Calculate the parameters of a BetaPERT distribution
Description
The BetaPERT methodology allows to parametrize a generalized Beta distribution based on expert opinion regarding
a pessimistic estimate (minimum value), a most likely estimate (mode),
and an optimistic estimate (maximum value). The betaPERT
function incorporates two methods of
calculating the parameters of a BetaPERT distribution, designated "classic"
and "vose"
.
Usage
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Arguments
a 
Pessimistic estimate (Minimum value) 
m 
Most likely estimate (Mode) 
b 
Optimistic estimate (Maximum value) 
k 
Scale parameter 
method 

x 
Object of class 
y 
Currently ignored 
conf.level 
Confidence level used in printing quantiles of resulting BetaPERT distribution 
... 
Other arguments to pass to function 
Details
The BetaPERT methodology was developed in the context of Program Evaluation and Review Technique (PERT). Based on a pessimistic estimate (minimum value), a most likely estimate (mode), and an optimistic estimate (maximum value), typically derived through expert elicitation, the parameters of a Beta distribution can be calculated. The BetaPERT distribution is used in stochastic modeling and risk assessment studies to reflect uncertainty regarding specific parameters.
Different methods exist in literature for defining the parameters of a Beta distribution based on PERT. The two most common methods are included in the BetaPERT
function:
 Classic:
The standard formulas for mean, standard deviation, α and β, are as follows:
mean = (a + k*m + b) / (k + 2)
sd = (b  a) / (k + 2)
α = { (mean  a) / (b  a) } * { (mean  a) * (b  mean) / sd^{2}  1 }
β = α * (b  mean) / (mean  a)
The resulting distribution is a 4parameter Beta distribution: Beta(α, β, a, b).
 Vose:
Vose (2000) describes a different formula for α:
(mean  a) * (2 * m  a  b) / { (m  mean) * (b  a) }
Mean and β are calculated using the standard formulas; as for the classical PERT, the resulting distribution is a 4parameter Beta distribution: Beta(α, β, a, b).
Note: If m = mean, α is calculated as 1 + k/2, in accordance with the mc2d package (see 'Note').
Value
A list of class "betaPERT"
:
alpha 
Parameter α (shape1) of the Beta distribution 
beta 
Parameter β (shape2) of the Beta distribution 
a 
Pessimistic estimate (Minimum value) 
m 
Most likely estimate (Mode) 
b 
Optimistic estimate (Maximum value) 
method 
Applied method 
Available generic functions for class "betaPERT"
are print
and plot
.
Note
The mc2d package provides
the probability density function, cumulative distribution function, quantile function and random number generation function
for the PERT distribution, parametrized by the "vose"
method.
Author(s)
Brecht Devleesschauwer <brechtdv@gmail.com>
References
 Classic:

Malcolm DG, Roseboom JH, Clark CE, Fazar W (1959) Application of a technique for research and development program evaluation. Oper Res 7(5):646669.
 Vose:

David Vose. Risk analysis, a quantitative guide, 2nd edition. Wiley and Sons, 2000.
PERT distribution in ModelRisk (Vose software)
See Also
betaExpert
, for modelling a standard Beta distribution based on expert opinion
Examples
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