dcomper_mv {dsfa} R Documentation

## Multivariate Composed-Error distribution

### Description

Probablitiy density function, distribution, quantile function and random number generation for the multivariate composed-error distribution

### Usage

dcomper_mv(
x,
mu = matrix(c(0, 0), ncol = 2),
sigma_v = matrix(c(1, 1), ncol = 2),
sigma_u = matrix(c(1, 1), ncol = 2),
delta = matrix(0, nrow = 1),
s = c(-1, -1),
distr = c("normhnorm", "normhnorm", "normal"),
rot = 0,
deriv_order = 0,
tri = NULL,
log.p = FALSE
)

pcomper_mv(
q,
mu = matrix(c(0, 0), ncol = 2),
sigma_v = matrix(c(1, 1), ncol = 2),
sigma_u = matrix(c(1, 1), ncol = 2),
delta = 0,
s = c(-1, -1),
distr = c("normhnorm", "normhnorm", "normal"),
rot = 0,
deriv_order = 0,
tri = NULL,
log.p = FALSE
)

rcomper_mv(
n,
mu = matrix(c(0, 0), ncol = 2),
sigma_v = matrix(c(1, 1), ncol = 2),
sigma_u = matrix(c(1, 1), ncol = 2),
delta = matrix(0, nrow = 1),
s = c(-1, -1),
distr = c("normhnorm", "normhnorm", "normal"),
rot = 0
)


### Arguments

 x numeric matrix of quantiles. Must have two columns. mu numeric matrix of \mu. Must have two columns. sigma_v numeric matrix of \sigma_V. Must be positive and have two columns. sigma_u numeric matrix of \sigma_U. Must be positive and have two columns. delta numeric vector of copula parameter \delta. s integer vector of length two; each element corresponds to one marginal. distr string vector of length three; the first two elements determine the distribution of the marginals. Available are: 'normhnorm', Normal-halfnormal distribution 'normexp', Normal-exponential distribution The last element determines the distribution of the copula: 'independent', Independence copula 'normal', Gaussian copula 'clayton', Clayton copula 'gumbel', Gumbel copula 'frank', Frank copula 'joe', Joe copula 'amh', Ali-Mikhail-Haq copula rot integer determining the rotation for Archimedian copulas. Can be 90, 180 or 270. deriv_order integer; maximum order of derivative. Available are 0,2 and 4. tri optional; List of objects generated by [trind_generator()]. log.p logical; if TRUE, probabilities p are given as log(p). q numeric matrix of probabilities. n positive integer; number of observations.

### Details

A bivariate random vector (X_1,X_2)=\boldsymbol{X} follows a multivariate composed-errordistribution f_{X_1,X_2}(x_1,x_2), which can be rewritten using Sklars' theorem via a copula

f_{X_1,X_2}(y_1,y_2)=c(F_{X_1}(x_1),F_{X_2}(x_2),\delta) \cdot f_{X_1}(x_1) f_{X_2}(x_2) \qquad,

where c(\cdot) is the density of the copula and F_{X_m}(x_m),f_{X_m}(x_m) are the marginal cdfs and pdfs respectively for m \in \{1,2\}. \delta is the copula parameter.

### Value

dcomper_mv gives the density, pcomper_mv give the distribution function, and rcomper_mv generates random numbers, with given parameters. If the derivatives are calculated the output is a derivs object.

### Functions

• pcomper_mv(): distribution function for the multivariate composed-error distribution.

• rcomper_mv(): random number generation for the multivariate composed-error 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(), dnormexp(), dnormhnorm()

### Examples

pdf <- dcomper_mv(x=matrix(c(0,10),ncol=2), mu=matrix(c(1,2),ncol=2),
sigma_v=matrix(c(3,4),ncol=2), sigma_u=matrix(c(5,6),ncol=2),
delta=c(0.5), s=c(-1,-1), distr=c("normhnorm","normhnorm","normal"),
deriv=2 ,
tri=list(trind_generator(3),trind_generator(3),trind_generator(1),
trind_generator(6),trind_generator(7)),
log.p=TRUE)
cdf <- pcomper_mv(q=matrix(c(0,10),ncol=2), mu=matrix(c(1,2),ncol=2),
sigma_v=matrix(c(3,4),ncol=2), sigma_u=matrix(c(5,6),ncol=2),
delta=c(0.5), s=c(-1,-1), distr=c("normhnorm","normhnorm","normal"))
r <- rcomper_mv(n=10, mu=matrix(c(1,2),ncol=2),
sigma_v=matrix(c(3,4),ncol=2), sigma_u=matrix(c(5,6),ncol=2),
delta=c(0.5), s=c(-1,-1), distr=c("normhnorm","normhnorm","normal"))



[Package dsfa version 2.0.2 Index]