bluuebunny/crossref_metadata_2025 huggingface dataset:
Extreme values of independent stochastic processes
Authors: Brown Bruce M. , Resnick Sidney I.
Published: July 2016

The maxima of independent Weiner processes spatially normalized with time scales compressed is considered and it is shown that a weak limit process exists. This limit process is stationary, and its one-dimensional distributions are of standard extreme-value type. The method of proof involves showing convergence of related point processes to a limit Poisson point process. The method is extended to handle the maxima of independent Ornstein–Uhlenbeck processes.
The average noise from a Poisson stream of vehicles
Authors: Andersen Per K. , Andersen Søren , Lauritzen Steffen L.
Published: July 2016

The distribution of the average noise power from a Poisson stream of vehicles is, properly normalised, shown to converge to a normal distribution although the corresponding stationary process is deterministic. The speed of convergence is estimated. Finally the asymptotic efficiency of a sampling procedure is discussed.
Two theorems on optimal stopping with backward solicitation
Authors: Karni Edi , Schwartz Aba
Published: July 2016

This paper deals with optimal stopping rules for a sampling process with uncertain recall, i.e. the probability that a past observation is currently available declines exponentially with the time elapsed since it was last observed. The main result of this paper is that for such sampling processes, and for any utility function, if a solicitation of a past observation incurs the same cost as a new draw, then it is never optimal to continue the sampling when the observation solicited is found to be available. This result applies to both bounded and unbounded sequential decision procedures.
Moments of the exit-time distribution for carboxyhemoglobin due to cigarette smoking
Authors: Marcus Allan H. , Czajkowski Stanley
Published: July 2016

Linear and non-linear secondary Poisson process models are developed for the level of carboxyhemoglobin in cigarette smoking. The moments of the time of first exit above a critical threshold are expanded as power series in the initial state of the process. Recurrence relations for the coefficients in the moment series are solved numerically. An explicit formula involving Poisson probability functions is obtained for the mean exit time of shot noise.
Continuous-time stochastic models of a multigrade population
Authors: McClean Sally I.
Published: July 2016

The continuous-time Markov model of a multigrade organization is extended in several ways. Firstly the internal transitions and the leaving process are generalized to a semi-Markov formulation which allows for the inclusion of well-authenticated leaving distributions such as the mixed exponential distribution. The previous assumption of Poisson recruitment is then generalized to allow for a time-dependent Poisson arrival distribution in which the instantaneous probability of an arrival is a mixture of exponential terms. Finally we extend the capital-related manpower model to describe a multigrade organization.
Criteria for strong ergodicity of Markov chains
Authors: Isaacson Dean , Tweedie Richard L.
Published: July 2016

For finite Markov chains, the concepts of ergodicity and strong ergodicity are equivalent, but this is not necessarily the case when the state space is infinite. In this note we give some new characterizations of strong ergodicity. These lead to simple necessary or sufficient criteria for strong ergodicity, which readily enable us to classify a number of examples.
An asymptotic formula for the transition density of random genetic drift
Authors: Antonelli Peter L.
Published: July 2016

Stochastic models in population genetics which lead to diffusion equations are considered. A geometric formula for the asymptotic expansions of the fundamental solutions of these equations is presented. Specifically, the random genetic drift process of one-locus theory and the Ohta–Kimura model of two-locus di-allelic systems with linkage are studied. Agreement with the work of Keller and Voronka for the two-allele one-locus case is obtained. For the general n-allele problem, the formulas obtained here are apparently new.
Comment on ‘Corrected diffusion approximations in certain random walk problems'
Authors: Hogan Michael L.
Published: July 2016

Correction terms for the diffusion approximation to the maximum and ruin probabilities for a random walk with small negative drift, obtained by Siegmund (1979) in the exponential family case, are extended by different methods to some non-exponential family cases.
Optimum replacement of a system subject to shocks
Authors: Abdel-Hameed Mohamed
Published: July 2016

A system is subject to shocks. Each shock weakens the system and makes it more expensive to run. It is desirable to determine a replacement time for the system. Boland and Proschan [4] consider periodic replacement of the system and give sufficient conditions for the existence of an optimal finite period, assuming that the shock process is a non-homogeneous Poisson process and the cost structure does not depend on time. Block et al. [3] establish similar results assuming that cost structure is time dependent, still requiring that the shock process is a non-homogeneous Poisson process. We show via a sample path argument that the results of [3] and [4] hold for any counting process whose jump size is of one unit magnitude.
A correlated queue with infinitely many servers
Authors: Langaris C.
Published: July 2016

This work analyses a queueing mechanism with infinitely many servers in which the interarrival interval T preceding the arrival of a customer and his service time S are assumed correlated. A bivariate distribution with negative exponential marginals is used and the Laplace transforms pn
(z) of the system probabilities in transient state are obtained. Closed-form expressions for the steady-state probabilities pn
and their moments are given too. Finally the output process is investigated and conclusions are drawn from numerical calculations.
The substability and ergodicity of complicated queueing systems
Authors: Nakatsuka Toshinao
Published: July 2016

