# Shelemyahu ZacksBinghamton University | SUNY Binghamton · Department of Mathematical Sciences

Shelemyahu Zacks

Ph.D., Columbia University, 1962

## About

306

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Introduction

Additional affiliations

January 1993 - December 1999

January 1985 - December 2011

March 1983 - August 1983

## Publications

Publications (306)

Goodness-of-fit is used for the evaluation a model. They are commonly used to compare among competing models. The material is mostly classic. For more on the subject the reader is referred to the References including the two revised volumes Bickel and Docksum (2016).

There are two types of software reliability model: time domain models and data domain models. Time domain models provide survival functions or hazard functions of time, which depend on parameters that have to be estimated. Data domain models are not time‐dependent models, but sampling models from finite populations that provide estimates of the num...

We investigate the one-dimensional telegraph random process in the presence of an elastic boundary at the origin. This process describes a finite-velocity random motion that alternates between two possible directions of motion (positive or negative). When the particle hits the origin, it is either absorbed, with probability α, or reflected upwards,...

This monograph is focused on the derivations of exact distributions of first boundary crossing times of Poisson processes, compound Poisson processes, and more general renewal processes. The content is limited to the distributions of first boundary crossing times and their applications to various stochastic models. This book provides the theory and...

A growth-collapse process is one which grows linearly between random partial collapse times. The jump down of the process at a collapse time has a random size, following some distribution which is conditional on the level of the process at that time. There are many application of such models in geophysics, population growth, insurance models, inven...

In this chapter we discuss distributions of first crossing linear boundaries. First crossing of concave boundaries by Poisson processes is studied in Chapter 5.

A rendezvous time is a time at which two different stochastic processes intersect (meet). In this section we discuss the first such rendezvous time of a Brownian motion and an independent compound Poisson process. The reader is referred to Perry et al. (2004) and Che and Dassios (2013).

Poisson processes are Markov jump processes, having jumps of equal (deterministic) size. Without loss of generality, we assume that the jump size is d = 1. We give here a constructive definition of the Poisson process, based on its properties. For the postulates on which it is derived, see Kao (1997, ch. 2) or Resnick (2005, ch. 4).

In this chapter we discuss reliability models for systems under stress, which are deteriorating.

Telegraph processes are ON and OFF processes, which change intermittently, according to alternating renewal processes. In physics these processes describe the movement of a particle on a line. The following is a simple example from physics.

In this chapter we illustrate the use of the methodology of sample path analysis for deriving the exact distributions of sampling size (stopping times) in two-stage and sequential estimation of the parameters of distributions, with precision requirements, like fixed-width confidence intervals or bounded risk estimators. We show here two cases, the...

In this chapter we develop the distribution function of the first crossing times of compound Poisson processes, with different types of linear boundaries. The results have applications in inventory theory, in queuing theory, in insurance, reliability and more. Examples of applications will be given in the appropriate sections. One of the first pape...

The paper is focused on the problem of estimating the probability $p$ of individual contaminated sample, under group testing. The precision of the estimator is given by the probability of proportional closeness, a concept defined in the Introduction. Two-stage and sequential sampling procedures are characterized. An adaptive procedure is examined.

In a normal distribution with its mean unknown, we have developed Stein-type (1945,1949) two-stage and Chow and Robbins-type (1965) purely sequential strategies to estimate the unknown variance under a modified Linex loss function. We control the associated risk function per unit cost by bounding it from above with a fixed preassigned positive numb...

Statistical data analysis includes several phases. First, there is the phase of data collection. Second, there is the phase of analysis and inference. The two phases are interconnected. There are two types of data analysis. One type is called parametric and the other type is nonparametric. In the present paper, we discuss parametric inference. In p...

Much was written on generalized linear models. The reader is referred to the following books: P. McCullagh and J. A. Nelder, Generalized Linear Models, 2nd edn. (Chapman & Hall, 1989); A. J. Dobson and A. Barnett, An Introduction to Generalized Linear Models, 3 (Chapman & Hall, 2008); C. E. McCulloch and S. R. Searle, Generalized Linear and Mixed M...

In this article, we propose two-stage and purely sequential procedures to construct bounded width and prescribed proportional closeness confidence intervals for the unknown parameter N of B(N,p) distribution where the parameter p is assumed to be known. The exact distributions of the stopping variables and the estimators of N at stopping are derive...

Professor Nikiforov gave an excellent introduction to the modern theory and applications of detection/isolation. We discuss possible consequences of the “ultra-minimax” (working against the worst-worst-worst case) approach as well as some extensions and modifications of the problem.

