Nothing
R/internal_PIOS_galambos.R
In gofCopula: Goodness-of-Fit Tests for Copulae
Defines functions
.galambos.12.S
.galambos.12.V
.galambos.12.density
# Internal functions for the derivation of the galambos copula for the PIOSRn and
# PIOSTn test statistics. Called by .Rn and .Tn, the internal functions for
# their test statistics.
# 2d density
.galambos.12.density <- function(theta, u) {
if (is.null(dim(u))) {
u1 <- u[1]
u2 <- u[2]
} else {
u1 <- u[, 1]
u2 <- u[, 2]
}
lu1 <- -log(u1)
lu2 <- -log(u2)
lu1t <- (lu1^theta)
lu2t <- (lu2^theta)
(exp((((lu1^(-theta)) + (lu2^(-theta)))^(-1 / theta))) * (-(lu1 *
(((lu1^(-theta)) + (lu2^(-theta)))^(1 / theta)) * (lu1t + lu2t) *
(lu1t - lu2 * (((lu1^(-theta)) + (lu2^(-theta)))^(1 / theta)) *
(lu1t + lu2t))) + lu2t * (-((lu2^(1 + theta)) * (((lu1^(-theta)) +
(lu2^(-theta)))^(1 / theta))) + lu1t * (1 + (((lu1^(-theta)) +
(lu2^(-theta)))^(1 / theta)) * (1 - lu2 + theta))))) /
(lu1 * lu2 * (((lu1^(-theta)) + (lu2^(-theta)))^(2 / theta)) *
((lu1t + lu2t)^2))
}
# 2d first derivative wrt theta of the log-density
.galambos.12.V <- function(theta, u1, u2) {
lu1 <- -log(u1)
lu2 <- -log(u2)
plu1mt <- lu1^(-theta)
plu2mt <- lu2^(-theta)
plu12invt <- (plu1mt + plu2mt)^(1 / theta)
logplu <- log(plu1mt + plu2mt)
llu1 <- log(lu1)
llu2 <- log(lu2)
((lu2^(1 + 3 * theta)) * plu12invt * (-1 + (-1 + lu1) * plu12invt) *
(theta * llu1 + logplu) + (lu1^(1 + 3 * theta)) * plu12invt * (-1 +
(-1 + lu2) * plu12invt) * (theta * llu2 + logplu) + (lu1^(2 * theta)) *
(lu2^theta) * (theta * llu2 + logplu - plu12invt * (theta * (lu1 + theta) *
llu1 + (-3 + lu1 + lu2 - 2 * theta) * theta * llu2 + (-3 + 2 * lu1 + lu2 -
theta) * logplu) + (plu12invt^2) * (theta * (theta + (-1 + lu2 - theta) *
(lu1 + theta) * llu1 + ((1 + theta) * (1 - lu2 + theta) + lu1 * (-1 + 2 *
lu2 + theta)) * llu2) + (1 - lu2 + lu1 * (-2 + 3 * lu2) + theta) *
logplu)) + (lu1^theta) * (lu2^(2 * theta)) * (theta * llu1 + logplu +
plu12invt * (theta * (3 - lu1 - lu2 + 2 * theta) * llu1 - theta *
(lu2 + theta) * llu2 + (3 - lu1 - 2 * lu2 + theta) * logplu) +
(plu12invt^2) * (theta * (theta + (lu1 * (-1 + 2 * lu2 - theta) +
lu2 * (-1 + theta) + ((1 + theta)^2)) * llu1 + (-1 + lu1 - theta) *
(lu2 + theta) * llu2) + (1 - 2 * lu2 + lu1 * (-1 + 3 * lu2) + theta) *
logplu))) / (plu12invt * ((lu1^theta) + (lu2^theta)) * ((lu2^(1 + 2 *
theta)) * plu12invt * (-1 + lu1 * plu12invt) + (lu1^(1 + 2 * theta)) *
plu12invt * (-1 + lu2 * plu12invt) + (lu1^theta) * (lu2^theta) *
(1 + 2 * lu1 * lu2 * (plu12invt^2) - plu12invt *
(-1 + lu1 + lu2 - theta))) * (theta^2))
}
# 2d second derivative wrt theta of the log-density
.galambos.12.S <- function(theta, u1, u2) {
lu1 <- -log(u1)