The substability and the ergodicity of various queueing models are discussed. This paper considers the vector-valued queueing processXrn= (xrn,1,xrn,2, · ··) with non-negative components and a constant initial valueXrr= a.For this, the substability is derived under simple conditions by showing the finiteness ofWith respect to the ergodicity, in order to make use of Borovkov's theorem, we additionally assume that the distribution of the interarrival of customers has non-bounded tail for any given past sequence.
Return probabilities for correlated random walks
Authors: Iossif Gillian
Published: July 2016

A correlated random walk on a d-dimensional integer lattice is studied in which, at any stage, the probabilities of the next step being in the various possible directions depend on the direction of the previous step. Using a renewal argument, asymptotic estimates are obtained for the probability of return to the origin after n steps.
A characterization of the Poisson process
Authors: Gupta Pushpa Lata , Gupta Ramesh C.
Published: July 2016

Denoting by v(t) the residual life of a component in a renewal process, Çinlar and Jagers (1973) and Holmes (1974) have shown that if E(v(t)) is independent of t for all t, then the process is Poisson. In this note we prove, under mild conditions, that if E(G(v(t))) is constant, then the process is Poisson. In particular if E((v(t))r) for some specific real number r ≧ 1 is independent of t, then the process is Poisson.
A Pólya urn model and the coalescent
Authors: Trieb Gudrun
Published: July 2016

In recent papers by Hoppe and Donnelly it has been shown that a Pólya urn model generating the Ewens sampling formula (population genetics) parallels a construction of Kingman using a Poisson–Dirichlet ‘paintbox'. Even the jump chain of Kingman's n-coalescent can be constructed using the urn. The properties of a certain process based on the coalescent also are derived. This process was introduced by Hoppe.
Gaussian integrals on Wiener spaces
Authors: Helfer Adam D. , Zhao Zhongxin
Published: July 2016

We consider integrals on Wiener space of the forms E(exp K(x)) and E(exp K(x) |L(x) = l) where K is a quadratic form and L a system of linear forms. We give explicit formulas for these integrals in terms of the operators K and L, in the case that these arise from quasilinear functions in the sense of Zhao (1981). As examples, we recover Lévy's area formula in the plane, and derive new formulas for the probability density of the radius of gyration tensor for Brownian paths.
Discrete minification processes and reversibility
Authors: Littlejohn R. P.
Published: July 2016

Discrete minification processes are introduced and it is proved that the discrete first-order autoregression of McKenzie (1986) and the discrete minification process are mutually time-reversible if and only if they have common marginal geometric distribution, corresponding to a result for continuous processes given by Chernick et al. (1988). It is also proved that a discrete minification process is time-reversible if and only if it has marginal Bernoulli distribution.
Optimal timing of antiviral therapy in HIV infection
Authors: Berman Simeon M.
Published: July 2016

A three-stage real diffusion process is used as a model of the T-cell count of an HIV-positive individual who is to receive antiviral therapy such as AZT. The ‘quality of life' of such a person is identified as the sojourn time of the diffusion process above a certain critical T-cell level c. The time of introducing therapy is defined as the first-passage time of the diffusion to a prescribed level z > c. The distribution of the sojourn time of the diffusion above the level c depends on the level z at which therapy is initiated. The expected sojourn time is explicitly computed as a function of z for the particular diffusion process defining the model. There is a simple criterion for determining when to start therapy as early as possible.
The work of Lajos Takács on probability theory
Authors: Bingham N. H.
Published: July 2016

In more than four decades of prolific scientific activity, Lajos Takács has produced so much that no one contributor to this Festschrift can hope to cover all of it in a balanced way. I thus propose to make a virtue of necessity, and concentrate on those aspects of Takács' work which have particularly interested or influenced me, and on the impact Takács' ideas in these areas have had on the subsequent development of the subject.
Minimax one-sided Kolmogorov-type distribution-free tests
Authors: Pyke Ronald
Published: July 2016

A family of distribution-free statistics along with related tests is defined and properties of its members are studied. These statistics, one of which is the Smirnov-Wald and Wolfowitz statistic, D

n

+ yield tests of the one-sided hypotheses. The minimax and maximin tests in this family against a restricted class of alternatives of minimum power are obtained. The connection to a confidence bound of Stringer for the mean is also remarked.
Replacement of train wheels: an application of dynamic reversal of a Markov process
Authors: Gates David , Westcott Mark
Published: July 2016

