Showing papers in "Journal of Applied Probability in 1975"

Soft Modelling by Latent Variables: The Non-Linear Iterative Partial Least Squares (NIPALS) Approach

Abstract: The NIPALS approach is applied to the ‘soft’ type of model that has come to the fore in sociology and other social sciences in the last five or ten years, namely path models that involve latent variables which serve as proxies for blocks of directly observed variables. Such models are seen as hybrids of the ‘hard’ models of econometrics where all variables are directly observed (path models in the form of simultaneous equations systems) and the ‘soft’ models of psychology where the human mind is described in terms of latent variables and their directly observed indicators. For hybrid models that involve one or two latent variables the NIPALS approach has been developed in [38], [41] and [42]. The present paper extends the NIPALS approach to path models with three or more latent variables. Each new latent variable brings a rapid increase in the pluralism of possible model designs, and new problems arise in the parameter estimation of the models. Iterative procedures are given for the point estimation of the parameters. With a view to cases when the iterative estimation does not converge, a device of range estimation is developed, where high profile versus low profile estimates give ranges for the parameter estimates.

Time-reversibility of linear stochastic processes

Abstract: Time-reversibility is defined for a process X(t) as the property that {X(t), - - -, X(t.)} and {X(- t), - -, X(- t.)} have the same joint probability distribution. It is shown that, for discrete mixed autoregressive moving-average processes, this is a unique property of Gaussian processes. TIME-REVERSIBILITY; SHOT NOISE; CHARACTERISATIONS OF THE NORMAL DISTRIBUTION; TIME SERIES; STOCHASTIC PROCESSES

Longest common subsequences of two random sequences.

Abstract: Given two random k-ary sequences of length n, what is f(n,k), the expected length of their longest common subsequence? This problem arises in the study of molecular evolution. We calculate f(n,k) for all k, where n $\leq$ 5 , and f(n,2) where n $\leq$ 10. We study the limiting behavior of $n^{-1}$f(n,k) and derive upper and lower bounds on these limits for all k. Finally we estimate by Monte-Carlo methods f(100,k), f(1000,2) and f(5000,2).

On the tails of waiting-time distributions

Abstract: Results are given which relate the tail behaviour of the service and limiting waiting time distributions of a GI/G/1 queue. A limit theorem for the maxima of waiting times is given.

Networks of queues with customers of different types

Abstract: The behaviour in equilibrium of networks of queues in which customers may be of different types is studied. The type of a customer is allowed to influence his choice of path through the network and, under certain conditions, his service time distribution at each queue. The model assumed will usually cause each service time distribution to be of a form related to the negative exponential distribution. Theorems 1 and 2 establish the equilibrium distribution for the basic model in the closed and open cases; in the open case the individual queues are independent in equilibrium. In Section 4 similar results are obtained for other models, models which include processes better described as networks of colonies or as networks of stacks. In Section 5 the effect of time reversal upon certain processes is used to obtain further information about the equilibrium behaviour of those processes.

Determination of the order of a Markov chain by Akaike's information criterion

Abstract: Using Akaike's information criterion, we have presented an objective procedure for the determination of the order of an ergodic Markov chain with a finite number of states. The procedure exploits the asymptotic properties of the maximum likelihood ratio statistics and Kullback and Leibler's mean information for the discrimination between two distributions. Numerical illustrations are given, using data from Bartlett (1966), Good and Gover (1967) and some weather records.

The Galton-Watson process conditioned on the total progeny

Abstract: Let Zk denote the number in the kth generation of a Galton-Watson process initiated by one individual and let N be the total progeny, i.e., As n → ∞ the limiting behaviour of the process {Zk, 0 ≦ k ≦ n} conditioned on the event {N =n} is studied. The results obtained are of exactly the same form for the subcritical, critical and supercritical cases. This is in marked contrast to the analogous situation got by conditioning on non-extinction by the nth generation and letting n → ∞. In the latter case the limiting results differ in form for the critical and non-critical cases.

