Johan René van Dorp

Beyond beta : other continuous families of distributions with bounded support and applications

The Triangular Distribution Some Early Extensions of the Triangular Distribution The Standard Two-Sided Power Distribution The Two-Sided Power Distribution The Generalized Trapezoidal Distribution Uneven Two-Sided Power Distributions The Reflected Generalized Topp and Leone Distribution A Generalized Framework for Two-Sided Distributions.

A Risk Management Procedure for the Washington State Ferries

The state of Washington operates the largest passenger vessel ferry system in the United States. In part due to the introduction of high-speed ferries, the state of Washington established an independent blue-ribbon panel to assess the adequacy of requirements for passenger and crew safety aboard the Washington state ferries. On July 9, 1998, the Blue Ribbon Panel on Washington State Ferry Safety engaged a consultant team from The George Washington University and Rensselaer Polytechnic Institute/Le Moyne College to assess the adequacy of passenger and crew safety in the Washington state ferry (WSF) system, to evaluate the level of risk present in the WSF system, and to develop recommendations for prioritized risk reduction measures, which, once implemented, can improve the level of safety in the WSF system. The probability of ferry collisions in the WSF system was assessed using a dynamic simulation methodology that extends the scope of available data with expert judgment. The potential consequences of collisions were modeled in order to determine the requirements for onboard and external emergency response procedures and equipment. The methodology was used to evaluate potential risk reduction measures and to make detailed risk management recommendations to the blue-ribbon panel and the Washington State Transportation Commission.

Two-sided generalized Topp and Leone (TS-GTL) distributions

Over 50 years ago, in a 1955 issue of JASA, a paper on a bounded continuous distribution by Topp and Leone [C.W. Topp and F.C. Leone, A family of J-shaped frequency functions, J. Am. Stat. Assoc. 50(269) (1955), pp. 209–219] appeared (the subject was dormant for over 40 years but recently the family was resurrected). Here, we shall investigate the so-called Two-Sided Generalized Topp and Leone (TS-GTL) distributions. This family of distributions is constructed by extending the Generalized Two-Sided Power (GTSP) family to a new two-sided framework of distributions, where the first (second) branch arises from the distribution of the largest (smallest) order statistic. The TS-GTL distribution is generated from this framework by sampling from a slope (reflected slope) distribution for the first (second) branch. The resulting five-parameter TS-GTL family of distributions turns out to be flexible, encompassing the uniform, triangular, GTSP and two-sided slope distributions into a single family. In addition, the probability density functions may have bimodal shapes or admitting shapes with a jump discontinuity at the ‘threshold’ parameter. We will discuss some properties of the TS-GTL family and describe a maximum likelihood estimation (MLE) procedure. A numerical example of the MLE procedure is provided by means of a bimodal Galaxy M87 data set concerning V–I color indices of 80 globular clusters. A comparison with a Gaussian mixture fit is presented.

An Elicitation Procedure for the Generalized Trapezoidal Distribution with a Uniform Central Stage

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.

Revisiting the PERT mean and variance

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 activitys duration, while retaining the original PERT mean expression. Moreover, our approach ensures that an activitys elicited most likely value aligns with the beta distributions 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.

The generalized two-sided power distribution

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.

Generalized Diagonal Band Copulas with Two-Sided Generating Densities

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. Simulation57(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. Papers36(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 Gora, Zielona Gora, Poland). Similar to Archimedean copulas, GDB copula construction requires a generator function. Bojarskis 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 Methods32(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.

On a bounded bimodal two-sided distribution fitted to the Old-Faithful geyser data

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.

Indirect Parameter Elicitation Procedures for Some Distributions with Bounded Support - with Applications in Program Evaluation and Review Technique (PERT)

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 activitys completion time. These expressions are specified, given a lower and upper bound estimates a and b and a most likely estimate θ for the activitys 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.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 activitys completion time. These expressions are specified, given a lower and upper bound estimates a and b and a most likely estimate θ for the activitys 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, ...

A Bayesian expert judgement model to determine lifetime distributions for maintenance optimisation

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.