A problem of regrinding and recycling worn train wheels leads to a Markov population process with distinctive properties, including a product-form equilibrium distribution. A convenient framework for analyzing this process is via the notion of dynamic reversal, a natural extension of ordinary (time) reversal. The dynamically reversed process is of the same type as the original process, which allows a simple derivation of some important properties. The process seems not to belong to any class of Markov processes for which stationary distributions are known.
On the arc-sine laws for Lévy processes
Authors: Getoor R. K. , Sharpe M. J.
Published: July 2016

Let X be a Lévy process on the real line, and let Fc
denote the generalized arcsine law on [0, 1] with parameter c. Then t
−1 ⨍0
t

P

0(X

s
> 0) ds → c as t → ∞ is a necessary and sufficient condition for t
—1 ⨍0

t
1{Xs
>0}
ds to converge in P
0 law to Fc. Moreover, P
0(Xt
> 0) = c for all t > 0 is a necessary and sufficient condition for t
—1 ⨍0

t
1{Xs
>0}
ds under P
0 to have law Fc
for all t > 0. We give an elementary proof of these results, and show how to derive Spitzer's theorem for random walks in a simple way from the Lévy process version.
Multivariate self-similar processes: second-order theory
Authors: Bladt M.
Published: July 2016

The aim of this paper is to extend the existing theory of second-order self-similar processes as defined by Cox (1984) from the univariate case to higher dimensions. Multivariate self-similar processes defined in terms of second-order theory for stationary time series can be used as models for long-range dependent observations when the marginal observations are long-range dependent. An interesting question concerns the correlation structure within the processes when the marginal processes are correlated. We show that the self-similarity requirement, as defined in this article, implies a cross-correlation structure similar to that for the marginal processes. This occurs both in the time domain and in the frequency domain. This fact can be used to obtain generalized least squares estimates for the long-range dependence parameters. We discuss some difficulties concerning estimation based on simulations.
The variance of a truncated mixed exponential process
Authors: Hebert Jaimie L. , Seaman John W.
Published: July 2016

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.
On a best-choice problem by dependent criteria
Authors: Gnedin Alexander V.
Published: July 2016

We study the problem of maximizing the probability of stopping at an object which is best in at least one of a given set of criteria, using only stopping rules based on the knowledge of whether the current object is relatively best in each of the criteria. The asymptotic results for the case of independent criteria are shown to hold in certain cases where the componentwise maxima are, pairwise, either asymptotically independent or asymptotically full dependent.

An example of the former is a random sample from a bivariate correlated normal distribution; thus our results settle a question posed recently by T. S. Ferguson.
Distribution of the final extent of a rumour process
Authors: Lefevre Claude , Picard Philippe
Published: July 2016

A rumour model due to Maki and Thompson (1973) is slightly modified to incorporate a continuous-time random contact process and varying individual behaviours in front of the rumour. Two important measures of the final extent of the rumour are provided by the ultimate number of people who have heard the rumour, and the total personal time units during which the rumour is spread. Our purpose in this note is to derive the exact joint distribution of these two statistics. That will be done by constructing a family of martingales for the rumour process and then using a particular family of Gontcharoff polynomials.
A new policy iteration scheme for Markov decision processes using Schweitzer's formula
Authors: Lasserre J. B.
Published: July 2016

Given a family of Markov chains with a single recurrent class, we present a potential application of Schweitzer's exact formula relating the steady-state probability and fundamental matrices of any two chains in the family. We propose a new policy iteration scheme for Markov decision processes where in contrast to policy iteration, the new criterion for selecting an action ensures the maximal one-step average cost improvement. Its computational complexity and storage requirement are analysed.
Generalized semi-Markov decision processes
Authors: Doshi Bharat T.
Published: July 2016

Various authors have derived the necessary and sufficient conditions for optimality in semi-Markov decision processes in which the state remains constant between jumps. In this paper similar results are presented for a generalized semi-Markov decision process in which the state varies between jumps according to a Markov process with continuous sample paths. These results are specialized to a general storage model and an application to the service rate control in a GI/G/1 queue is indicated.
A solution of the carrier-borne epidemic
Authors: Henderson W.
Published: July 2016

An alternative approach to the derivation of a general class of results in carrier-borne epidemic theory is presented. Some insight can be gained into the basic structure of carrier-borne epidemics and the method indicates which other problems of a similar type could yield to analysis.
A note on exchangeable events
Authors: Hsu Yu-Sheng
Published: July 2016

Kendall [2] gave a thorough discussion of De Finetti constants for a sequence (finite or infinite) of exchangeable events. Galambos [1] and Ridler-Rowe [3] also found some interesting results in this area. In this paper, we intend to give a necessary and sufficient condition for a finite sequence of exchangeable events to be extendable to an infinite sequence of exchangeable events.
Bellman–Harris branching processes with state-dependent immigration
Authors: Mitov K. V. , Yanev N. M.
Published: July 2016

We consider critical Bellman-Harris processes which admit an immigration component only in the state 0. The asymptotic behaviour of the probability of extinction and of the first two moments is investigated and a limit theorem is also proved.
On a class of continuous-time Markov processes
Authors: Gani J. , Tin Pyke
Published: July 2016

