Difficulties with the interpretation of the parameters of the beta distribution let Malcolm et al. (1959) to suggest in the Program Evaluation and Review Technique (PERT) their by now classical expressions for the mean and variance for activity completion for practical applications. In this note, we shall provide an alternative for the PERT variance expression addressing a concern raised by Hahn (2008) regarding the constant PERT variance assumption given the range for an activity’s duration, while retaining the original PERT mean expression. Moreover, our approach ensures that an activity’s elicited most likely value aligns with the beta distribution’s mode. While this was the original intent of Malcolm et al. (1959), their method of selecting beta parameters via the PERT mean and variance is not consistent in this manner.

Copulas are joint continuous distributions with uniform marginals and have been proposed to capture probabilistic dependence between random variables. Maximum-entropy copulas introduced by Bedford and Meeuwissen (Bedford, T., A. M. H. Meeuwissen. 1997. Minimally informative distributions with given rank correlations for use in uncertainty analysis. J. Statist. Comput. Simulation 57 (1-4) 143-175) provide the option of making minimally informative assumptions given a degree-of-dependence constraint between two random variables. Unfortunately, their distribution functions are not available in a closed form, and their application requires the use of numerical methods. In this paper, we study a subfamily of generalized diagonal band (GDB) copulas, separately introduced by Ferguson (Ferguson, T. F. 1995. A class of symmetric bivariate uniform distributions. Statist. Papers 36 (1) 31-40) and Bojarski (Bojarski, J. 2001. A new class of band copulas-Distributions with uniform marginals. Technical report, Institute of Mathematics, Technical University of Zielona Góra, Zielona Góra, Poland). Similar to Archimedean copulas, GDB copula construction requires a generator function. Bojarski's GDB copula generator functions are symmetric probability density functions. In this paper, symmetric members of a two-sided framework of distributions introduced by van Dorp and Kotz (van Dorp, J. R., S. Kotz. 2003. Generalizations of two-sided power distributions and their convolution. Comm. Statist.: Theory and Methods 32 (9) 1703-1723) shall be considered. This flexible setup allows for derivations of GDB copula properties resulting in novel convenient expressions. A straightforward elicitation procedure for the GDB copula dependence parameter is proposed. Closed-form expressions for specific examples in the subfamily of GDB copulas are presented, which enhance their transparency and facilitate their application. These examples closely approximate the entropy of maximum-entropy copulas. Application of GDB copulas is illustrated via a value-of-information decision analysis example.

•Statistical dependence in a project network is modeled using a Bayesian network.•Remaining project uncertainty is monitored coherently as the project progresses.•Faster learning in terms of project completion uncertainty is demonstrated using the Bayesian paradigm.•Faster learning enables timely corrective action avoiding potentially costly schedule slippage. [Display omitted] The Program Evaluation and Review Technique (PERT) dates back to 1959. This method evaluates the uncertainty distribution of a project’s completion time given the uncertain completion times of the activities/tasks comprised within it. Each activity’s uncertainty was defined originally by a unique two parameter beta PERT distribution satisfying what is known to be the PERT mean and PERT variance. In this paper, a three-parameter PERT family of bounded distributions is introduced satisfying that same mean and variance, generalizing the beta PERT distribution. Their additional flexibility allows for the modeling of statistical dependence in a continuous Bayesian network, generalizing in turn the traditional PERT procedure where statistical independence is assumed among beta PERT activity durations. Through currently available Bayesian network software and the construction of that PERT family herein, the coherent monitoring of remaining project completion time uncertainty given partial completion of a project may become more accessible to PERT analysts. An illustrative example demonstrating the benefit of monitoring of remaining project completion time uncertainty as activities complete in that Bayesian fashion shall be presented, including expressions and algorithms for the specification of the three prior parameters for each activity in the project network to adhere to classical the PERT mean and PERT variance and a degree of statistical dependence between them.

•A novel Generalized Trapezoidal Ogive distribution (GTO) is developed.•A new flexible fatality curve model was developed utilizing the GTO distribution.•The fatality curve GTO model is fitted in stages sequentially.•The fatality GTO model avoids a right tail fit under sole availability of left tail fatality data.•GTO model was fitted to COVID-19 Italy fatality data. [Display omitted] The construction of a continuous family of distributions on a compact set is demonstrated by concatenating, in a continuous manner, three probability density functions with bounded support using a modified mixture technique. The construction technique is similar to that of generalized trapezoidal (GT) distributions, but contrary to GT distributions, the resulting density function is smooth within its bounded domain. The construction of Generalized Trapezoidal Ogive (GTO) distributions was motivated by the COVID-19 epidemic, where smoothness of an infection rate curve may be a desirable property combined with the ability to separately model three stages and their durations as the epidemic progresses, being: (1) an increasing infection rate stage, (2) an infection rate stage of some stability and (3) a decreasing infection rate stage. The resulting model allows for asymmetry of the infection rate curve opposite to, for example, the Gaussian Error Infection (GEI) rate curve utilized early on for COVID-19 epidemic projections by the Institute for Health Metrics and Evaluation (IHME). While other asymmetric distributions too allow for the modeling of asymmetry, the ability to separately model the above three stages of an epidemic’s progression is a distinct feature of the model proposed. The latter avoids unrealistic projections of an epidemic’s right-tail in the absence of right tail data, which is an artifact of any fatality rate model where a left-tail fit determines its right-tail behavior.

