MinimumQuantileInformation() R function from [mc2d]

Minimum Quantile Information Distribution

Density, distribution function, quantile function and random generation for Minimum Quantile Information distribution.


dmqi(x, 
  mqi, 
  mqi.quantile = c(0.05, 0.5, 0.95),
  realization = NULL, 
  k = 0.1, 
  intrinsic = NA,
  log = FALSE)

pmqi(q, 
  mqi, 
  mqi.quantile = c(0.05, 0.5, 0.95),
  realization = NULL,
  k = 0.1,
  intrinsic = NA,
  lower.tail = TRUE,
  log.p = FALSE
)

qmqi(p, 
  mqi, 
  mqi.quantile = c(0.05, 0.5, 0.95),
  realization = NULL, 
  k = 0.1, 
  intrinsic = NA,
  lower.tail = TRUE, 
  log.p = FALSE
)

rmqi(n, 
  mqi, 
  mqi.quantile = c(0.05, 0.5, 0.95),
  realization = NULL, 
  k=0.1, 
  intrinsic = NA
)

pmqi(
  q,
  mqi,
  mqi.quantile = c(0.05, 0.5, 0.95),
  realization = NULL,
  k = 0.1,
  intrinsic = NA,
  lower.tail = TRUE,
  log.p = FALSE
)

qmqi(
  p,
  mqi,
  mqi.quantile = c(0.05, 0.5, 0.95),
  realization = NULL,
  k = 0.1,
  intrinsic = NA,
  lower.tail = TRUE,
  log.p = FALSE
)

rmqi(
  n,
  mqi,
  mqi.quantile = c(0.05, 0.5, 0.95),
  realization = NULL,
  k = 0.1,
  intrinsic = NA
)

Arguments

x, q: Vector of quantiles
mqi: Minimum quantile information
mqi.quantile: The quantile of mqi. It's a vector of length 3. Default is c(0.05, 0.5, 0.95), that is the 5th, 50th and 95th.
realization: Default is NULL. If not NULL, used to define L or U (see details).
k: Overshot, default value is 0.1.
intrinsic: Use to specify a prior bounds of the intrinsic range. Default = NA.
log, log.p: Logical; if TRUE, probabilities p are given as log(p).
lower.tail: Logical; if TRUE (default), probabilities are P[X <= x] otherwise, P[X > x].
p: Vector of probabilities.
n: Number of observations.

Details

$p_1$ , $p_2$ , and $p_3$ are percentiles of a distribution with $p_1 < p_2 < p_3$ . The interval $[L,U]$ is given with:

L = x_{p_{1}}L = x_p_1

U = x_{p_{3}}U = x_p_3

The support of minimum quantile information distribution is determined by the intrinsic range:

[L^{*}, U^{*}] = [L - k \times (U - L), U + k \times (U - L)][L^*, U^*] = [L - k*(U - L), U + k*(U - L)]

where $k$ denotes an overshoot and is chosen by the analyst (usually $k = 10%$ , which is the default value).

Given the three values of quantile, $x_{p_1}$ , $x_{p_2}$ and $x_{p_3}$ , and define $p_0 = 0$ , $p_4 = 1$ , $x_{p_0} = L^{*}$ and $x_{p_4} = U^{*}$

the minimum quantile information distribution is given by:

Probability density function

f(x) = \frac{p_{i}-p_{i-1}}{x_{p_{i}}-x_{p_{i-1}}} \text{ for } x_{p_{i-1}} \le x < x_{p_{i}},i = 1,\dots,4f(x)=(p_i-p_(i-1))/(x_p_i-x_p_(i-1)) for x_p_(i-1)\le x_p_i, i=1,\dots,4

f(x) = 0, \text{ otherwise}f(x) = 0, otherwise

Cumulative distribution function

F(x) = 0 \text{ for } x < x_{p_{0}}F(x) = 0 for x < x_p_0

F(x) = \frac{p_{i}-p_{i-1}}{x_{p_{i}}-x_{p_{i-1}}}*(x-x_{p_{i-1}})+p_{i-1} \text{ for } x_{p_{i-1}} \le x < x_{p_{i}}, i = 1,\dots,4F(x) = (p_i-p_(i-1))/(x_p_i-x_p_(i-1))*(x-x_p_(i-1))+p_(i-1) for x_p_(i-1)\le x_p_i, i=1,\dots,4

F(x) = 1 \text{ for } x_{p_{4}}\le xF(x) = 1 for x_p_(4) \le x

This distribution is usually used for expert elicitation. If experts have realization information, then the range $[L,U]$ is given by:

L = \min(x_{p_{1}}, realization)L = min(x_p_1, realization)

U = \max(x_{p_{3}}, realization)U = max(x_p_3, realization)

For some questions, experts may have information for the intrinsic range and set a prior intrinsic range ( $L^*$ and $U^*$ ).

NOTE that the function is vectorized only for x, q, p, n. As a consequence, it can't be used for variable other parameters.

Examples


curve(dmqi(x, mqi=c(40,50,60), intrinsic=c(0,100)), from=0, to=100, type = "l", xlab="x",ylab="pdf")
curve(pmqi(x, mqi=c(40,50,60), intrinsic=c(0,100)), from=0, to=100, type = "l", xlab="x",ylab="cdf")
rmqi(n = 10, mqi=c(555, 575, 586))

References

Hanea, A. M., & Nane, G. F. (2021). An in-depth perspective on the classical model. In International Series in Operations Research & Management Science (pp. 225–256). Springer International Publishing.

Author(s)

Yu Chen and Arie Havelaar

mc2d package Read PDF manual

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

Useful links

MinimumQuantileInformation function