dnormhnorm {dsfa} R Documentation

## Normal-halfnormal distribution

### Description

Probablitiy density function, distribution, quantile function and random number generation for the normal-halfnormal distribution

### Usage

dnormhnorm(
x,
mu = 0,
sigma_v = 1,
sigma_u = 1,
s = -1,
deriv_order = 0,
tri = NULL,
log.p = FALSE
)

pnormhnorm(
q,
mu = 0,
sigma_v = 1,
sigma_u = 1,
s = -1,
deriv_order = 0,
tri = NULL,
log.p = FALSE
)

qnormhnorm(p, mu = 0, sigma_v = 1, sigma_u = 1, s = -1)

rnormhnorm(n, mu = 0, sigma_v = 1, sigma_u = 1, s = -1)


### Arguments

 x numeric vector of quantiles. mu numeric vector of \mu. sigma_v numeric vector of \sigma_V. Must be positive. sigma_u numeric vector of \sigma_U. Must be positive. s integer; s=-1 for production and s=1 for cost function. deriv_order integer; maximum order of derivative. Available are 0,2 and 4. tri optional; index matrix for upper triangular, generated by trind_generator. log.p logical; if TRUE, probabilities p are given as log(p). q numeric vector of quantiles. p numeric vector of probabilities. n positive integer; number of observations.

### Details

A random variable X follows a normal-halfnormal distribution if X = V + s \cdot U , where V \sim N(\mu, \sigma_V^2) and U \sim HN(\sigma_U^2). The density is given by

f_X(x)=\frac{1}{\sqrt{\sigma_V^2+\sigma_U^2}} \phi(\frac{x-\mu}{\sqrt{\sigma_V^2+\sigma_U^2}}) \Phi(s \frac{\sigma_U}{\sigma_V} \frac{x-\mu}{\sqrt{\sigma_V^2+\sigma_U^2}}) \qquad,

where s=-1 for production and s=1 for cost function.

### Value

dnormhnorm() gives the density, pnormhnorm() give the distribution function, qnormhnorm() gives the quantile function, and rnormhnorm() generates random numbers, with given parameters. dnormhnorm() and pnormhnorm() return a derivs object. For more details see trind and trind_generator.

### Functions

• pnormhnorm(): distribution function for the normal-halfnormal distribution.

• qnormhnorm(): quantile function for the normal-halfnormal distribution.

• rnormhnorm(): random number generation for the normal-halfnormal distribution.

### References

• Aigner D, Lovell CK, Schmidt P (1977). “Formulation and estimation of stochastic frontier production function models.” Journal of econometrics, 6(1), 21–37.

• Kumbhakar SC, Wang H, Horncastle AP (2015). A practitioner's guide to stochastic frontier analysis using Stata. Cambridge University Press.

• Schmidt R, Kneib T (2020). “Analytic expressions for the Cumulative Distribution Function of the Composed Error Term in Stochastic Frontier Analysis with Truncated Normal and Exponential Inefficiencies.” arXiv preprint arXiv:2006.03459.

• Gradshteyn IS, Ryzhik IM (2014). Table of integrals, series, and products. Academic press.

• Azzalini A (2013). The skew-normal and related families, volume 3. Cambridge University Press.

Other distribution: dcomper_mv(), dcomper(), dnormexp()
pdf <- dnormhnorm(x=5, mu=1, sigma_v=2, sigma_u=3, s=-1)