MinimumQuantileInformation {mc2d}R Documentation

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

p1p_1, p2p_2, and p3p_3 are percentiles of a distribution with p1<p2<p3p_1 < p_2 < p_3. The interval [L,U][L,U] is given with:

L=xp1L = x_{p_{1}}

U=xp3U = x_{p_{3}}

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

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

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

Given the three values of quantile, xp1x_{p_1}, xp2x_{p_2} and xp3x_{p_3}, and define p0=0p_0 = 0, p4=1p_4 = 1, xp0=Lx_{p_0} = L^{*} and xp4=Ux_{p_4} = U^{*} the minimum quantile information distribution is given by:

Probability density function

f(x)=pipi1xpixpi1 for xpi1x<xpi,i=1,,4f(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, otherwisef(x) = 0, \text{ otherwise}

Cumulative distribution function

F(x)=0 for x<xp0F(x) = 0 \text{ for } x < x_{p_{0}}

F(x)=pipi1xpixpi1(xxpi1)+pi1 for xpi1x<xpi,i=1,,4F(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 for xp4xF(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][L,U] is given by:

L=min(xp1,realization)L = \min(x_{p_{1}}, realization)

U=max(xp3,realization)U = \max(x_{p_{3}}, realization)

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

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))

[Package mc2d version 0.2.1 Index]