A compound Poisson process whose randomized time is an independent Poisson process is called a compound Poisson process with Poisson subordinator. We provide its probability distribution, which is expressed in terms of the Bell polynomials, and investigate in detail both the special cases in which the compound Poisson process has exponential jumps...

The present article reviews several published papers in which the distributions of stopping variables, the expected values, risk functions, and coverage probabilities of estimators at stopping were derived analytically for two-stage Stein-like procedures. The reviewed papers deal with fixed-width and bounded risk estimation of the location and scal...

Alternating renewal processes have been widely used to model social and scientific phenomenal where independent “on” and “off” states alternate. In this paper, we study a model where the value of a process cumulates and declines according to two modes of compound Poisson processes with respect to an underlying alternating renewal process. The model...

A compound Poisson process whose randomized time is an independent Poisson process is called compound Poisson process with Poisson subordinator. We give its probability distribution, which is expressed in terms of the Bell polynomials, and investigate in detail both the special cases in which the compound Poisson process has exponential jumps and n...

Truncated sequential test procedures are proposed for testing the mean time between failures of a system with exponential life distribution. The exact distributions of the stopping times for the sequential tests are derived by investigating boundary crossing times of homogeneous Poisson processes. The exact formulas for the expected values of the s...

A compound Poisson process whose randomized time is an independent Poisson process is called compound Poisson process with Poisson subordinator. We provide its probability distribution, which is expressed in terms of the Bell polynomials, and investigate in detail both the special cases in which the compound Poisson process has exponential jumps an...

The article presents stochastic processes, which may lead to catastrophic events, like ruin of an insurance company or failure of systems due to cumulative damage. We focus attention on compound renewal processes, and in particular on compound Poisson processes. Distribution of stopping times associated with these risk processes are derived.

We consider a stochastic fluid EOQ-type model with demand rates operating in a two-state random environment. This environment alternates between exponentially distributed periods of high demand and generally distributed periods of low demand. The inventory level starts at some level q, and decreases linearly at rate β H during the periods of high d...

A basic model in mathematical finance theory is the celebrated geometric Brownian motion. Moreover, the geometric telegraph process is a simpler model to describe the alternating dynamics of the price of risky assets. In this note we consider a more general stochastic process that combines the characteristics of such two models. Precisely, we deal...

This chapter demonstrates how to design and analyze experiments which are aimed at testing scientific or technological hypotheses. Blocking and randomization are devices aimed at increasing the precision of the outcome and ensuring the validity of the inference. Blocking is used to reduce errors. Randomization within each block is important to vali...

This chapter focuses on computer experiments and specific design and analysis methods relevant to such experiments. The major difference between computer numerical experiments and physical experiments is the logical difficulty in specifying a source of randomness for computer experiments. A specific case where randomness is introduced into computer...

Industrial phenomena are characterized by the fact that measurements performed on them are often not constant but reveal a certain degree of variability. This chapter presents methods to analyze this variability, in order to understand the variability structure and enhance our ability to control, improve and predict future behavior of such phenomen...

This chapter introduces basic concepts and methods of statistical inference. It discusses inference on parameters of infinite populations using classical point estimation, confidence intervals, tolerance intervals and hypothesis testing. Properties of point estimators such as moment equation estimators and maximum likelihood estimators are discusse...

This chapter presents the construction of multivariate control charts with the multivariate data. Multivariate statistical process control (MSPC) requires applications of methods and tools of multivariate data analysis, and the first section of the chapter reviews such methods and tools. A section focuses on multivariate data and describes several...

This chapter discusses methods and tools compatible with the top of the Quality Ladder. It covers both the Genichi Taguchi methods and the application of Quality by Design (QbD) in pharmaceutical companies. A section discusses optimization strategy that solves the problem of controlling both accuracy and variability by minimizing various loss funct...

Based on pathwise duality constructions, several new results on
truncated queues and storage systems of the G/M/1 type
are derived by transforming the workload (content) processes into
certain `dual' M/G/1-type processes. We consider queueing
systems in which (a) any service requirement that would increase the
total workload beyond the capacity is...

The present article presents two-stage procedures for fixed-width interval estimators of the common variance of equi-correlated normal distributions. This study is a continuation of those of Zacks and Ramig (19874.
Zacks , S. and
Ramig , P. ( 1987 ). Confidence Intervals for the Common Variance of Equicorrelated Normal Random Variables , in Contr...

