MinimumQuantileInformation: 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.

Author

Yu Chen and Arie Havelaar

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.

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

Run this code


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))

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