This paper considers a certain class of continuous-time Markov processes, whose time-dependent and stationary distributions are studied. In the stationary case, the analogy with Whittle's relaxed Markov process is pointed out. The derivation of the probability generating functions of the general process provides useful results for the analysis of some population and queueing processes.
On regenerative and ergodic properties of the k-server queue with non-stationary Poisson arrivals
Authors: Thorisson Hermann
Published: July 2016

We consider the stable k-server queue with non-stationary Poisson arrivals and i.i.d. service times and show that the non-time-homogeneous Markov process Zt
= (the queue length and residual service times at time t) can be subordinated to a stable time-homogeneous regenerative process. As an application we show that if the system starts from given conditions at time s then the distribution of Zt
stabilizes (but depends on t) as s tends backwards to –∞. Also moment and stochastic domination results are established for the delay and recurrence times of the regenerative process leading to results on uniform rates of convergence.
First-passage-time moments of Markov processes
Authors: Yao David D.
Published: July 2016

We consider the first-passage times of continuous-time Markov chains. Based on the approach of generalized inverse, moments of all orders are derived and expressed in simple, explicit forms in terms of the ‘fundamental matrix'. The formulas are new and are also efficient for computation.
Individually optimal routing in parallel systems
Authors: Kumar P. R. , Walrand J.
Published: July 2016

Jobs arrive at a buffer from which there are several parallel routes to a destination. A socially optimal policy is one which minimizes the average delay of all jobs, whereas an individually optimal policy is one which, for each job, minimizes its own delay, with route preference given to jobs at the head of the buffer. If there is a socially optimal policy for a system with no arrivals, which can be implemented by each job following a policy γ in such a way that no job ever utilizes a previously declined route, then we show that such a γ is an individually optimal policy for each job. Moreover γ continues to be individually optimal even if the system has an arbitrary arrival process, subject only to the restriction that past arrivals are independent of future route-traversal times. Thus, γ is an individually optimal policy which is insensitive to the nature of the arrival process. In the particular case where the times to traverse the routes are exponentially distributed with a possibly different mean time for each of the parallel routes, then such an insensitive individually optimal policy does in fact exist and is moreover trivially determined by certain threshold numbers. A conjecture is also made about more general situations where such individually optimal policies exist.
On random genetic drift in a subdivided population
Authors: Pollak Edward
Published: July 2016

Generations are assumed to be non-overlapping. We consider a haploid population divided intoKparts, each of which containNadults in any generation. These are obtained by a random sampling of the offspring of the previous generation. We assume that the probability of an adult offspring of an individual in one subpopulation being in some other subpopulation is the same small positive number, no matter what two subpopulations are considered.

If the population initially has individuals of two types, A and a, it is of interest to study approximations, if n is large, to

(1)

the rate at which A or a is lost between generations n-1 and n,

(2)

the probability that A and a are still present in generation n,

(3)

the joint distribution of frequencies of A in the subpopulations.

A solution is given for the first problem. It is found that if the mean number of migrants per generation from one subpopulation to another is at least as large as 1, the population behaves almost as if it were not subdivided. But if this number is considerably less that 1, then the rate at which one or the other gene is lost is slower than in an undivided population. The other two problems are discussed for K = 2.
Some new results in storage theory
Authors: Prabhu N. U.
Published: July 2016

In the storage model proposed by Kendall [6], the input is assumed to be a stochastic process with stationary independent increments, and the release is continuous and is at a unit rate. The storage process arising from this model is known to reach a steady state if the expected net input per unit time is negative. In this paper we consider the situation where this expected net input is nonnegative and derive limiting distributions for the wet period, total dry period during a given time interval and the dam content.
On random sequential packing in the plane and a conjecture of palasti
Authors: Blaisdell B. Edwin , Solomon Herbert
Published: July 2016

The random packing of geometric objects in one-, two- or three-dimensions may afford useful insights into the structure of crystals, liquids, absorbates on crystals, and in higher dimensions, into problems of pattern recognition. Random packing has accordingly received increasing attention in recent years. Two principal packing procedures have been formulated and each gives rise to different packing ratios. In one case, all possible configurations of a sphere-packed volume are assumed to be equally likely. In the other and most widely reported case, there is random sequential addition of spheres to the volume until it is packed. This is the situation we study in this paper. Most of the work to date has been limited to the theoretical study of the one-dimensional lattice or to continuous cases particularly in the limit for long lines. The higher dimensional cases have resisted theoretical attack but have been studied by computer simulation by Palasti [12] and Solomon [14] and by physical simulation by Bernal and Scott (see [14]).
Occupation times of stationary gaussian processes
Authors: Berman Simeon M.
Published: July 2016