lu2 <- -log(u2)
plu1mt <- lu1^(-theta)
plu2mt <- lu2^(-theta)
lu1t <- lu1^theta
lu2t <- lu2^theta
plu12invt <- (plu1mt + plu2mt)^(1 / theta)
logplu <- log(plu1mt + plu2mt)
llu1 <- log(lu1)
llu2 <- log(lu2)
t2 <- theta^2
tp12 <- (1 + theta)^2
lu12t <- lu1^(1 + 2 * theta)
lu13t <- lu1^(1 + 3 * theta)
lu22t <- lu2^(1 + 2 * theta)
lu23t <- lu2^(1 + 3 * theta)
aaa <- lu2t * theta * llu1 + lu1t * theta * llu2 + (lu1t + lu2t) * logplu
(-2 * (lu1t + lu2t) * (lu22t * plu12invt * (-1 + lu1 * plu12invt) +
lu12t * plu12invt * (-1 + lu2 * plu12invt) + lu1t * lu2t * (1 + 2 *
lu1 * lu2 * (plu12invt^2) - plu12invt * (-1 + lu1 + lu2 - theta))) *
(lu23t * plu12invt * (-1 + (-1 + lu1) * plu12invt) * (theta * llu1 +
logplu) + lu13t * plu12invt * (-1 + (-1 + lu2) * plu12invt) *
(theta * llu2 + logplu) + (lu1^(2 * theta)) * lu2t * (theta * llu2 +
logplu - plu12invt * (theta * (lu1 + theta) * llu1 + (-3 + lu1 + lu2 -
2 * theta) * theta * llu2 + (-3 + 2 * lu1 + lu2 - theta) * logplu) +
(plu12invt^2) * (theta * (theta + (-1 + lu2 - theta) * (lu1 + theta) *
llu1 + ((1 + theta) * (1 - lu2 + theta) + lu1 * (-1 + 2 * lu2 + theta)) *
llu2) + (1 - lu2 + lu1 * (-2 + 3 * lu2) + theta) * logplu)) + lu1t *
(lu2^(2 * theta)) * (theta * llu1 + logplu + plu12invt * (theta *
(3 - lu1 - lu2 + 2 * theta) * llu1 - theta * (lu2 + theta) * llu2 +
(3 - lu1 - 2 * lu2 + theta) * logplu) + (plu12invt^2) * (theta *
(theta + (lu1 * (-1 + 2 * lu2 - theta) + lu2 * (-1 + theta) + tp12) *
llu1 + (-1 + lu1 - theta) * (lu2 + theta) * llu2) + (1 - 2 * lu2 + lu1 *
(-1 + 3 * lu2) + theta) * logplu))) - (lu22t * plu12invt *
(-1 + lu1 * plu12invt) + lu12t * plu12invt * (-1 + lu2 * plu12invt) +
lu1t * lu2t * (1 + 2 * lu1 * lu2 * (plu12invt^2) - plu12invt *
(-1 + lu1 + lu2 - theta))) * theta * (lu1t * llu1 + lu2t * llu2) *
(lu23t * plu12invt * (-1 + (-1 + lu1) * plu12invt) * (theta * llu1 +
logplu) + lu13t * plu12invt * (-1 + (-1 + lu2) * plu12invt) *
(theta * llu2 + logplu) + (lu1^(2 * theta)) * lu2t * (theta * llu2 +
logplu - plu12invt * (theta * (lu1 + theta) * llu1 + (-3 + lu1 + lu2 - 2 *
theta) * theta * llu2 + (-3 + 2 * lu1 + lu2 - theta) * logplu) +
(plu12invt^2) * (theta * (theta + (-1 + lu2 - theta) *
(lu1 + theta) * llu1 + ((1 + theta) * (1 - lu2 + theta) + lu1 *
(-1 + 2 * lu2 + theta)) * llu2) + (1 - lu2 + lu1 * (-2 + 3 * lu2) +
theta) * logplu)) + lu1t * (lu2^(2 * theta)) * (theta * llu1 + logplu +
plu12invt * (theta * (3 - lu1 - lu2 + 2 * theta) * llu1 - theta *
(lu2 + theta) * llu2 + (3 - lu1 - 2 * lu2 + theta) * logplu) +
(plu12invt^2) * (theta * (theta + (lu1 * (-1 + 2 * lu2 - theta) + lu2 *
(-1 + theta) + ((1 + theta)^2)) * llu1 + (-1 + lu1 - theta) *
