# Triangular: The Triangular Distribution. In ExtDist: Extending the Range of Functions for Probability Distributions

## Description

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

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```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, ...) ```

## 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.

## 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.

## 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.

## Author(s)

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

## 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.