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,
  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)

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-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. '

Value

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.

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

• 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.

See Also

Other distribution: dcomper_mv(), dcomper(), dnormhnorm()

Examples

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)

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