(lu2 + theta) * llu2) + (1 - 2 * lu2 + lu1 * (-1 + 3 * lu2) + theta) *
logplu))) + ((lu22t * plu12invt * (-1 + lu1 * plu12invt) + lu12t *
plu12invt * (-1 + lu2 * plu12invt) + lu1t * lu2t * (1 + 2 * lu1 * lu2 *
(plu12invt^2) - plu12invt * (-1 + lu1 + lu2 - theta))) * aaa * (lu23t *
plu12invt * (-1 + (-1 + lu1) * plu12invt) * (theta * llu1 + logplu) +
lu13t * plu12invt * (-1 + (-1 + lu2) * plu12invt) * (theta * llu2 +
logplu) + (lu1^(2 * theta)) * lu2t * (theta * llu2 + logplu - plu12invt *
(theta * (lu1 + theta) * llu1 + (-3 + lu1 + lu2 - 2 * theta) * theta *
llu2 + (-3 + 2 * lu1 + lu2 - theta) * logplu) + (plu12invt^2) *
(theta * (theta + (-1 + lu2 - theta) * (lu1 + theta) * llu1 +
((1 + theta) * (1 - lu2 + theta) + lu1 * (-1 + 2 * lu2 + theta)) * llu2) +
(1 - lu2 + lu1 * (-2 + 3 * lu2) + theta) * logplu)) + lu1t *
(lu2^(2 * theta)) * (theta * llu1 + logplu + plu12invt * (theta *
(3 - lu1 - lu2 + 2 * theta) * llu1 - theta * (lu2 + theta) * llu2 +
(3 - lu1 - 2 * lu2 + theta) * logplu) + (plu12invt^2) * (theta *
(theta + (lu1 * (-1 + 2 * lu2 - theta) + lu2 * (-1 + theta) + tp12) *
llu1 + (-1 + lu1 - theta) * (lu2 + theta) * llu2) + (1 - 2 * lu2 + lu1 *
(-1 + 3 * lu2) + theta) * logplu)))) / theta - ((2 * lu12t * plu12invt *
(lu1t + lu2t) * (-1 + lu2 * plu12invt) * t2 * llu1 + lu1t * lu2t *
(lu1t + lu2t) * (1 + 2 * lu1 * lu2 * (plu12invt^2) - plu12invt *
(-1 + lu1 + lu2 - theta)) * t2 * llu1 + 2 * lu22t * plu12invt *
(lu1t + lu2t) * (-1 + lu1 * plu12invt) * t2 * llu2 + lu1t * lu2t *
(lu1t + lu2t) * (1 + 2 * lu1 * lu2 * (plu12invt^2) - plu12invt *
(-1 + lu1 + lu2 - theta)) * t2 * llu2 - lu12t * lu2 * (plu12invt^2) *
(lu2t * theta * llu1 + lu1t * theta * llu2 + (lu1t + lu2t) * logplu) -
lu1 * lu22t * (plu12invt^2) * aaa + lu22t * plu12invt *
(1 - lu1 * plu12invt) * aaa + lu12t * plu12invt * (1 - lu2 * plu12invt) *
aaa + lu1t * lu2t * plu12invt * ((lu1t + lu2t) * t2 - 4 * lu1 * lu2 *
plu12invt * aaa - (1 - lu1 - lu2 + theta) * (lu2t * theta * llu1 + lu1t *
theta * llu2 + (lu1t + lu2t) * logplu))) * (lu23t * plu12invt *
(-1 + (-1 + lu1) * plu12invt) * (theta * llu1 + logplu) + lu13t *
plu12invt * (-1 + (-1 + lu2) * plu12invt) * (theta * llu2 + logplu) +
(lu1^(2 * theta)) * lu2t * (theta * llu2 + logplu - plu12invt *
(theta * (lu1 + theta) * llu1 + (-3 + lu1 + lu2 - 2 * theta) * theta *
llu2 + (-3 + 2 * lu1 + lu2 - theta) * logplu) + (plu12invt^2) *
(theta * (theta + (-1 + lu2 - theta) * (lu1 + theta) * llu1 +
((1 + theta) * (1 - lu2 + theta) + lu1 * (-1 + 2 * lu2 + theta)) * llu2) +
(1 - lu2 + lu1 * (-2 + 3 * lu2) + theta) * logplu)) + lu1t *
(lu2^(2 * theta)) * (theta * llu1 + logplu + plu12invt *
(theta * (3 - lu1 - lu2 + 2 * theta) * llu1 - theta * (lu2 + theta) *