Let X(t), t ≧ 0, be a stationary Gaussian process with zero mean, unit variance and continuous covariance function r(t). Suppose that, for some ε > 0

so that there is a version of the process whose sample functions are continuous [1].
Weak convergence of the average of flag processes
Authors: Iglehart Donald L. , Kennedy Douglas P.
Published: July 2016

Let be a sequence of non-negative random variables and the associated point process defined by
On invariant measures for simple branching processes
Authors: Seneta E.
Published: July 2016

This paper was initially motivated by a problem raised earlier by the author (Seneta (1969), Section 5.3) viz. that of the existence and uniqueness of an invariant measure for a supercritical Galton-Watson process with immigration; and, indeed, in the sequel we show that such a measure always exists, but is not in general unique.
Bisexual Galton–Watson branching processes with superadditive mating functions
Authors: Daley D. J. , Hull David M. , Taylor James M.
Published: July 2016

For a bisexual Galton–Watson branching process with superadditive mating function there is a simple criterion for determining whether or not the process becomes extinct with probability 1, namely, that the asymptotic growth rate r should not exceed 1. When extinction is not certain (equivalently, r > 1), simple upper and lower bounds are established for the extinction probabilities. An example suggests that in the critical case that r = 1, some condition like superadditivity is essential for ultimate extinction to be certain. Some illustrative numerical comparisons of particular mating functions are made using a Poisson offspring distribution.
The maximum in critical Galton–Watson and birth and death processes
Authors: Kämmerle K. , Schuh H.-J.
Published: July 2016

In this paper recent results by Weiner [10] on Mn
:= max{Z
0, · ··, Zn
} are strengthened and generalized, where (Zn
)
n
is a critical Galton–Watson branching process with finite and positive offspring variance and Z
0 ≡ 1. It is shown that

Explicit asymptotic bounds are given for with . If (Zn
)
n
has a linear fractional offspring distribution, it can be embedded in a critical birth and death process (Ẑ t
)
t
. Using martingale methods one obtains thereof.

These results generalize to the case Z
0 ≡ k.
On the possible form of size distributions for Gibbsian processes of mutually non-intersecting balls
Authors: Mase Shigeru
Published: July 2016

We construct a model of mutually non-intersecting balls using a generalized notion of Gibbsian processes and characterize its size, i.e. radius, distribution. The explicit form of this distribution is given in the one-dimensional case, that is, mutually non-intersecting segment processes on the line.
Bounding the stochastic performance of continuum structure functions. I
Authors: Baxter Laurence A. , Kim Chul
Published: July 2016

A continuum structure function γ is a non-decreasing mapping from the unit hypercube to the unit interval. Minimal path (cut) sets of upper (lower) simple continuum structure functions are introduced and are used to determine bounds on the distribution of γ (Χ) when X is a vector of associated random variables and when γ is right (left)-continuous. It is shown that, if γ admits of a modular decomposition, improved bounds may be obtained.
Random sequential coding by Hamming distance
Authors: Itoh Yoshiaki , Solomon Herbert
Published: July 2016

Here we introduce two simple models: simple cubic random packing and random packing by Hamming distance. Consider the packing density γ d
of dimension d by cubic random packing. From computer simulations up to dimension 11, γ d

+1/γ d
seems to approach 1. Also, we give simulation results for random packing by Hamming distance and discuss the behavior of packing density when dimensionality is increased. For the case of Hamming distances of 2 or 3, d–α
fits the simulation results of packing density where α is an empirical constant. The variance of packing density is larger when k is even and smaller when k is odd, where k represents Hamming distance.
Aspects of positive ageing
Authors: Deshpande Jayant V. , Kochar Subhash C. , Singh Harshinder
Published: July 2016

The concept of positive ageing describes the adverse effects of age on the lifetime of units. Various aspects of this concept are described in terms of conditional probability distributions of residual lifetimes, failure rates, equilibrium distributions, etc. In this paper we further analyse this concept and relate it to stochastic dominance of first and higher orders. In the process we gain many insights and are able to define several new kinds of ageing criteria which supplement those existing in the literature.
Scheduling jobs with stochastically ordered processing times on parallel machines to minimize expected flowtime
Authors: Weber R. R. , Varaiya P. , Walrand J.
Published: July 2016

A number of jobs are to be processed using a number of identical machines which operate in parallel. The processing times of the jobs are stochastic, but have known distributions which are stochastically ordered. A reward r(t) is acquired when a job is completed at time t. The function r(t) is assumed to be convex and decreasing in t. It is shown that within the class of non-preemptive scheduling strategies the strategy SEPT maximizes the expected total reward. This strategy is one which whenever a machine becomes available starts processing the remaining job with the shortest expected processing time. In particular, for r(t) = – t, this strategy minimizes the expected flowtime.
Power variation and stochastic volatility: a review and some new results
Authors: Barndorff-Nielsen Ole E. , Graversen Svend Erik , Shephard Neil
Published: July 2016

