MinimumQuantileInformation {mc2d}

Minimum Quantile Information Distribution

Description

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

Usage

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}}

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)]

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,4

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

Cumulative distribution function

F(x) = 0 \text{ 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,4

F(x) = 1 \text{ 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)

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.

Author(s)

Yu Chen and Arie Havelaar

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.

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))