llu2 + (3 - lu1 - 2 * lu2 + theta) * logplu) + (plu12invt^2) *
(theta * (theta + (lu1 * (-1 + 2 * lu2 - theta) + lu2 * (-1 + theta) +
tp12) * llu1 + (-1 + lu1 - theta) * (lu2 + theta) * llu2) +
(1 - 2 * lu2 + lu1 * (-1 + 3 * lu2) + theta) * logplu)))) /
theta + (lu1t + lu2t) * (lu22t * plu12invt * (-1 + lu1 * plu12invt) +
lu12t * plu12invt * (-1 + lu2 * plu12invt) + lu1t * lu2t * (1 + 2 *
lu1 * lu2 * (plu12invt^2) - plu12invt * (-1 + lu1 + lu2 - theta))) *
theta * ((lu1t * lu23t * plu12invt * (-1 - plu12invt + lu1 * plu12invt) *
(llu1 - llu2)) / (lu1t + lu2t) + (lu13t * lu2t *
plu12invt * (-1 - plu12invt + lu2 * plu12invt) * (-llu1 + llu2)) /
(lu1t + lu2t) + 3 * lu23t * plu12invt * (-1 + (-1 + lu1) * plu12invt) *
llu2 * (theta * llu1 + logplu) + 3 * lu13t * plu12invt * (-1 + (-1 + lu2) *
plu12invt) * llu1 * (theta * llu2 + logplu) - ((-1 + lu1) * lu23t *
(plu12invt^2) * (theta * llu1 + logplu) * aaa) /
((lu1t + lu2t) * t2) - (lu23t * plu12invt * (-1 + (-1 + lu1) * plu12invt) *
(theta * llu1 + logplu) * (lu2t * theta * llu1 + lu1t * theta * llu2 +
(lu1t + lu2t) * logplu)) / ((lu1t + lu2t) * t2) - (lu13t * (-1 + lu2) *
(plu12invt^2) * (theta * llu2 + logplu) * (lu2t * theta * llu1 + lu1t *
theta * llu2 + (lu1t + lu2t) * logplu)) / ((lu1t + lu2t) * t2) - (lu13t *
plu12invt * (-1 + (-1 + lu2) * plu12invt) * (theta * llu2 + logplu) *
aaa) / ((lu1t + lu2t) * t2) + 2 * (lu1^(2 * theta)) * lu2t * llu1 *
(theta * llu2 + logplu - plu12invt * (theta * (lu1 + theta) * llu1 +
(-3 + lu1 + lu2 - 2 * theta) * theta * llu2 + (-3 + 2 * lu1 + lu2 -
theta) * logplu) + (plu12invt^2) * (theta * (theta + (-1 + lu2 - theta) *
(lu1 + theta) * llu1 + ((1 + theta) * (1 - lu2 + theta) + lu1 *
(-1 + 2 * lu2 + theta)) * llu2) + (1 - lu2 + lu1 * (-2 + 3 * lu2) +
theta) * logplu)) + (lu1^(2 * theta)) * lu2t * llu2 * (theta * llu2 +
logplu - plu12invt * (theta * (lu1 + theta) * llu1 + (-3 + lu1 + lu2 - 2 *
theta) * theta * llu2 + (-3 + 2 * lu1 + lu2 - theta) * logplu) +
(plu12invt^2) * (theta * (theta + (-1 + lu2 - theta) * (lu1 + theta) *
llu1 + ((1 + theta) * (1 - lu2 + theta) + lu1 * (-1 + 2 * lu2 + theta)) *
llu2) + (1 - lu2 + lu1 * (-2 + 3 * lu2) + theta) * logplu)) +
(lu1^(2 * theta)) * lu2t * (llu2 - (lu2t * llu1 + lu1t * llu2) /
(lu1t + lu2t) + plu12invt * (-(theta * llu1) - (lu1 + theta) * llu1 -
(-3 + lu1 + lu2 - 2 * theta) * llu2 + 2 * theta * llu2 + ((-3 + 2 * lu1 +
lu2 - theta) * (lu2t * llu1 + lu1t * llu2)) /
(lu1t + lu2t) + logplu) + (plu12invt^2) * (theta + (-1 + lu2 - theta) *
(lu1 + theta) * llu1 + ((1 + theta) * (1 - lu2 + theta) + lu1 * (-1 + 2 *
lu2 + theta)) * llu2 - ((1 - lu2 + lu1 * (-2 + 3 * lu2) + theta) *
(lu2t * llu1 + lu1t * llu2)) /