In this paper we review some recent work on limit results on realised power variation, that is, sums of powers of absolute increments of various semimartingales. A special case of this analysis is realised variance and its probability limit, quadratic variation. Such quantities often appear in financial econometrics in the analysis of volatility. The paper also provides some new results and discusses open issues.
Option bounds
Authors: De La Peña Victor H. , Ibragimov Rustam , Jordan Steve
Published: July 2016

In this paper, we obtain sharp estimates for the expected payoffs and prices of European call options on an asset with an absolutely continuous price in terms of the price density characteristics. These techniques and results complement other approaches to the derivative pricing problem. Exact analytical solutions to option-pricing problems and to Monte-Carlo techniques make strong assumptions on the underlying asset's distribution. In contrast, our results are semi-parametric. This allows the derivation of results without knowing the entire distribution of the underlying asset's returns. Our results can be used to test different modelling assumptions. Finally, we derive bounds in the multiperiod binomial option-pricing model with time-varying moments. Our bounds reduce the multiperiod setup to a two-period setting, which is advantageous from a computational perspective.
Box–Cox transformations and heavy-tailed distributions
Authors: Teugels Jef L. , Vanroelen Giovanni
Published: July 2016

It is a stylized fact that estimators in extreme-value theory suffer from serious bias. Moreover, graphical representations of extremal data often show erratic behaviour. In the statistical literature it is advised to use a Box–Cox transformation in order to make data more suitable for statistical analysis. We provide some of the theoretical background to see the effect of such transformations and to investigate under what circumstances they might be helpful.
Some effects of trimming on the law of the iterated logarithm
Authors: Kesten Harry , Maller Ross
Published: July 2016

We investigate some effects that the ‘light' trimming of a sum Sn = X
1 + X
2 + · ·· + Xn
of independent and identically distributed random variables has on behaviour of iterated logarithm type. Light trimming is defined as removing a constant number of summands from Sn. We consider two versions: (r)
Sn, which is obtained by deleting the r largest Xi
from Sn
, and , which is obtained by deleting the r variables Xi
which are largest in absolute value from Sn. We summarise some relevant results from Rogozin (1968), Heyde (1969), and later writers concerning the untrimmed sum, and add some new results concerning trimmed sums. Among other things we show that a general form of the law of the iterated logarithm holds for but not (completely) for .
Random-allocation and urn models
Authors: Gani J.
Published: July 2016

We review some urn and random-allocation models, mostly using probability generating function (PGF) methods. We begin by formulating a basic problem which can be thought of as either an urn or a random-allocation model; a PGF solution to it is outlined. When the compartments in the latter model are no longer homogeneous, the multivariate PGF can still be derived, though the algebra becomes cumbersome. Some results are given for the case where there are two types of compartment and for the case where there are two types of ball. Some comments are offered on the Frobenius–Harper property of PGFs.
Averaging analysis of a point process adaptive algorithm
Authors: Solo Victor
Published: July 2016

Motivated by a problem in neural encoding, we introduce an adaptive (or real-time) parameter estimation algorithm driven by a counting process. Despite the long history of adaptive algorithms, this kind of algorithm is relatively new. We develop a finite-time averaging analysis which is nonstandard partly because of the point process setting and partly because we have sought to avoid requiring mixing conditions. This is significant since mixing conditions often place restrictive history-dependent requirements on algorithm convergence.
Asymptotics of the sample mean and sample covariance of long-range-dependent series
Authors: Dai Wen
Published: July 2016

An asymptotic distribution is given for the partial sums of a stationary time-series with long-range dependence. The law of large numbers for the sample covariance of the series is also derived. The results differ from those given elsewhere in relaxing the assumption of the independence of the innovations of the series.
On diagnostics in conditionally heteroskedastic time series models under elliptical distributions
Authors: Liu Shuangzhe
Published: July 2016

In statistical diagnostics and sensitivity analysis, the local influence method plays an important rôle. In the present paper, we use this method to study financial time series data and conditionally heteroskedastic models under elliptical distributions. We start with a likelihood displacement, and consider data- and model-perturbation schemes. We obtain corresponding matrices of derivatives, and measures of slope and normal curvature, and then discuss the assessment of local influence.
The teaching of statistical consulting
Authors: Russell K. G.
Published: July 2016

This paper considers how students might be taught statistical consulting. Desirable characteristics of a good consultant are described. While skill in statistical techniques is considered, it is just one of several areas of expertise that are discussed. Suggestions are made as to what should be included in a consulting course to equip a student with these characteristics.
Similar Markov chains
Authors: Pollett P. K.
Published: July 2016

