A relationship between a family of waiting time distributions and the asymptotic residual waiting time distribution

For a class of renewal process waiting time distributions defined herein, one may describe the distribution of asymptotic residual waiting times. The relationship between the two distributions characterizes the class, which includes the gamma distribution. Possible consequences for hypothesis testing are discussed.     

Some results on residual life and inactivity time at random time

In this article, we enhance the study of residual life at random time (RLRT) and inactivity time at random time (ITRT). To this aim, first we provide some stochastic orderings results among ITRT in two-sample problems when they failed at two different random times. Then, we develop some sufficient conditions which lead to the stochastic comparisons of RLRT and ITRT based on variance residual life order. The results are expected to be useful in reliability theory, forensic science, queue theory, and actuarial science.     

Maximizing survival time in a random walk on an interval

A gambler buys N tokens that enable him to play N rounds of the following game. A symmetric random walk on a discrete interval { ? r, …, r} starts from the point 0. The gambler knows only the number of steps made so far, but is unaware of the current position of the walk. Once the walk hits one of the barriers ? r or r for the first time in the current round, the round ends with no payoff. The gambler can start a new round by inserting a new token, if there are any tokens left. The gambler can end the game at any time getting the payoff equal to the number of steps made in the current round. We find the optimal stopping strategy for this game and calculate the expected payoff once the optimal strategy is applied.     

The existence of waiting time moments for single server queues with mixed renewal and poisson inputs

The following queuing system is considered: Two independent recurrent input streams (streams 1 and 2) arrive at a server. It is assumed that stream 1 is of Poisson type. Three priority disciplines are studied in case that customers of type 1 have priority: head-of-the-line, preemptive-resume, and preemptive-repeat discipline. For all three cases, the limiting distribution function of actual waiting times of low-priority customers is considered, and conditions are given for the existence of moments related to these limiting distributions.     

Waiting time paradox and size biased sampling

Feller's (1971) 'Waiting time for the bus paradox1 is explained by the fact that the long Interarrival times have a better chance of covering an arbitrary point in time than do short interarrival times, This gives rise to size biased sampling and the length of the Interarrival time that contains the arbitrary point has a moment distribution, This paper deals with the characterization of the exponential distribution based on this 'Waiting time paradox'. Similar characterizations of the binomial, the negative binomial, the Poisson and the geometric distributions are obtained.     

Waiting time distributions for a run with additional constraints

This paper proposes a method for obtaining the exact probability of occurrence of the first success run of specified length with the additional constraint that at every trial until the occurrence of the first success run the number of successes up to the trial exceeds that of failures. For the sake of the additional constraint, the problem cannot be solved by the usual method of conditional probability generating functions. An idea of a kind of truncation is introduced and studied in order to solve the problem. Concrete methods for obtaining the probability in the cases of Bernoulli trials and time-homogeneous {0,1}$\{0,1\}$-valued Markov dependent trials are given. As an application of the results, a modification of the start-up demonstration test is studied. Numerical examples which illustrate the feasibility of the results are also given.     

Discrete time cold standby repairable system: Combinatorial analysis

ABSTRACT

In this article, we obtain exact expression for the distribution of the time to failure of discrete time cold standby repairable system under the classical assumptions that both working time and repair time of components are geometric. Our method is based on alternative representation of lifetime as a waiting time random variable on a binary sequence, and combinatorial arguments. Such an exact expression for the time to failure distribution is new in the literature. Furthermore, we obtain the probability generating function and the first two moments of the lifetime random variable.     

Preservation of generalized ageing classes on the residual life at random time

In this paper, we provide some new preservation properties of generalized ageing classes (s-IFR) on the residual life at random time, where s is a nonnegative integer. We also obtain bounds of the residual life at exponential random time. Results are expected to be useful in the reliability, queue theory and actuarial science.     

