BetaSubjective() R function from [mc2d]

The BetaSubjective Distribution

Density, distribution function, quantile function and random generation for the "Beta Subjective" distribution UTF-8


dbetasubj(x, 
  min,
  mode,
  mean,
  max, 
  log = FALSE)

pbetasubj(q, 
  min,
  mode,
  mean,
  max, 
  lower.tail = TRUE,
  log.p = FALSE
)

qbetasubj(p, 
  min,
  mode,
  mean,
  max, 
  lower.tail = TRUE, 
  log.p = FALSE
)

rbetasubj(n, 
  min,
  mode,
  mean,
  max
)

pbetasubj(q, min, mode, mean, max, lower.tail = TRUE, log.p = FALSE)

qbetasubj(p, min, mode, mean, max, lower.tail = TRUE, log.p = FALSE)

rbetasubj(n, min, mode, mean, max)

Arguments

x, q: Vector of quantiles.
min: continuous boundary parameter min < max
mode: continuous parameter $min < mode < max$ and $mode \ne mean$ .
mean: continuous parameter min < mean < max
max: continuous boundary parameter
log, log.p: Logical; if TRUE, probabilities p are given as log(p).
lower.tail: Logical; if TRUE (default), probabilities are $P[X \le x]$ otherwise, $P[X > x]$ .
p: Vector of probabilities.
n: Number of observations.

Details

The Subjective beta distribution specifies a [stats::dbeta()] distribution defined by the minimum, most likely (mode), mean and maximum values and can be used for fitting data for a variable that is bounded to the interval $[min, max]$ . The shape parameters are calculated from the mode value and mean parameters. It can also be used to represent uncertainty in subjective expert estimates.

Define

mid=(min+max)/2mid=(min+max)/2

a_{1}=2*\frac{(mean-min)*(mid-mode)}{((mean-mode)*(max-min))}a_1=2*(mean-min)*(mid-mode)/((mean-mode)*(max-min))

a_{2}=a_{1}*\frac{(max-mean)}{(mean-min)}a_2=a_1*(max-mean)/(mean-min)

The subject beta distribution is a [stats::dbeta()] distribution defined on the $[min, max]$ domain with parameter $shape1 = a_{1}$ and $shape2 = a_{2}$ .

Hence, it has density

f(x)=(x-min)^{(a_{1}-1)}*(max-x)^{(a_{2}-1)} / (B(a_{1},a_{2})*(max-min)^{(a_{1}+a_{2}-1)})f(x)=(x-min)^(a_1-1)*(max-x)^(a_2-1)/(B(a_1,a_2)*(max-min)^(a_1+a_2-1))

The cumulative distribution function is

F(x)=B_{z}(a_{1},a_{2})/B(a_{1},a_{2})=I_{z}(a_{1},a_{2})F(x)=B_z(a_1,a_2)/B(a_1,a_2)=I_z(a_1,a_2)

where $z=(x-min)/(max-min)$ . Here B is the beta function and $B_z$ is the incomplete beta function.

The parameter restrictions are:

min <= mode <= maxmin <= mode <= max

min <= mean <= maxmin <= mean <= max

If $mode > mean$ then $mode > mid$ , else $mode < mid$ .

Examples


curve(dbetasubj(x, min=0, mode=1, mean=2, max=5), from=-1,to=6) 
pbetasubj(q = seq(0,5,0.01), 0, 1, 2, 5)
qbetasubj(p = seq(0,1,0.01), 0, 1, 2, 5)
rbetasubj(n = 1e7, 0, 1, 2, 5)

Author(s)

Yu Chen

mc2d package Read PDF manual

Maintainer: Regis Pouillot
License: GPL (>= 2)
Last published: 2024-06-05

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BetaSubjective function