is obtained which seems to explain many of their known asymptotic properties. The process of inter-record times is shown to be approximately non-homogeneous Poisson.