Lenin et al. (2000) recently introduced the idea of similarity in the context of birth-death processes. This paper examines the extent to which their results can be extended to arbitrary Markov chains. It is proved that, under a variety of conditions, similar chains are strongly similar in a sense which is described, and it is shown that minimal chains are strongly similar if and only if the corresponding transition-rate matrices are strongly similar. A general framework is given for constructing families of strongly similar chains; it permits the construction of all such chains in the irreducible case.
The detection of words and an ordering for Markov chains
Authors: Irle Albrecht , Gani Joseph
Published: July 2016

This paper considers the occurrence of patterns in sequences of independent trials from a finite alphabet; Gani and Irle (1999) have described a finite state automaton which identifies exactly those sequences of symbols containing the specific pattern, which may be thought of as the word of interest. Each word generates a particular Markov chain. Motivated by a result of Guibas and Odlyzko (1981) on stochastic monotonicity for the random times when a particular word is completed for the first time, a new level-crossing ordering is introduced for stochastic processes. A process {Yn
: n = 0, 1, …} is slower in level-crossing than a process {Zn
}, if it takes {Yn
} stochastically longer than {Zn
} to exceed any given level. This relation is shown to be useful for the comparison of stochastic automata, and is used to investigate this ordering for Markov chains in discrete time.
Fitting hidden semi-Markov models to breakpoint rainfall data
Authors: Sansom John , Thomson Peter
Published: July 2016

The paper proposes a hidden semi-Markov model for breakpoint rainfall data that consist of both the times at which rain-rate changes and the steady rates between such changes. The model builds on and extends the seminal work of Ferguson (1980) on variable duration models for speech. For the rainfall data the observations are modelled as mixtures of log-normal distributions within unobserved states where the states evolve in time according to a semi-Markov process. For the latter, parametric forms need to be specified for the state transition probabilities and dwell-time distributions.

Recursions for constructing the likelihood are developed and the EM algorithm used to fit the parameters of the model. The choice of dwell-time distribution is discussed with a mixture of distributions over disjoint domains providing a flexible alternative. The methods are also extended to deal with censored data. An application of the model to a large-scale bivariate dataset of breakpoint rainfall measurements at Wellington, New Zealand, is discussed.
Automatic methods for generating seismic intensity maps
Authors: Brillinger David R. , Chiann Chang , Irizarry Rafael A. , Morettin Pedro A.
Published: July 2016

For many years the modified Mercalli (MM) scale has been used to describe earthquake damage and effects observed at scattered locations. In the next stage of an analysis involving MM data, isoseismal lines based on the observations have been added to maps by hand, i.e. subjectively. However a few objective methods have been proposed (by e.g. De Rubeis et al., Brillinger, Wald et al. and Pettenati et al.). The work presented here develops objective methods further. In particular the ordinal character of the MM scale is specifically taken into account. Numerical smoothing is basic to the approach and methods involving splines, local polynomial regression and wavelets are illustrated. The approach also allows the inclusion of explanatory variables, for example site effects. The procedure is implemented for data from the 17 October 1989 Loma Prieta earthquake.
Relative quiescence within the Jiashi swarm in Xinjiang, China: an application of the ETAS point process model
Authors: Ma Li , Zhuang Jiancang
Published: July 2016

This paper uses the epidemic-type aftershock sequence (ETAS) point process model to study certain seismicity features of the Jiashi swarm of certain earthquakes, investigating in particular whether there is relative quiescence prior to the quite big events within the Jiashi sequence. The seven earthquake sequences studied occurred in the region of Jiashi, south of Tianshan Mountain, Xinjiang, China. The particular ETAS model that is developed is consistent with the reality of seismic activity. The various features of Jiashi swarm activity can be described as focusing in different stages. There is obvious precursory quiescence prior to most big events with Ms
≥ 6.0 within the Jiashi swarm. Thus, checking for relative quiescence can be use for earthquake prediction.
An evaluation of Chinese annual earthquake predictions, 1990–1998
Authors: Shi Yaolin , Liu Jie , Zhang Guomin
Published: July 2016

The annual earthquake predictions of the China Seismological Bureau (CSB) are evaluated by means of an R score (an R score is approximately 0 for completely random guesses, and approximately 1 for completely successful predictions). The average R score of the annual predictions in China in the period 1990–1998 is about 0.184, significantly larger than 0.0. However, background seismicity is higher in seismically active regions. If a ‘random guess' prediction is chosen to be proportional to the background seismicity, the expected R score is 0.123, and the nine-year mean R score of 0.184 as observed is only marginally higher than this background value. Monte Carlo tests indicate that the probability of attaining an R score of actual prediction by background seismicity based on random guess is about . It is concluded that earthquake prediction in China is still in a very preliminary stage, barely above a pure chance level.
A dialysis system with one absorbing and one semi-reflecting state
Authors: Kastenbaum Marvin A.
Published: July 2016