(lu1t + lu2t) + theta * (1 - (1 + lu1 - lu2 + 2 * theta) * llu1 +
(2 + lu1 - lu2 + 2 * theta) * llu2) + logplu) + (plu12invt *
aaa * (theta * (lu1 + theta) * llu1 + (-3 + lu1 + lu2 - 2 * theta) *
theta * llu2 + (-3 + 2 * lu1 + lu2 - theta) * logplu)) /
((lu1t + lu2t) * t2) - (2 * (plu12invt^2) * aaa * (theta * (theta +
(-1 + lu2 - theta) * (lu1 + theta) * llu1 + ((1 + theta) *
(1 - lu2 + theta) + lu1 * (-1 + 2 * lu2 + theta)) * llu2) +
(1 - lu2 + lu1 * (-2 + 3 * lu2) + theta) * logplu)) /
((lu1t + lu2t) * t2)) + lu1t * (lu2^(2 * theta)) * llu1 *
(theta * llu1 + logplu + plu12invt * (theta * (3 - lu1 - lu2 + 2 * theta) *
llu1 - theta * (lu2 + theta) * llu2 + (3 - lu1 - 2 * lu2 + theta) *
logplu) + (plu12invt^2) * (theta * (theta + (lu1 * (-1 + 2 * lu2 - theta) +
lu2 * (-1 + theta) + tp12) * llu1 + (-1 + lu1 - theta) *
(lu2 + theta) * llu2) + (1 - 2 * lu2 + lu1 * (-1 + 3 * lu2) + theta) *
logplu)) + 2 * lu1t * (lu2^(2 * theta)) * llu2 * (theta * llu1 +
logplu + plu12invt * (theta * (3 - lu1 - lu2 + 2 * theta) * llu1 - theta *
(lu2 + theta) * llu2 + (3 - lu1 - 2 * lu2 + theta) * logplu) +
(plu12invt^2) * (theta * (theta + (lu1 * (-1 + 2 * lu2 - theta) + lu2 *
(-1 + theta) + tp12) * llu1 + (-1 + lu1 - theta) * (lu2 + theta) * llu2) +
(1 - 2 * lu2 + lu1 * (-1 + 3 * lu2) + theta) * logplu)) + lu1t *
(lu2^(2 * theta)) * (llu1 - (lu2t * llu1 + lu1t * llu2) / (lu1t + lu2t) +
(plu12invt^2) * (theta + (lu1 * (-1 + 2 * lu2 - theta) + lu2 *
(-1 + theta) + tp12) * llu1 + (-1 + lu1 - theta) * (lu2 + theta) * llu2 -
((1 - 2 * lu2 + lu1 * (-1 + 3 * lu2) + theta) *
(lu2t * llu1 + lu1t * llu2)) / (lu1t + lu2t) +
theta * (1 + (2 - lu1 + lu2 + 2 * theta) * llu1 + (-1 + lu1 - lu2 -
2 * theta) * llu2) + logplu) + (plu12invt * ((-(lu1^(1 + theta)) + lu2t *
(lu2 + 3 * theta) + lu1t * (3 - lu2 + 4 * theta)) * llu1 +
((lu1^(1 + theta)) + lu1t * (-3 + lu2 - 3 * theta) - lu2t * (lu2 + 2 *
theta)) * llu2 + (lu1t + lu2t) * logplu)) /
(lu1t + lu2t) - (plu12invt * aaa * (theta * (3 - lu1 - lu2 + 2 * theta) *
llu1 - theta * (lu2 + theta) * llu2 + (3 - lu1 - 2 * lu2 + theta) *
logplu)) / ((lu1t + lu2t) * t2) - (2 * (plu12invt^2) *
aaa * (theta * (theta + (lu1 * (-1 + 2 * lu2 - theta) + lu2 *
(-1 + theta) + ((1 + theta)^2)) * llu1 + (-1 + lu1 - theta) *
(lu2 + theta) * llu2) + (1 - 2 * lu2 + lu1 * (-1 + 3 * lu2) + theta) *
logplu)) / ((lu1t + lu2t) * t2)))) / (plu12invt *
((lu1t + lu2t)^2) * ((lu22t * plu12invt * (-1 + lu1 * plu12invt) +
lu12t * plu12invt * (-1 + lu2 * plu12invt) + lu1t * lu2t *
(1 + 2 * lu1 * lu2 * (plu12invt^2) - plu12invt *
(-1 + lu1 + lu2 - theta)))^2) * (theta^3))
}
Try the gofCopula package in your browser
Any scripts or data that you put into this service are public.