The semiconductor industry ranges from the design and production of semiconductors on silicon wafers to automatic placement robots that insert semiconductor devices on hybrid microcircuits.Wafers consist of electronic circuits or chips that are characterized by electrical and mechanical characteristics. Process modeling and simulations provide pred...

We consider a generalized telegraph process which follows an
alternating renewal process and is subject to random jumps. More
specifically, consider a particle at the origin of the real line at
time t=0. Then it goes along two alternating velocities with opposite
directions, and performs a random jump toward the alternating direction
at each veloci...

We consider a standard Brownian motion whose drift alternates randomly between a positive and a negative value, according to a generalized telegraph process. We first investigate the distribution of the occupation time, i.e. the fraction of time when the motion moves with positive drift. This allows to obtain explicitly the probability law and the...

In this paper we study the distribution of the location, at time t, of a
particle moving U time units upwards, V time units downwards, and
W time units of no movement (idle). These are repeated cyclically,
according to independent alternating renewals. The distributions of U,
V, and W are absolutely continuous. The velocities are
v = +1 upwards, v...

This chapter reviews various important methods in Phase I, II and III of clinical trials. In Phase I trials, the main objective is to find the maximum tolerated dose (MTD) of a new agent (drug). The chapter discusses and evaluates the up-and-down methods, the continuous reassessment method (CRM) and the escalation with overdose control (EWOC) metho...

We consider alternating renewal processes which change their mode intermittently, like on and off of a system. The distribution of the total on time for an interval (0, t] is developed explicitly in terms of the distributions F and G of the alternating renewals. This distribution is then applied to determine the distributions of the locations of te...

The geometric telegrapher’s process has been proposed in 2002 as a model to describe the dynamics of the price of risky assets. In this contribution we consider a related stochastic process, whose trajectories have two alternating slopes, for which the random times between consecutive slope changes have exponential distribution with linearly increa...

Two-stage and sequential procedures are developed for fixed-width interval estimation of the parameter β in a gamma distribution G(α, β) when α is known. Exact properties are obtained for the two-stage procedure and some asymptotics and approximations are given for the operating characteristics of the sequential procedure and some numerical computa...

Zacks (Failure distribution associated with general renewal damage processes. In: Nikulin M, Commenges D, Haber C (eds) Probability
statistics and modelling in public health. Springer, Berlin, pp 465–475, 2006) studied the reliability function, the hazard function and the distribution of the failure time when a system is subject
to a cumulative, co...

The discussion concentrates on the material of Section 2 of [A. N. Shiryaev, ibid. 29, No. 4, 345–385 (2010; Zbl 1203.62137)], and shows how to apply the Wald martingale to obtain exact formulae for certain functionals R W (A,B), T ∞ (Z), etc.

The present chapter derives the reliability functions and hazard functions, when the threshold for deterioration is an increasing
function of time. Four cases are considered. Case I: The threshold is a step function with K jumps at known points. Case II: The threshold is a step function with K jumps, where the location of jumps are random, followin...

We consider an M/G/1 queue in which an arriving customer doesn't enter the system when-ever its virtual waiting time, i.e., the amount of work seen upon arrival, is larger than a certain random patience time. We determine the busy period distribution for various choices of the patience time distribution. The main cases under consideration are expon...

In this article, we consider the bounded risk point estimation problem for the mean μ in a N(μ, σ2) distribution under a LINEX loss function. We have proposed both two-stage and sequential procedures with a goal that the associated risk functions approximately fall under a preassigned risk-bound ω (>0). Our two-stage and sequential procedures and t...

The present article reviews various criteria and approaches for an early detection of abrupt changes in the parameters of stochastic sequences. The discussion focuses on two issues: (i) the distributions of the stopping variables employed; and (ii) the probability of false alarm and the conditional expected delay characteristics.

Stage-Wise Adaptive Designs presents the theory and methodology of stage-wise adaptive design across various areas of study within the field of statistics, from sampling surveys and time series analysis to generalized linear models and decision theory. Providing the necessary background material along with illustrative S-PLUS functions, this book s...

We consider single-server queues of the M/G/1 kind with a special kind of partial customer rejection called quasi-restricted accessibility (QRA). Under QRA, the actual service time assigned to an arriving customer depends on his service requirement, say x, the current workload, say w, and a prespecified threshold b. If x + w ≤ b the customer is ful...

The asymptotic properties of two-stage and sequential procedures for fixed-width, 2δ, confidence interval estimation of the mean β of an exponential distribution are well known. In the present paper we derive the exact operating characteristics of these procedures for any given δ > 0. The methodology is similar to that of Zacks and Mukhopadhyay (20...

