tpn (version 1.8)

utpn: Truncated positive normal

Description

Density, distribution function and random generation for the unit truncated positive normal (utpn) type 1 or 2 discussed in Gomez, Gallardo and Santoro (2021).

Usage

dutpn(x, sigma = 1, lambda = 0, type = 1, log = FALSE)
putpn(x, sigma = 1, lambda = 0, type = 1, lower.tail = TRUE, log = FALSE)
qutpn(p, sigma = 1, lambda = 0, type = 1)
rutpn(n, sigma = 1, lambda = 0, type = 1)

Value

dutpn gives the density, putpn gives the distribution function, qutpn provides the quantile function and rutpn generates random deviates.

The length of the result is determined by n for rtpn, and is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

A variable has utpn distribution with scale parameter \(\sigma>0\) and shape parameter \(\lambda \in\) R if its probability density function can be written as $$ f(y; \sigma, \lambda) = \frac{\phi\left(\frac{1-y}{\sigma y}-\lambda\right)}{\sigma y^2\Phi(\lambda)}, y>0, \mbox{(type 1),} $$

$$ f(y; \sigma, \lambda) = \frac{\phi\left(\frac{y}{\sigma (1-y)}-\lambda\right)}{\sigma (1-y)^2\Phi(\lambda)}, y>0, \mbox{(type 2),} $$

$$ f(y; \sigma, \lambda) = \frac{\phi\left(\frac{\log(y)}{\sigma}+\lambda\right)}{\sigma y\Phi(\lambda)}, y>0, \mbox{(type 3),} $$

$$ f(y; \sigma, \lambda) = \frac{\phi\left(\frac{\log(1-y)}{\sigma}+\lambda\right)}{\sigma (1-y)\Phi(\lambda)}, y>0, \mbox{(type 4),} $$

where \(\phi(\cdot)\) and \(\Phi(\cdot)\) denote the density and cumulative distribution functions for the standard normal distribution.

Arguments

x

vector of quantiles

n

number of observations

p

vector of probabilities

sigma

scale parameter for the distribution

lambda

shape parameter for the distribution

type

to distinguish the type of the utpn model: 1 (default) or 2.

log

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].

Author

Gallardo, D.I.

Details

Random generation is based on the inverse transformation method.

References

Gomez, H.J., Olmos, N.M., Varela, H., Bolfarine, H. (2018). Inference for a truncated positive normal distribution. Applied Mathemetical Journal of Chinese Universities, 33, 163-176.

Examples

dutpn(c(0.1,0.2), sigma=1, lambda=-1)
putpn(c(0.1,0.2), sigma=1, lambda=-1)
rutpn(n=10, sigma=1, lambda=-1)