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)