Simple diagnostic tools for inverstigating linear trends in time series

This paper presents a simple diagnostic tool for time series. Based on a coefficient α that veries between 1 and 0, the tool measures the approximation of a time series to an arithmetic progression (i.e., a linear function of time). The proposed α is based on the ratio of the average squared second difference to the average squared first difference of the ginven series. As such, α reduces to the Von Neumann ratio η of the series of first differences, namely, α = 1-η/4. For an arithmetic progression α = 1, and deviations therefrom cause it to decrease. Unlike the correlation coefficient (between the entries and the indics), α is sensitive to local, or piecewise, linearity. Here α is evaluated for an assortment of simple time series models such as random walk, AR(1) and MA(1). Large-sample distribution yields a number of commonly used stochastic models including non-normal process. For most standard deterministic and stochastic models, α stabilizes as n approaches infinity, and provides a statistic that is capable of distinguishing between many different standard random and deterministic models. A further measure τ, which together with α distinguisches between random walks and deterministic trend plus i.i.d., is also suggested. Some examples based on empirical data are also studied.     

Approximating probabilities of first passage in a particular gaussian process

Shepp (1971) derives the distribution of waiting times of first passage for a particular Gaussian process. However, Shepp notes that for moderate to large waiting tines the expressions for the probability cannot be evaluated either numerically or by asymptotic estimation. Me present a useful approximation for the distribution and expected waiting time for the conditional and unconditional versions of this first passage problem. The probabilities play a role in bounds by Adler (1984) for the probability distribution of the supremum of a particular two-parameter Gaussian field, a detection problem (Lai, 197 3) and the study of signal shape problems in radars (Zakai Ziv, 1969).     

Promotion Time Cure Rate Model with Bivariate Random Effects

In this article, we consider the inclusion of random effects in both the survival function for at-risk subjects and the cure probability assuming a bivariate normal distribution for those effects in each cluster. For parameter estimation, we implemented the restricted maximum likelihood (REML) approach. We consider Weibull and Piecewise Exponential distributions to model the survival function for non-cured individuals. Simulation studies are performed, and based on a real database we evaluate the performance of our proposed model. Effect of different follow-up times and the effect of considering independent random effects instead of bivariate random effects are also studied.     

A statistic to check model adequacy in time series

A simple statistic is suggested to examine if the assumptions on variances in a fitted time series model is valid or not. The properties of the statistic are discussed and examples are considered.     

A note on the passage time of finite-state Markov chains

Consider a Markov chain with finite state {0, 1, …, d}. We give the generation functions (or Laplace transforms) of absorbing (passage) time in the following two situations: (1) the absorbing time of state d when the chain starts from any state i and absorbing at state d; (2) the passage time of any state i when the chain starts from the stationary distribution supposed the chain is time reversible and ergodic. Example shows that it is more convenient compared with the existing methods, especially we can calculate the expectation of the absorbing time directly.     

Book review

G.Bamberg:Statistische Entseheldungstheorle, Physica-Verlag, Würzburg-Wien 1972, 149 S., DM 20,–.

J.M.Mendel:Discrete Techniques of Parameter Estimation. Marcel Dekker, Inc., New York 1973, XIV, 385 S., $ 19.50.

P.Eykhoff:System Identification. Parameter and State Estimation. John Wiley & Sons, London 1974, XX, 555 pp., £ 12.50.

G.Härtler:Versuchsplanung und Statistische Datenanalyse, Akademie-Verlag, Berlin 1976, 103 S., 6 Abb., 12,– M.

P.Mertens(Hrsg.):Prognoserechnung. Physica-Verlag, Würzburg-Wien 1973; 295 S., DM 48,–

E.Schaich, D. Kohle, W.Schweitzer, F.Wegner:Statistik I (für Volkswirte, Betriebswirte und Soziologen.) Verlag Franz Vahlen GrnbH, München 1974, 160 S., DM 22,80.

A.Rosengard: Probabilltés et Statistique en Recherche Scientifique. Dunod, Paris 1972, XIV, 310 S.

D. Zschocke:Betriebsökonometrie.Physica-Verlag, Würzburg-Wien 1974, 287 S., DM 90.–.

J.C.G. Boot, E.B. Cox:Statistical Analysis for Managerial Decisions. McGraw-Hill, Inc., New York 1970, XI, 651 S.

H.Richter, V. Mammitzsch: Methode der kleinsten Quallrate (mit Übungen und Aufgaben).Verlag Berliner Union GmbH, Stuttgart, und Verlag W. Kohlharnmer GmbH, Stuttgart-Berlin-Köln-Mainz 1973, 144 S., DM 24,–.