A novel smooth, three-parameter, asymmetric two-sided distribution with bounded support is constructed via a half-symmetric beta distribution. Denoting that novel asymmetric distribution the generalized two-sided beta (GTSB) distribution, it is characterized by a mode (or anti-mode) parameter together with two branch power parameters. GTSB distributions serve as a smooth alternative for generalized two-sided power distributions. Some properties of the GTSB family of distributions shall be derived. Two separate algorithms to solve for the power parameters of the GTSB distribution shall be presented. The first algorithm ensures matching of the most likely value, specified through expert judgment, as well as the PERT mean and PERT variance, popular in project management. The latter result is a novel PERT contribution by itself. The second algorithm solves for the power parameters from a lower and upper quantile constraint. The application of the GTSB distribution shall be demonstrated in illustrative PERT example(s).

Complex computational engineering uncertainty analyses have become more prevalent. When input parameters of such engineering models are uncertain, the output metric's uncertainty distribution is of an unknown parametric form. Since Wilks' method, named after the seminal paper by SS Wilks in 1941 entitled “Determination of sample sizes for setting tolerance limits”, is a nonparametric statistical procedure, it has received renewed interest, in particular in nuclear and chemical safety engineering. Unfortunately, the prevailing Wilks' method applied relies on arbitrary specification of order statistics' ranks with undue influence on the sample size recommendations that follow. Herein, a novel modification of Wilks' method involving two quantiles is proposed resolving that arbitrary rank selection. Together with a confidence level to be exceeded, these quantiles uniquely determine the parameters of an order statistics' beta distribution which drive the selection of symmetric tolerance limits. The modified procedure is demonstrated in two illustrative engineering uncertainty analysis examples drawn from the nuclear and chemical engineering domains.

The idea of transforming one random variate to another with a more convenient density has been developed in the first half of the 20th century. In his thesis, Norman L. Johnson (1917-2004) developed a pioneering system of transformations of the standard normal distribution which gained substantial popularity in the second half of the 20th century and beyond. In Johnson's 1949 Biometrika paper entitled Systems of frequency curves generated by methods of translation, summarizing that thesis, one of his primary interests was the behavior of the shape of the probability density functions as their parameter values change. Herein, we attempt to further elucidate this behavior through a series of geometric expositions of that transformation process. In these expositions insight is obtained into the behavior of Johnson's density functions, and their skewness and kurtosis, as they converge to their limiting distributions, a topic which received little attention.

Recent advances in computation technology for decision/simulation and uncertainty analyses have revived interest in the triangular distribution and its use to describe uncertainty of bounded input phenomena. The trapezoidal distribution is a generalization of the triangular distribution that allows for the specification of the modal value by means of a range of values rather than a single point estimate. Whereas the trapezoidal and the triangular distributions are restricted to linear geometric forms in the successive stages of the distribution, the generalized trapezoidal (GT) distribution allows for a nonlinear behavior at its tails and a linear incline (or decline) in the central stage. In this paper we develop two novel elicitation procedures for the parameters of a special case of the GT family by restricting ourselves to a uniform (horizontal) central stage in accordance with the central stage of the original trapezoidal distribution.

In this paper, we shall develop a novel family of bimodal univariate distributions (also allowing for unimodal shapes) and demonstrate its use utilizing the well-known and almost classical data set involving durations and waiting times of eruptions of the Old-Faithful geyser in Yellowstone park. Specifically, we shall analyze the Old-Faithful data set with 272 data points provided in Dekking et al. [3]. In the process, we develop a bivariate distribution using a copula technique and compare its fit to a mixture of bivariate normal distributions also fitted to the same bivariate data set. We believe the fit-analysis and comparison is primarily illustrative from an educational perspective for distribution theory modelers, since in the process a variety of statistical techniques are demonstrated. We do not claim one model as preferred over the other.

