dnormexp {dsfa} R Documentation

## Normal-exponential distribution

### Description

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

### Usage

dnormexp(
x,
mu = 0,
sigma_v = 1,
lambda = 1,
s = -1,
deriv = 0,
tri = NULL,
log.p = FALSE,
check = TRUE
)

pnormexp(
q,
mu = 0,
sigma_v = 1,
lambda = 1,
s = -1,
deriv = 0,
tri = NULL,
lower.tail = TRUE,
log.p = FALSE,
check = TRUE
)

qnormexp(
p,
mu = 0,
sigma_v = 1,
lambda = 1,
s = -1,
lower.tail = TRUE,
log.p = FALSE,
check = TRUE
)

rnormexp(n, mu = 0, sigma_v = 1, lambda = 1, s = -1, check = TRUE)


### Arguments

 x vector of quantiles. mu vector of \mu sigma_v vector of \sigma_V. Must be positive. lambda vector of \lambda. Must be positive. s s=-1 for production and s=1 for cost function. deriv derivative of order deriv of the log density. Available are 0,2 and 4. tri optional, index arrays for upper triangular matrices, generated by trind.generator() and supplied to chainrule(). log.p logical; if TRUE, probabilities p are given as log(p). check logical; if TRUE, check inputs. q vector of probabilities. lower.tail logical; if TRUE (default), probabilities are P[X \le x] otherwise, P[X > x]. p vector of quantiles. n number of observations.

### Details

A random variable \mathcal{E} follows a normal-exponential distribution if \mathcal{E} = V + s \cdot U , where V \sim N(\mu, \sigma_V^2) and U \sim Exp(\lambda). The density is given by

f_\mathcal{E}(\epsilon)=\frac{\lambda}{2} \exp \{\lambda (s \mu) + \frac{1}{2} \lambda^2 \sigma_V^2-\lambda (s \epsilon) \} 2 \Phi(\frac{1}{\sigma_V} (-s \mu)-\lambda \sigma_V+\frac{1}{\sigma_V}(s \epsilon)) \qquad,

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

### Value

dnormexp gives the density, pnormexp give the distribution function, qnormexp gives the quantile function, and rnormexp generates random numbers, with given parameters. If the derivatives are calculated these are provided as the attributes gradient, hessian, l3 and l4 of the output of the density.

### Functions

• pnormexp(): distribution function for the normal-exponential distribution.

• qnormexp(): quantile function for the normal-exponential distribution.

• rnormexp(): random number generation for the normal-exponential distribution.

### References

• Meeusen W, van Den Broeck J (1977). “Efficiency estimation from Cobb-Douglas production functions with composed error.” International economic review, 435–444.

• 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.

### Examples

pdf <- dnormexp(x=seq(-3, 3, by=0.1), mu=1, sigma_v=2, lambda=1/3, s=1)
cdf <- pnormexp(q=seq(-3, 3, by=0.1), mu=1, sigma_v=2, lambda=1/3, s=1)
q <- qnormexp(p=seq(0.1, 0.9, by=0.1), mu=1, sigma_v=2, lambda=1/3, s=1)
r <- rnormexp(n=100, mu=1, sigma_v=2, lambda=1/3, s=1)



[Package dsfa version 1.0.1 Index]