Shock Models with Underlying Birth Process

Abstract: : The paper extends results of Esary, Marshall, and Proschan (1973), Ann. of Probability, and Abdel-Hameed and Proschan (1972), FSU Statistics Report M243. The authors consider the life distribution of a device subject to a sequence of shocks occurring randomly in time according to a nonstationary pure birth process; given that k shocks have occurred in (0,t), the probability of a shock occurring in (t,t+delta) is (lambda sub k) lambda (t)delta + o(delta). The authors show that various fundamental classes of life distributions (such as those with increasing failure rate, or those with the 'new better than used' property, etc.) are obtained under appropriate assumptions on (lambda sub k), lambda (t), and on the probability of surviving a given number of shocks. (Author)

A supplement to the strong law of large numbers

Abstract: The strong law of large numbers for independent and identically distributed random variables Xi, i = 1,2,3, …, with finite mean µ can be stated as, for any ∊ > 0, the number of integers n such that |n −1 Σ i=1 n X i − μ| > ∊, N (∊), is finite a.s. It is known, furthermore, that EN (∊) < ∞ if and only if EX 1 2 < ∞. Here it is shown that if EX 1 2 < ∞ then ∊ 2 EN (∊) → var X 1 as ∊ → 0.

The markovian self-exciting process

Abstract: The self-exciting point process with exponential exciting function is investigated using the immigration birth representation of the process. The counting distribution is derived explicitly and some simpler interval properties are given. In particular, it is shown that the serial covariances of the interval sequence decrease monotonically to zero.

On the extinction times of varying and random environment branching processes

Abstract: Bounds are derived for the probability of extinction by the nth generation for a branching process in a varying environment. From these bounds, necessary and sufficient conditions are established for such a process to become extinct with probability one. The extinction time of a random environment branching process in which the environmental random variables are independent but not necessarily identically distributed is stochastically bounded by the extinction times of two varying environment processes.

Estimation of a Distribution Function from Incomplete Observations

Abstract: The product-limit estimator for a distribution function, appropriate to observations which are variably censored, was introduced by Kaplan and Meier in 1958; it has provided a basis for study of more complex problems by Cox and by others. Its properties in the case of random censoring have been studied by Efron and later writers. The basic properties of the product-limit estimator are here shown to be closely parallel to the properties of the empirical distribution function in the general case of variably and arbitrarily censored observations.

Asymptotic likelihood theory for diffusion processes

Abstract: We investigate the large-sample behaviour of maximum likelihood estimates (MLE's) of the parameters of a diffusion process, which is observed throughout continuous time. The results (limit normal distribution for the MLE and an asymptotic chi-squared likelihood ratio test) correspond exactly to classical asymptotic likelihood results, and follow easily from a central limit theorem for stochastic integrals. DIFFUSION PROCESSES; MAXIMUM LIKELIHOOD ESTIMATES; BROWNIAN MOTION; STATIONARITY; ERGODICITY; STOCHASTIC INTEGRALS; RADON-NIKODYM DERIVATIVES; MARTINGALE CENTRAL LIMIT THEOREM; LIKELIHOOD RATIO TEST

Prediction Intervals and Empirical Bayes Confidence Intervals

Abstract: Approximate parametric prediction intervals are obtained for an unobserved random variable when the amount of data on which to base the estimation is large. Applications include the construction of approximate confidence intervals in empirical Bayes estimation.

On the distribution of the inter-record times in an increasing population

Abstract: It is shown in this note that if the population increases geometrically, then the asymptotic distribution for the inter-record times is also geometric. The records in Olympic games are used as an example. Also, it is noted that the rapid breaking of Olympic records is not due mainly to the increase in population.

A secretary problem with uncertain employment

Abstract: A ‘Secretary Problem’ with no recall but which allows the applicant to refuse an offer of employment with a fixed probability 1 – p, (0 < p < 1), is considered. The optimal stopping rule and the maximum probability of employing the best applicant are derived.

Some mean first-passage time approximations for the ornstein-uhlenbeck process

Abstract: This paper describes an accurate method of approximating the mean of the first-passage time distribution for an Ornstein-Uhlenbeck process with a single absorbing barrier. The accuracy of the approximation is demonstrated through some numerical comparisons. ORNSTEIN-UHLENBECK PROCESS; DIFFUSION APPROXIMATIONS; FIRST-PASSAGE TIMES

Limits of Compound and Thinned Point Processes.