O.Anderson, W. Popp, M. Schaffranek, D. Steinmetz, H. Stenger: Schätzen und Testen. Eine Einfhrung in die Wahrscheinlichkeitsrechnung und schließeude Statistik.Springer-Verlag, Berlin-Heidelberg-New York 1976, XI, 385 S., 68 Abb., 56 Tab., DM 19,80.

H.Bauer:Wahrscheinlichkeitstheorie und Grundzilge der Maßtheorie. Walter de Gruyter, Berlin-New York 1974, 407 S., DM 48,–.

A.M.Mathai, P.N. Rathie:Basic Concepts in Information and Statistics(Axioluatic Foundations and Applications). Wiley Eastern Limited, New Delhi 1975, X, 137 S., £ 2.75; $ 5.50.

D.F.Merriam (Hrsg.): Random Processes in Geology.Springer-Verlag, Berlin-Heidelberg-New York 1976, VIII, 168 S., 63 Abb., 18 Tab., DM 34,50.

G.Matheron: Random Sets and Integral Geometry. Wiley Series in Probability and Mathematical Statistics, J. Wiley and Sons, New York 1975, XXIII, 261 S., £ 10.60; $ 21.20.

L.Kleinrock: Queueing Systems. Volume I: Theory. J. Wiley and Sons, New York-London-Sydney-Toronto 1975, XVIII, 417 S.; £ 13.55; $ 24.30.

Bartsch, H.-J.:Mathematische Formeln, VEB Fachbuchverlag, Leipzig 1976,15. Auflage, 510 S., 362 Abb., 9,80 M.     

Generating random numbers of prescribed distribution using physical sources

When constructing uniform random numbers in [0, 1] from the output of a physical device, usually n independent and unbiased bits B _j are extracted and combined into the machine number . In order to reduce the number of data used to build one real number, we observe that for independent and exponentially distributed random variables X _n (which arise for example as waiting times between two consecutive impulses of a Geiger counter) the variable U _n : = X _{2n – 1}/(X _{2n – 1} + X _2n ) is uniform in [0, 1]. In the practical application X _n can only be measured up to a given precision (in terms of the expectation of the X _n ); it is shown that the distribution function obtained by calculating U _n from these measurements differs from the uniform by less than /2.We compare this deviation with the error resulting from the use of biased bits B _j with P {B _j = 1{ = (where ] – [) in the construction of Y above. The influence of a bias is given by the estimate that in the p-total variation norm Q ^TV _p = ( |Q()| ^p )^1/p (p 1) we have P _Y – P ⁰ _Y ^TV _p (c _n · )^1/p with c _n p for n . For the distribution function F _Y – F ⁰ _Y 2(1 – 2^–n )|| holds.     

Retrospective change detection in categorical time series

The aim of this paper is to propose methods of detecting change in the coefficients of a multinomial logistic regression model for categorical time series offline. The alternatives to the null hypothesis of stationarity can be either the hypothesis that it is not true, or that there is a temporary change in the sequence. We use the efficient score vector of the partial likelihood function. This has several advantages. First, the alternative value of the parameter does not have to be estimated; hence, we have a procedure that has a simple structure with only one parameter estimation using all available observations. This is in contrast with the generalized likelihood ratio-based change point tests. The efficient score vector is used in various ways. As a vector, its components correspond to the different components of the multinomial logistic regression model’s parameter vector. Using its quadratic form a test can be defined, where the presence of a change in any or all parameters is tested for. If there are too many parameters one can test for any subset while treating the rest as nuisance parameters. Our motivating example is a DNA sequence of four categories, and our test result shows that in the published data the distribution of the four categories is not stationary.     

Wavelet estimation in time-varying coefficient time series models with measurement errors

The article studies a time-varying coefficient time series model in which some of the covariates are measured with additive errors. In order to overcome the bias of estimator of the coefficient functions when measurement errors are ignored, we propose a modified least squares estimator based on wavelet procedures. The advantage of the wavelet method is to avoid the restrictive smoothness requirement for varying-coefficient functions of the traditional smoothing approaches, such as kernel and local polynomial methods. The asymptotic properties of the proposed wavelet estimators are established under the α-mixing conditions and without specifying the error distribution. These results can be used to make asymptotically valid statistical inference.