The present paper deals with the problem of testing new treatments against a control in two stages. The objective is to select treatments whose probability of success is greater by a specified quantity from that of the control. I propose a simpler selection rule by improvising upon the customary chi-squared test.

Prediction theory for sampling surveys can be considered as a general framework for statistical inference on the characteristics of finite populations. Well—known estimators of population totals or population variances encountered in the classical theory, as expansion, ratio, regression, and other estimators, can be obtained as predictors in a gene...

In addition to the ARL∞, the discussion emphasizes the importance of determining PFA() and CED(). An example of such computations is given. When the parameters before and after the change-point are unknown, it is recommended to employ the full Bayesian framework.

Sequential Testing (SPRT) Characteristics of Sequential Procedures in Reliability Estimation and Testing Some Comments on Sequential Design of Experiments Sequential Testing of Software Reliability

Stein's Two-Stage Procedure Modifications to Attain Asymptotic Efficiency Two-Stage Sampling from Exponential Distributions Sequential Fixed-Width Interval Estimation Distributions of Stopping Variables of Sequential Sampling from Exponential Distributions Sequential Fixed-Width Intervals for the Log-Odds in Bernoulli Trials Bayesian Sequential Est...

Bayesian Detection When the Distributions Before and After the Change are Known Bayesian Detection When the Distributions Before and After the Change are Unknown CUSUM Procedures for Sequential Detection Tracking Algorithms for Processes with Change Points Recursive Estimation with Change Points Additional Theoretical Contributions

Basic Tools of Time Series Analysis Linear Predictors for Covariance Stationary T.S. Quadratic LSE Predictors for Nonstationary T.S. Moving Average Predictors for Nonstationary T.S. Predictors for General Trends with Exponential Discounting Dynamic Linear Models Asymptotic Behavior of DLM Linear Control of DLM

Up-and-Down Adaptive Designs Bayesian Adaptive Search: The Continuous Reassessment Method Efficient Dose Escalation with Overdose Control Patient-Specific Dosing Toxicity versus Efficacy

Bernoulli Bandits Gittins Dynamic Allocation Indices Sequential Allocations in Clinical Trials Bernoulli Bandits with Change Point Sequential Designs for Estimating the Common Mean of Two Normal Distributions: One Variance Known

Basic Theory Two-Stage and Sequential Estimation of the Population Mean Adaptive Allocation of Stratified SRS Adaptive Search for Special Units Adaptive Estimation of the Size of a Finite Population Applications in Software Reliability Sampling Inspection Schemes Dynamic Bayesian Prediction

Exponential Example Adaptive Designs for the Fisher Information Adaptive Bayesian Designs Adaptive Designs for Inverse Regression Stochastic Approximation

Randomization in Clinical Trials Adaptive Randomization Procedures Fixed-Width Sequential Estimation of the Success Probability in Bernoulli Trials Sequential Procedure for Estimating the Probability of Success in Bernoulli Trials with Prescribed Proportional Closeness Sequential Comparison of Success Probabilities Group Sequential Methods Dynamic...

We develop the exact distribution of the stopping variable of a sequential procedure that was originally given by Robbins and Siegmund (1974). The stopping variable was designed for estimating the log-odds in a sequence of Bernoulli trials. Using our exact distribution of the stopping variable, we give explicit formulas for the expected value and m...

We study an M/G/1 system with finite workload capacity that can be switched between two capacity levels v * <v ** . Whenever the workload process {V(t)|t≥0} is about to exceed the current capacity, the excess service time is truncated. We consider hysteretic control policies with two trigger points 0<v L <v U <v * , where switching from v * to v **...

The paper reviews recent results of D. Perry, W. Stadje and S. Zacks, on functionals of stopping times and the associated
compound Poisson process with lower and upper linear boundaries. In particular, formulae of these functionals are explicitly
developed for the total expected discounted cost of discarded service in an M/G/1 queue with restricted...

We derive a new formula for the probability that a compound Poisson process with positive jumps hits a lower straight line before it crosses a parallel upper line. This yields a new approach to determine the distribution of the cycle maximum of the queue. Moreover, we express the hitting probabilities in terms of the corresponding ruin probabilitie...

Distributions of the first-exit times from a region with concave upper boundary are discussed for ordinary and compound Poisson
processes. Explicit formulae are developed for the case of ordinary Poisson processes. Recursive formulae are given for the
compound Poisson case, where the jumps are positive, having discrete or continuous distributions w...