The introduction of the Program Evaluation and Review Technique (PERT) dates back to the 1960s and has found wide application since then in the planning of construction projects. Difficulties with the interpretation of the parameters of the beta distribution let Malcolm et al. (Malcolm, D.G., et al., 1959. Application of a technique for research and development program evaluation. Operations Research, 7, 646-649) to elicit them indirectly via the classical expressions for the PERT mean and variance for an activity's completion time. These expressions are specified, given a lower and upper bound estimates a and b and a most likely estimate θ for the activity's duration. Despite a by now 50 -year controversy exemplified by current articles still questioning the PERT mean and variance elicitation approach, their use is still prevalent in current operations research and industrial engineering college text books. In this article an overview is presented of some alternative approaches that have been suggested, including a recent approach that allows for a direct modal range estimation combined with an indirect elicitation of bound and tail parameters of generalised trapezoidal uniform distributions describing activity uncertainty. The potential effect of eliciting a modal range, rather than the narrower point estimate of a single most likely value, shall be demonstrated in an illustrative Monte Carlo analysis for the completion time of an 18-node activity network.

Preventive maintenance of mechanical equipment subject to random failures requires a lifetime distribution to establish an optimal maintenance interval. Typically, the optimal maintenance interval under an age replacement regimen is obtained by minimising the long term average cost of the maintenance activity. Only when the cost of maintaining the equipment preventively is less than the cost of failure of the equipment, can preventive maintenance be worthwhile. In practical contexts with an effective preventive maintenance policy, the estimation of such a lifetime distribution is complicated due to a lack of failure time data despite an abundance of right censored data, i.e. survival data of the component up to the time it was preventively maintained. Herein, we shall present a model for eliciting lifetime distributions via a histogram technique reminiscent of the method proposed by Van Noortwijk et al. (Van Noortwijk, J.A., et al., 1992. On the use of expert opinion for maintenance optimization. IEEE Transactions in Reliability, 41, 427-432). The elicited discrete distribution is used to estimate the prior parameters of a Dirichlet Process (DP). This DP is next updated using the failure time and maintenance data in a Bayesian fashion. The resulting lifetime distribution follows the posterior estimate of the DP process. Utilising the posterior lifetime distribution estimate, a maintenance interval can be established graphically by plotting the long term average cost as a function of the preventive maintenance frequency. An illustrative calculation example is presented.

This paper presents research conducted in modeling specific exposure metrics of communities in the vicinity of public use, nontowered airports to aviation accidents that result in crash sites outside the immediate confines of a runway (termed external airport risk). Two exposure metrics are explored: a relative exposure (termed crash hazard), defined as the probability that a crash site will be located in a specific area if a crash were to occur at an airport, and an absolute exposure (termed crash risk), defined as the expected number of crashes per year within any defined area. Results of this research are presented as a series of choro-pleth maps that define boundaries as contours, within which these metrics exceed some defined threshold. A specific application of this research is presented within the context of a model airport.

The generalized standard two-sided power (GTSP) distribution was mentioned only in passing by Kotz and van Dorp Beyond Beta, Other Continuous Families of Distributions with Bounded Support and Applications, World Scientific Press, Singapore, 2004. In this paper, we shall further investigate this three-parameter distribution by presenting some novel properties and use its more general form to contrast the chronology of developments of various authors on the two-parameter TSP distribution since its initial introduction. GTSP distributions allow for J-shaped forms of its pdf, whereas TSP distributions are limited to U-shaped and unimodal forms. Hence, GTSP distributions possess the same three distributional shapes as the classical beta distributions. A novel method and algorithm for the indirect elicitation of the two-power parameters of the GTSP distribution is developed. We present a Project Evaluation Review Technique example that utilizes this algorithm and demonstrates the benefit of separate powers for the two branches of activity GTSP distributions for project completion time uncertainty estimation.

Recent advances in computation technology for simulation/uncertainty analyses have shed new light on the triangular distribution and its use to describe the uncertainty of bounded input phenomena. Herein, we develop a novel fitting procedure for a continuous unimodal (four-parameter) family of distributions on a bounded domain, utilizing three properly selected quantile estimates and an estimate of the most likely value. The family in question is the two-sided power family of which the triangular distribution is a member. We analyze some of the procedure's fitting characteristics and use them to estimate the waiting time distribution in a stationary M/G/1 queuing system and the completion time distribution of a small project network example taken from the shipbuilding domain.

Targeted malicious emails (TME) for computer network exploitation have become more insidious and more widely documented in recent years. Beyond spam or phishing designed to trick users into revealing personal information, TME can exploit computer networks and gather sensitive information. They can consist of coordinated and persistent campaigns that can span years. A new email-filtering technique based on email's persistent-threat and recipient-oriented features with a random forest classifier outperforms two traditional detection methods, SpamAssassin and ClamAV, while maintaining reasonable false positive rates.