dnormexp {dsfa} | R Documentation |
Probablitiy density function, distribution, quantile function and random number generation for the normal-exponential distribution
dnormexp(
x,
mu = 0,
sigma_v = 1,
sigma_u = 1,
s = -1,
deriv_order = 0,
tri = NULL,
log.p = FALSE
)
pnormexp(
q,
mu = 0,
sigma_v = 1,
sigma_u = 1,
s = -1,
deriv_order = 0,
tri = NULL,
log.p = FALSE
)
qnormexp(p, mu = 0, sigma_v = 1, sigma_u = 1, s = -1, log.p = FALSE)
rnormexp(n, mu = 0, sigma_v = 1, sigma_u = 1, s = -1)
x |
numeric vector of quantiles. |
mu |
numeric vector of |
sigma_v |
numeric vector of |
sigma_u |
numeric vector of |
s |
integer; |
deriv_order |
integer; maximum order of derivative. Available are |
tri |
optional; index matrix for upper triangular, generated by |
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. |
A random variable X
follows a normal-exponential distribution if X = V + s \cdot U
, where V \sim N(\mu, \sigma_V^2)
and U \sim Exp(\sigma_u)
.
The density is given by
f_X(x)=\frac{\sigma_U}{2} \exp \{\sigma_U (s \mu) + \frac{1}{2} \sigma_U^2 \sigma_V^2-\sigma_U (s x) \} 2 \Phi(\frac{1}{\sigma_V} (-s \mu)-\sigma_U \sigma_V+\frac{1}{\sigma_V}(s x)) \qquad,
where s=-1
for production and s=1
for cost function. In the latter case the distribution is equivalent to the Exponentially modified Gaussian distribution.
dnormexp()
gives the density, pnormexp()
give the distribution function, qnormexp()
gives the quantile function, and rnormexp()
generates random numbers, with given parameters.
dnormexp()
and pnormexp()
return a derivs
object. For more details see trind()
and trind_generator()
.
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.
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()
,
dnormhnorm()
pdf <- dnormexp(x=5, mu=1, sigma_v=2, sigma_u=3, s=-1)
cdf <- pnormexp(q=5, mu=1, sigma_v=2, sigma_u=3, s=-1)
q <- qnormexp(p=seq(0.1, 0.9, by=0.1), mu=1, sigma_v=2, sigma_u=3, s=-1)
r <- rnormexp(n=10, mu=1, sigma_v=2, sigma_u=3, s=-1)