Let be a real-valued, homogeneous, and isotropic random field indexed in . When restricted to those indices with , the Euclidean length of , equal to r (a positive constant), then the random field resides on the surface of a sphere of radius r. Using a modified stratified spherical sampling plan (Brown (1993)) on the sphere, define to be a realization of the random process and to be the cardinality of . Without specifying the dependence structure of nor the marginal distribution of the , conditions for asymptotic normality of the standardized sample mean, , are given. The conditions on and are motivated by the ideas and results for dependent stationary sequences.

In this paper we establish a characterization theorem for a general class of life-testing models based on a relationship between conditional expectation and the failure rate function. As a simple application of the theorem, we characterize the gamma, Weibull, and Gompertz distributions, since they have many probabilistic and statistical properties useful in both biometry and engineering reliability.

In the present paper we study the number of occurrences of non-overlapping success runs of length in a sequence of (not necessarily identical) Bernoulli trials arranged on a circle. An exact formula is given for the probability function, along with some sharp bounds which turn out to be very useful in establishing limiting (Poisson convergence) results. Certain applications to statistical run tests and reliability theory are also discussed.

Consider a sequence of possibly dependent random variables having the same marginal distribution F, whose tail 1−F is regularly varying at infinity with an unknown index − α < 0 which is to be estimated. For i.i.d. data or for dependent sequences with the same marginal satisfying mixing conditions, it is well known that Hill's estimator is consistent for α−1 and asymptotically normally distributed. The purpose of this paper is to emphasize the central role played by the tail empirical process for the problem of consistency. This approach allows us to easily prove Hill's estimator is consistent for infinite order moving averages of independent random variables. Our method also suffices to prove that, for the case of an AR model, the unknown index can be estimated using the residuals generated by the estimation of the autoregressive parameters.

In this paper we study the distribution associated with paired size-and-shape for two sets of correlated points having an underlying Gaussian distribution in the plane. This extends work previously done just for paired shape. Special cases are studied, and various properties described. In addition we give further results for the paired shape density.

We prove that if the renewal function M(t) corresponding to a life distribution F is convex (concave) then F is NBU (NWU), and hence answer two questions posed by Shaked and Zhu (1992). Moreover, based-on the renewal function, some characterizations of the exponential distribution within certain classes of life distributions are given.

We make some comments on recent papers involving the integrated Cauchy functional equation or specialized versions of it, and reveal in particular that these papers give an inaccurate picture of the current state of the literature on the topic.

Let F be the gamma distribution function with parameters a > 0 and α > 0 and let Gs be the negative binomial distribution function with parameters α and a/s, s > 0. By combining both probabilistic and approximation-theoretic methods, we obtain sharp upper and lower bounds for . In particular, we show that the exact order of uniform convergence is s–p , where p = min(1, α). Various kinds of applications concerning charged multiplicity distributions, the Yule birth process and Bernstein-type operators are also given.

Consider a workstation with one server, performing jobs with a service time, Y, having distribution function, G(t). Assume that the station is unreliable, in that it occasionally breaks down. The station is instantaneously repaired, and the server restarts the uncompleted job from the beginning. Let T denote the time it takes to complete each job. If G(t) is exponential with parameter A, then because of the lack-of-memory property of the exponential, P (T > t) = Ḡ(t) =exp(−γt), irrespective of when and how the failures occur. This property also characterizes the exponential distribution.

Consider a spatial point pattern realized from an inhomogeneous Poisson process on a bounded Borel set , with intensity function λ (s; θ), where . In this article, we show that the maximum likelihood estimator and the Bayes estimator are consistent, asymptotically normal, and asymptotically efficient as the sample region . These results extend asymptotic results of Kutoyants (1984), proved for an inhomogeneous Poisson process on [0, T] , where T →∞. They also formalize (and extend to the multiparameter case) results announced by Krickeberg (1982), for the spatial domain . Furthermore, a Cramér–Rao lower bound is found for any estimator of θ. The asymptotic properties of and are considered for modulated (Cox (1972)), and linear Poisson processes.

Mullooly (1988) provides sufficient conditions under which the variance of a left-truncated, non-negative random variable will be greater than the variance of the original variable. We consider this problem for the class of exponential mixtures, and provide an explicit expression for the inflation in variance in terms of the mixing density.

