A novel method for fitting unimodal continuous distributions on a bounded domain utilizing expert judgment estimates

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.

[1]  H. Raiffa,et al.  Judgment under uncertainty: A progress report on the training of probability assessors , 1982 .

[2]  William H. Press,et al.  Numerical Recipes in FORTRAN - The Art of Scientific Computing, 2nd Edition , 1987 .

[3]  T. Modis,et al.  Experts in uncertainty , 1993 .

[4]  S. Kotz,et al.  The Standard Two-Sided Power Distribution and its Properties , 2002 .

[5]  Robert Taggart.  Ship design and construction , 1980 .

[6]  Simaan M. AbouRizk,et al.  Visual Interactive Fitting of Beta Distributions , 1991 .

[7]  Salah E. Elmaghraby,et al.  Activity networks: Project planning and control by network models , 1977 .

[8]  J. E. Selvidge,et al.  ASSESSING THE EXTREMES OF PROBABILITY DISTRIBUTIONS BY THE FRACTILE METHOD , 1980 .

[9]  T. Ferguson.  A Bayesian Analysis of Some Nonparametric Problems , 1973 .

[10]  Jerzy Kamburowski,et al.  New validations of PERT times , 1997 .

[11]  Tayfur Altiok,et al.  Simulation Modeling and Analysis with ARENA , 2007 .

[12]  D. Malcolm,et al.  Application of a Technique for Research and Development Program Evaluation , 1959 .

[13]  James R. Wilson,et al.  Using bivariate Bézier distributions to model simulation input processes , 1994, Winter Simulation Conference.

[14]  Terry Williams,et al.  Practical Use of Distributions in Network Analysis , 1992 .

[15]  Thomas Simpson.  XIX. A letter to the Right Honourable George Earl of Macclesfield, President of the Royal Society, on the advantage of taking the mean of a number of observations, in practical Astronomy , 1755, Philosophical Transactions of the Royal Society of London.

[16]  Christos Alexopoulos,et al.  Output Data Analysis , 2007 .

[17]  Kaisa Simola,et al.  An expert panel approach to support risk-informed decision making , 2000 .

[18]  Taylor Francis Online,et al.  The American statistician , 1947 .

[19]  David Vose,et al.  Quantitative Risk Analysis: A Guide to Monte Carlo Simulation Modelling , 1996 .

[20]  David Johnson,et al.  The triangular distribution as a proxy for the beta distribution in risk analysis , 1997 .

[21]  José García Pérez,et al.  A note on the reasonableness of PERT hypotheses , 2003, Oper. Res. Lett..

[22]  Albert-László Barabási,et al.  Linked - how everything is connected to everything else and what it means for business, science, and everyday life , 2003 .

[23]  Joseph J. Moder,et al.  Judgment Estimates of the Moments of Pert Type Distributions , 1968 .

[24]  Charles E. Clark,et al.  Letter to the Editor—The PERT Model for the Distribution of an Activity Time , 1962 .

[25]  A. Macdonald.  A Statistician , 1921 .

[26]  D. Keefer,et al.  Better estimation of PERT activity time parameters , 1993 .

[27]  Amy Hing-Ling Lau,et al.  Improved Moment-Estimation Formulas Using More Than Three Subjective Fractiles , 1998 .

[28]  J. Banks,et al.  Discrete-Event System Simulation , 1995 .

[29]  Barry L. Nelson,et al.  Statistical Analysis of Simulation Results , 2007 .

[30]  Lynn B. Davidson,et al.  Implementing Effective Risk Analysis at Getty Oil Company , 1980 .

[31]  Charles T. Jahren,et al.  Journal of Construction Engineering and Management , 1983 .

[32]  Averill M. Law,et al.  Simulation Modeling and Analysis , 1982 .

[33]  William H. Press,et al.  Numerical recipes in C. The art of scientific computing , 1987 .

[34]  M. D. Wilkinson,et al.  Management science , 1989, British Dental Journal.

[35]  Eric R. Zieyel.  Operations research : applications and algorithms , 1988 .

[36]  James R. Wilson,et al.  Graphical interactive simulation input modeling with bivariate Bézier distributions , 1995, TOMC.

[37]  The two-sided power distribution for the treatment of the uncertainty in PERT , 2005 .

[38]  James R. Wilson,et al.  Using Univariate Bezier Distributions to Model Simulation Input Processes , 1996, Proceedings of 1993 Winter Simulation Conference - (WSC '93).

[39]  Frank E. Grubbs.  Letter to the Editor---Attempts to Validate Certain PERT Statistics or “Picking on PERT” , 1962 .

[40]  W. Fleming.  Functions of Several Variables , 1965 .

[41]  Samuel Kotz,et al.  A novel extension of the triangular distribution and its parameter estimation , 2002 .

[42]  Salvador Cruz Rambaud,et al.  The two-sided power distribution for the treatment of the uncertainty in PERT , 2005, Stat. Methods Appl..

[43]  A. Tversky,et al.  Judgment under Uncertainty: Heuristics and Biases , 1974, Science.

[44]  James R. Wilson,et al.  Visual interactive fitting of bounded Johnson distributions , 1989, Simul..

[45]  이성수,et al.  Simulation , 2006, Healthcare Simulation at a Glance.

[46]  D. V. Lindley,et al.  The theory of queues with a single server , 1952, Mathematical Proceedings of the Cambridge Philosophical Society.

[47]  W. Press,et al.  Numerical Recipes in Fortran: The Art of Scientific Computing.@@@Numerical Recipes in C: The Art of Scientific Computing. , 1994 .

[48]  H. Anton,et al.  Functions of several variables , 2021, Thermal Physics of the Atmosphere.