Triangular: The Triangular Distribution.

Description

Density, distribution, quantile, random number generation and parameter estimation functions for the triangular distribution with support \([a,b]\) and shape parameter \(\theta\). Parameter estimation can be based on a weighted or unweighted i.i.d. sample and can be performed numerically.

Usage

dTriangular(x, a = 0, b = 1, theta = 0.5, params = list(a, b, theta), ...)
pTriangular(q, a = 0, b = 1, theta = 0.5, params = list(a, b, theta), ...)
qTriangular(p, a = 0, b = 1, theta = 0.5, params = list(a, b, theta), ...)
rTriangular(n, a = 0, b = 1, theta = 0.5, params = list(a, b, theta), ...)
eTriangular(X, w, method = "numerical.MLE", ...)
lTriangular(
  X,
  w,
  a = 0,
  b = 1,
  theta = 0.5,
  params = list(a, b, theta),
  logL = TRUE,
  ...
)

Value

dTriangular gives the density, pTriangular the distribution function, qTriangular the quantile function, rTriangular generates random variables, and eTriangular estimates the parameters. lTriangular provides the log-likelihood function.

Arguments

x, q: A vector of quantiles.
a, b: Boundary parameters.
theta: Shape parameters.
params: A list that includes all named parameters.
...: Additional parameters.
p: A vector of probabilities.
n: Number of observations.
X: Sample observations.
w: An optional vector of sample weights.
method: Parameter estimation method.
logL: logical, it is assumed that the log-likelihood is desired. Set to FALSE if the likelihood is wanted.

Author

Haizhen Wu and A. Jonathan R. Godfrey.
Updates and bug fixes by Sarah Pirikahu.

Details

If a, b or theta are not specified they assume the default values of 0, 1 and 0.5 respectively.

The dTriangle(), pTriangle(), qTriangle(),and rTriangle() functions serve as wrappers of the dtriangle, ptriangle, qtriangle, and rtriangle functions in the VGAM package. They allow for the parameters to be declared not only as individual numerical values, but also as a list so parameter estimation can be carried out.

The triangular distribution has a probability density function, defined in Forbes et.al (2010), that consists of two lines joined at \(theta\), where \(theta\) is the location of the mode.

References

Kotz, S. and van Dorp, J. R. (2004). Beyond Beta: Other Continuous Families of Distributions with Bounded Support and Applications. Chapter 1. World Scientific: Singapore.

Forbes, C., Evans, M., Hastings, N. and Peacock, B. (2010) Triangular Distribution, in Statistical Distributions, Fourth Edition, John Wiley & Sons, Inc., Hoboken, NJ, USA.