When a random electrical network has the structure of a rooted tree and the edge resistances are either inverse Gaussian or reciprocal inverse Gaussian random variables then, subject to some restrictions, the overall resistance of the network is shown to follow a reciprocal inverse Gaussian distribution.

This article investigates the accuracy of approximations for the distribution of ordered m-spacings for i.i.d. uniform observations in the interval (0, 1). Several Poisson approximations and a compound Poisson approximation are studied. The result of a simulation study is included to assess the accuracy of these approximations. A numerical procedure for evaluating the moments of the ordered m-spacings is developed and evaluated for the most accurate approximation.

The accuracy of the Poisson approximation to the distribution of the numbers of large and small m-spacings, when n points are placed at random on the circle, was analysed using the Stein–Chen method in Barbour et al. (1992b). The Poisson approximation for m≧2 was found not to be as good as for 1-spacings. In this paper, rates of approximation of these distributions to suitable compound Poisson distributions are worked out, using the CP–Stein–Chen method and an appropriate coupling argument. The rates are better than for Poisson approximation for m≧2, and are of order O((log n)2/n) for large m-spacings and of order O(1/n) for small m-spacings, for any fixed m≧2, if the expected number of spacings is held constant as n → ∞.

In this article, we generalize results by Dimitrov and Khalil (1990), Khalil et al. (1991), and van Harn and Steutel (1991) and obtain some characterizations of the exponential and geometric laws.

This paper considers the joint limiting behavior of sums and maxima of stationary discrete-valued processes. The asymptotic behavior is a cross between a central limit theorem and asymptotic bounds for the distribution of the maxima. Some applications and simulations are also included.

Suppose n possibly censored survival times are observed under an independent censoring model, in which the observed times are generated as the minimum of independent positive failure and censor random variables. A practical difficulty arises when the largest observation is censored since then the usual non-parametric estimator of the distribution of the survival time is improper. We calculate the probability that this occurs and give necessary and sufficient conditions for this probability to converge to 0 as n →∞. As an application, we show that if this probability is 0, asymptotically, then a consistent estimator for the mean failure time can be found. An almost sure version of the problem is also considered.

Shock models based on Poisson processes have been used to derive univariate and multivariate exponential distributions. But in many applications, Poisson processes are not realistic models of physical shock processes because they have independent increments; expanded models that allow for possibly dependent increments are of interest. In this paper, univariate and bivariate Pólya urn schemes are used to derive models of shock sources. The life distributions obtained from these models form a large parametric family that includes the exponential distribution. Even in the univariate case these life distributions have not been widely used, though they form a large and flexible family. In the bivariate case, the family includes the bivariate exponential distributions of Marshall and Olkin as a special case.

Let {Fn }n ≧ 0 be a sequence of c.d.f. and let {Rn }n ≧ 1 be the sequence of record values in a non-stationary record model where after the (n − 1)th record the population is distributed according to Fn. Then the equidistribution of the nth population and the record increment Rn – Rn – 1 (i.e. Rn – Rn – 1~ Fn ) characterizes Fn to have an exponentially decreasing hazard function. To be more precise Fn is the exponential distribution if the support of Rn – 1 generates a dense subgroup in and otherwise the entity of all possible solutions can be obtained in the following way: let for simplicity the above additive subgroup be any c.d.f. F satisfying F(0) = 0, F(1) < 1 can be chosen arbitrarily. Setting λ = – log(1 – F(1)), Fn (x) = 1 – F(x – [x])exp(–λ [x]) is an admissible solution coinciding with F on the interval [0, 1] ([x] denotes the integer part of x). Simple additional assumptions ensuring that Fn is either exponential or geometric are given. Similar results for exponential or geometric tail distributions based on the independence of Rn – 1 and Rn – Rn – 1 are proved.

Let γ t and δ t denote the residual life at t and current life at t, respectively, of a renewal process , with the sequence of interarrival times. We prove that, given a function G, under mild conditions, as long as holds for a single positive integer n, then is a Poisson process. On the other hand, for a delayed renewal process with the residual life at t, we find that for some fixed positive integer n, if is independent of t, then is an arbitrarily delayed Poisson process. We also give some corresponding results about characterizing the common distribution function F of the interarrival times to be geometric when F is discrete. Finally, we obtain some characterization results based on the total life or independence of γ t and δ t.