Abstract
Techniques are presented for modelling and then randomly sampling many of the continuous univariate probabilistic input processes that drive discrete-event simulation experiments. Emphasis is given to the generalized beta distribution family, the Johnson translation system of distributions, and the Bézier distribution family because of the flexibility of these families to model a wide range of distributional shapes that arise in practical applications. Methods are described for rapidly fitting these distributions to data or to subjective information (expert opinion) and for randomly sampling from the fitted distributions. Also discussed are applications ranging from pharmaceutical manufacturing and medical decision analysis to smart-materials research and health-care systems analysis.
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Partial support for some of the research described in this article was provided by National Science Foundation Grant DMI-9900164.
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Appendix
Appendix
Exact computation of shape parameters for beta distribution fitted to user-specified mode and variance
To simplify the notation in this appendix, we let a, m, and b denote the user-specified minimum, mode, and maximum of the target distribution with a<b and m∈[a, b] as if these quantities were known exactly; in practice of course it is often necessary to use estimates â, m̂, and b̂ of these quantities in the following development. In this appendix, we provide exact computing formulas for the shape parameters α 1 and α 2 of the generalized beta distribution (1) on the interval [a, b] that has the user-specified mode m and the user-specified variance σ X 2=(b−a)2/ω.
If ω>12 (so that the desired beta distribution has a smaller variance than that of the uniform distribution on the interval [a, b]), then for any value of m∈[a, b], there is a unique generalized beta distribution on [a, b] with a unique mode at m. (If ω=12, then it can be shown that we must have α 1=α 2=1 so that the beta distribution with the given mode and variance coincides with the uniform distribution on [a, b]. Since the mode is assumed to be unique, this uninteresting case is eliminated from further consideration.) If we set the right-hand side of (4) equal to m and the right-hand side of (3) equal to (b−a)2/ω, then we obtain the following equivalent system of equations in terms of the asymmetry ratio r=(b−m)/(m−a), provided m>a so that r<∞:
where
Remark 6. In the case that m=a so that r=∞, we solve the ‘mirror image’ problem for which m=b and r=0; and then we interchange the resulting shape parameters to obtain a generalized beta distribution whose mode coincides with its minimum. See also Remark 7 below.
It can be proved that if ω>12, then for all r∈[0, ∞] the cubic equation in α 1 defined by (A1)–(A2) has a nonnegative discriminant
so that the cubic equation has three real roots {ζ j :j=1, 2, 3} such that:
As possible values of α 1, the roots ζ 2 and ζ 3 are unacceptable for the following reasons:
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i)
The assignment α 1∈(0, 1) yields a generalized beta distribution with an asymptote at its lower limit a, which seems intuitively problematic and is clearly unacceptable when the user-specified mode m exceeds the lower limit.
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ii)
The assignment α 1⩽0 does not define a legitimate generalized beta distribution.
We are therefore left with the unique assignment α 1=ζ 1; and a computing formula for α 1 can be derived from the explicit solution to a cubic equation as follows (see Sections 33–38 of Dickson, 1939). In terms of the auxiliary quantities
we have
Finally we take α 2=rα 1+1−r to complete the specification of the generalized beta distribution.
Remark 7. In general to avoid numerical difficulties that can occur with large values of r (that is, when r1), we recommend the following approach to the use of Equations (A1), (A2), (A3), (A4) and (A5). If (b−m)/(m−a)>1, then we solve the ‘mirror image’ problem for which r=(m−a)/(b−m)<1; and finally we interchange the resulting shape parameters to obtain a generalized beta distribution with the user-specified mode m.
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Kuhl, M., Ivy, J., Lada, E. et al. Univariate input models for stochastic simulation. J Simulation 4, 81–97 (2010). https://doi.org/10.1057/jos.2009.31
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DOI: https://doi.org/10.1057/jos.2009.31