The present article studies the two-stage sampling procedure for estimating the exponential parameters proposed by Mukhopadhyay and Pepe (200612.
Mukhopadhyay , N. and
Pepe , W. ( 2006 ). Exact Bounded Risk Estimation When the Terminal Sample Size and Estimator are Dependent: The Exponential Case , Sequential Analysis 25 : 85 – 101 . [CSA] [Taylo...

Peskir and Shiryaev [2002. Solving the Poisson disorder problem. In: Advances in Finance and Stochastics: Essays in Honor of Dieter Sonderman. Springer, New York, pp. 295-312] determined the optimal stopping rule for a problem of quick detection of a change-point in the intensity of a homogeneous ordinary Poisson process, when the cost per unit tim...

Under purely sequential sampling schemes, a theory is developed for the exact determination of the distributions of two classes of stopping variables (rules) in order to handle point estimation problems for the parametric functionals in an exponential distribution. Explicit formulae are derived for the expected value and risks of sequential estimat...

The discussion points on the inadequacy of the ARL as an index of efficiency of a detection procedure for change points. It is shown that the FAR=1/ARL might be small, while the probability of false alarm (PFA) is at the same time considerable. This is illustrated with simulation runs, using the Shiryayev–Roberts detection procedure. The need to de...

We consider growth-collapse processes (GCPs) that grow linearly between random partial collapse times, at which they jump down according to some distribution depending on their current level. The jump occurrences are governed by a state-dependent rate function r(x). We deal with the stationary distribution of such a GCP, (X
t
)
t≥0, and the distrib...

We consider growth-collapse processes (GCPs) that grow linearly between random partial collapse times, at which they jump down according to some distribution depending on their current level. The jump occurrences are governed by a state-dependent rate function r(x). We deal with the stationary distribution of such a GCP, (X<sub>t</sub>)<sub>t≥0</su...

The Pitman efficiency is an index for comparing test procedures or estimators. It is especially important for comparing procedures in large samples. If procedure P1 requires n1 observations to attain a certain power of a test, or a specified mean squared error, and procedure P2 requires n2 observations to achieve the same precision, the Pitman rela...

For a compound process with exponential jumps at renewal times, we
determine, in closed form, the density of the first time an upper linear
boundary is crossed. It is shown how simple formulas for the Laplace
transform and the first two moments can be directly derived from this
density.

Scale equivariant estimators of the common variance σ, of correlated normal random variables, have mean squared errors (MSE) which depend on the unknown correlations. For this reason, a scale equivariant estimator of σ which uniformly minimizes the MSE does not exist. For the equi-correlated case, we have developed three equivariant estimators of σ...

A production/inventory system is filled continuously at rate 1 and satisfies demands of i.i.d. random sizes that arrive at Poisson times. For this system we consider two clearing policies. Under sporadic review, clearing takes place after a random time independent of the content process. Under continuous review, the system is cleared as soon as the...

We consider the first-exit time of a compound Poisson process from a region that is bounded from below by an increasing straight line, while its upper boundary has positive jumps of i.i.d. sizes at Poisson times and increases linearly between jumps. An integral equation for the corresponding Laplace-Stieltjes transforms is derived and solved. The c...

A review of recent results is given, for the distribution of stopping times defined on compound Poisson processes and linear boundaries. Generalization of the results of Zacks (Commun. Statist. Stochastic Models 7 (1991) 233) is given for discrete compound Poisson processes. The main approach in the reviewed papers is that of sample path analysis,...

The structure of a CUSUM procedure on an ordinary Poisson jump process is analyzed in terms of stopping rules based on linear boundaries. The total run length is composed of a random number of renewal phases followed by a terminal phase. The exact distributions of the length of these phases are derived, as well as the Wald type approximations. The...

The ‘rendezvous time’ of two stochastic processes is the first time at which they cross or hit each other. We consider such times for a Brownian motion with drift, starting at some positive level, and a compound Poisson process or a process with one random jump at some random time. We also ask whether a rendezvous takes place before the Brownian mo...

The `rendezvous time' of two stochastic processes is the first time at which they cross or hit each other. We consider such times for a Brownian motion with drift, starting at some positive level, and a compound Poisson process or a process with one random jump at some random time. We also ask whether a rendezvous takes place before the Brownian mo...

We study a one-dimensional telegraph process (M
t
)
t≥0 describing the position of a particle moving at constant speed between Poisson times at which new velocities are chosen randomly. The exact distribution of M
t
and its first two moments are derived. We characterize the level hitting times of M
t
in terms of integro-differential equations which...

## Projects

Projects (3)