Abstract: Let r= Y _ be a point process on some space S and let B, l1, B2, ... be identically distributed non-negative rando variables which are mutually independent and independent of r. We can then form the compound point process = flj5 which is a random measure on S. The purpose of this paper is to study the limiting behaviour of ? as B -+ 0. In the particular case when B takes the values 1 and 0 with probabilities p and i-p respectively, becomes a p-thinning of 1 and our theorems contain some classical results by R6nyi and others on the thinnings of a fixed process, as well as a characterization by Mecke of the class of subordinated Poisson processes. COMPOUND AND THINNED POINT PROCESSES; INFINITELY DIVISIBLE RANDOM MEASURES; SUBORDINATED POISSON PROCESSES; CONVERGENCE IN DISTRIBUTION; REGULARITY AND DIFFUSENESS

A branching process with disasters

Abstract: A population process is considered where particles reproduce according to an age-dependent branching process, and are subjected to disasters which occur at the epochs of an independent renewal process. Each particle alive at the time of a disaster, survives it with probability p and the survival of any particle is assumed independent of the survival of any other particle. The asymptotic behavior of the mean of the process is determined and as a consequence, necessary and sufficient conditions are given for extinction.

An extension of watanabe's theorem of characterization of poisson processes over the positive real half line

Abstract: We give an elementary proof of the martingale characterization theorem for Poisson processes over the positive real half line. This theorem is due to Watanabe [8] in the case where the mean measure associated to the Poisson process is the Lebesgue measure. POISSON PROCESS; MARTINGALES

Uniform coupling of non-homogeneous Markov chains

Abstract: The Markov-Dobrushin condition for (weak) ergodicity of non-homogeneous discrete-time Markov chains, and an analogous criterion for continuous chains, are derived by means of coupling techniques. WEAK ERGODICITY; ERGODIC COEFFICIENT; NON-HOMOGENEOUS MARKOV CHAINS; COUPLING 0. Introduction A non-homogeneous Markov chain (6,) on a countable state space S will be called S-uniformly ergodic if lim sup Z IP(, = k i 4= i)P(?, = k i = j)i = 0 ti,jES kES for every to. This property was first studied by Markov [13] in the finite homogeneous case, and by Dobrushin [1] in the general case; a sufficient condition was given in terms of the ergodic. coefficient a (P) of a stochastic matrix. P = (pij),js, where (1) a (P): = inf [pik A Pjk i,j k (:= designates a defining notational equality, aAb = min {a, b}). Many authors (e.g. [7], [11], [14], [16]) have used a and similar coefficients to study uniform ergodicity. The usual approach is to derive the key inequality (2) 1 a(PO) -5 [1 a (P)] [1 a(Q)] for any stochastic matrices P, Q from Banach space considerations; this was the method of [1]. The probabilistic technique known as coupling, which dates back to Doeblin [3] in essence, has enjoyed a recent revival. Vasershtein [17], Dobrushin [2], Harris [8], Holley and Liggett [9], and others, have used coupling to prove ergodic theorems for Markovian lattice interactions. In [5] and [6] the author has applied the same strategy to study the tail or-algebra of homogeneous and Received in revised form 24 March 1975. 753 This content downloaded from 157.55.39.4 on Thu, 08 Sep 2016 05:14:15 UTC All use subject to http://about.jstor.org/terms

The G/M/m queue with finite waiting room

Abstract: The G/M/m queue with only s waiting places is studied. We start by studying the joint distribution of the number of customers present at time t and the time elapsing until the next arrival after t. This gives the asymptotic distribution of the number of customers at the moment of an arrival and at an arbitrary moment. Then waiting time and virtual waiting time distributions are easily obtained. For the G/M/1 queue also the transient solution is given. Finally the case s = ∞ is considered.

Maintainability of structures in Markov chain models under recruitment control

Abstract: We consider a fixed size Markov chain model which suffers losses and admits controlled recruitment. The family of n-step maintainable structures is described geometrically and examined.

Multivariate exponential distributions based on hierarchical successive damage

Abstract: A class of multivariate exponential distributions is suggested which includes those presented by Marshall and Olkin, Downton, and Hawkes. It is based on a concept of hierarchical successive damage. Recursive expressions for the Laplace transforms and for first and second moments are derived in the bivariate case. Related multivariate geometric distributions are described.

Diffusion Models with Relativity Effects

Abstract: In this paper models for diffusion in one dimension are obtained which are based on correlated random walks. The equations for diffusion with drift can be transformed into the equations for diffusion without drift (and conversely) by the transformations of Special Relativity Theory. The relationship of these equations to Maxwell's equations for electromagnetic phenomena is discussed.

The candidate problem with unknown population size

Abstract: The familiar problem of maximizing the probability of choosing the best from a group of N candidates, where N is known, is extended to the case of N unknown. An a priori distribution is assumed for N, and the case of a uniform distribution is examined. Let VN denote the probability of choosing the best from a group of at most N candidates, then it is shown that lim VN = 2e-2