The random walk between an absorbing and a reflecting or semi-reflecting barrier has been discussed in the literature. Indeed, a biological example of such a random walk, arising from a system of countercurrent dialysis, was described by this author. In this earlier paper a technique was suggested for obtaining the probability of being in a specific state after a given number of transitions, if the probability is unity of being in the reflecting state at time zero. The present paper extends this technique to yield the full matrix of higher transition probabilities.
Identities for passage times with applications to recurrent events and homogeneous differential functions
Authors: Darroch John N.
Published: July 2016

Let x be a non-negative, non-decreasing, right-continuous function of t defined on the interval [O, T) and let A = x(T–). Define the function y on [O, A] by

Then y is non-decreasing and left-continuous and

If t is a time variable it is natural to call y the passage time function.
Optimal procedures for stochastically failing equipment
Authors: Folkman Jon , Port Sidney
Published: July 2016

When a complex system such as the Apollo vehicle arrives on the Moon, certain parts may no longer function. To restore the system to a working state, these parts must be replaced by new ones; but the vehicle can carry only a limited number of spare parts and must also be able to leave at a specified time after arrival. This paper investigates an abstract (and simplified) version of the problem of the optimal maintenance procedure for such a system. As a bonus of the abstract formulation, the results obtained are applicable to a variety of specific systems.
A continuous time inventory model
Authors: Bather J. A.
Published: July 2016

This paper discusses an optimization problem arising in the theory of inventory control. Much of the previous work in this field has been focused on the Arrow-Harris-Marschak model, [1], [2], in which the inventory level can be modified only at the instants of discrete time. Here, we shall be concerned with a continuous time analogue of the model, in an attempt to avoid the difficulties experienced in solving the basic integral equations. The approach was suggested by recent investigations of a statistical decision problem, [3], [5], which exploited the advantages of a continuous treatment. Although the ideas discussed here are relatively straightforward and involve strong assumptions as to the behavior of the inventory, the explicit character of the optimal policy is encouraging and particular solutions might nevertheless provide useful restocking procedures.
Storage with deterministic outputs and inputs subject to breakdown
Authors: Wu P. M.
Published: July 2016

In this paper, we shall consider a storage model in which a bin of finite capacity is fed by specified r(r > 0) or zero input depending on the content of the bin and the operation of the input machines (i.e., whether or not they are working on the bin). These machines are allowed to fill the bin until its level reaches maximum and they are then used to produce some secondary products not relevant to the storage problem. When the content of the bin decreases to a given minimum level, these machines are again set to feed the bin. The whole process is then repeated. It is often desirable in practice to keep the minimum level as low as possible, to attain stability in the process, or for maintenance purposes. When the machines are subject to breakdowns, it becomes necessary to consider the reliability of the operational system.
Discrete queues with one server
Authors: Takács Lajos
Published: July 2016

In this paper mathematical methods are given for finding the distributions of the virtual waiting time and the occupation time for single-server queues in which the interarrival times and the service times are discrete random variables taking on positive or non-negative integral values. First we shall consider deterministic queues in which the interarrival times and the service times are numerical (non-random) quantities and deduce a fundamental relation for such queues. Then we shall consider random queues with recurrent input and general service times and study the time dependent behavior of such queues.
On a three-state sojourn time problem
Authors: Gibson A. E. , Conolly B. W.
Published: July 2016

Consider the real-valued stochastic process {S(t), 0 ≦ t < ∞} which assumes values in an arbitrary space X. For a given subset T ⊂ X we define

which represents the length in time of a visit to state T. We shall restrict ourselves to processes such that τT
is a random variable having a differentiable distribution function which is independent of the time t
0 at which the visit to state T begins.
Selective interaction of a Poisson and renewal process: the spectrum of the intervals between responses
Authors: Lawrance A. J.
Published: July 2016

This paper is concerned with the spectrum of the intervals between responses in the renewal inhibited Poisson process, and continues the author's earlier work on this type of process ((1970a), (1970b), (1971)). A generating function for all the pairwise joint distributions of the synchronous intervals following an average response is obtained and leads directly to the associated serial correlations. It is shown that these correlations are equivalent to those predicted on different assumptions by the general stationary point theory. The results are then used to obtain the interval spectrum, and to exhibit a relationship between the sum of the serial correlations and the variance-time function. Explicit results for the spectrum of the renewal inhibited Poisson process are given for gamma inhibitory distributions, and the qualitative behavior is determined. Possible further developments are briefly discussed.
Non-linear time series regression
Authors: Hannan E. J.
Published: July 2016

In Jennrich (1969) the modelis considered, wherex(n) is a sequence of i.i.d. (0,σ2) random variables andz(n;θ) is a continuous but possibly non-linear function ofθ∈Θ, Θ being a compact set inRp. We shall use a second subscript when referring to a particular coordinate ofθ0so thatθ0jis thejth coordinate. Jennrich establishes, under suitable conditions onz(n;θ) andx(n), the strong consistency and asymptotic normality of the least squares estimates ofθ.Our main purpose here is to extend these results to the case wherex(n) is generated by a stationary time series.