gofCopula documentation built on April 22, 2021, 5:10 p.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Defines functions .galambos.12.S .galambos.12.V .galambos.12.density
# Internal functions for the derivation of the galambos copula for the PIOSRn and
# PIOSTn test statistics. Called by .Rn and .Tn, the internal functions for
# their test statistics.
# 2d density
.galambos.12.density <- function(theta, u) {
if (is.null(dim(u))) {
u1 <- u[1]
u2 <- u[2]
} else {
u1 <- u[, 1]
u2 <- u[, 2]
}
lu1 <- -log(u1)
lu2 <- -log(u2)
lu1t <- (lu1^theta)
lu2t <- (lu2^theta)
(exp((((lu1^(-theta)) + (lu2^(-theta)))^(-1 / theta))) * (-(lu1 *
(((lu1^(-theta)) + (lu2^(-theta)))^(1 / theta)) * (lu1t + lu2t) *
(lu1t - lu2 * (((lu1^(-theta)) + (lu2^(-theta)))^(1 / theta)) *
(lu1t + lu2t))) + lu2t * (-((lu2^(1 + theta)) * (((lu1^(-theta)) +
(lu2^(-theta)))^(1 / theta))) + lu1t * (1 + (((lu1^(-theta)) +
(lu2^(-theta)))^(1 / theta)) * (1 - lu2 + theta))))) /
(lu1 * lu2 * (((lu1^(-theta)) + (lu2^(-theta)))^(2 / theta)) *
((lu1t + lu2t)^2))
}
# 2d first derivative wrt theta of the log-density
.galambos.12.V <- function(theta, u1, u2) {
lu1 <- -log(u1)
lu2 <- -log(u2)
plu1mt <- lu1^(-theta)
plu2mt <- lu2^(-theta)
plu12invt <- (plu1mt + plu2mt)^(1 / theta)
logplu <- log(plu1mt + plu2mt)
llu1 <- log(lu1)
llu2 <- log(lu2)
((lu2^(1 + 3 * theta)) * plu12invt * (-1 + (-1 + lu1) * plu12invt) *
(theta * llu1 + logplu) + (lu1^(1 + 3 * theta)) * plu12invt * (-1 +
(-1 + lu2) * plu12invt) * (theta * llu2 + logplu) + (lu1^(2 * theta)) *
(lu2^theta) * (theta * llu2 + logplu - plu12invt * (theta * (lu1 + theta) *
llu1 + (-3 + lu1 + lu2 - 2 * theta) * theta * llu2 + (-3 + 2 * lu1 + lu2 -
theta) * logplu) + (plu12invt^2) * (theta * (theta + (-1 + lu2 - theta) *
(lu1 + theta) * llu1 + ((1 + theta) * (1 - lu2 + theta) + lu1 * (-1 + 2 *
lu2 + theta)) * llu2) + (1 - lu2 + lu1 * (-2 + 3 * lu2) + theta) *
logplu)) + (lu1^theta) * (lu2^(2 * theta)) * (theta * llu1 + logplu +
plu12invt * (theta * (3 - lu1 - lu2 + 2 * theta) * llu1 - theta *
(lu2 + theta) * llu2 + (3 - lu1 - 2 * lu2 + theta) * logplu) +
(plu12invt^2) * (theta * (theta + (lu1 * (-1 + 2 * lu2 - theta) +
lu2 * (-1 + theta) + ((1 + theta)^2)) * llu1 + (-1 + lu1 - theta) *
(lu2 + theta) * llu2) + (1 - 2 * lu2 + lu1 * (-1 + 3 * lu2) + theta) *
logplu))) / (plu12invt * ((lu1^theta) + (lu2^theta)) * ((lu2^(1 + 2 *
theta)) * plu12invt * (-1 + lu1 * plu12invt) + (lu1^(1 + 2 * theta)) *
plu12invt * (-1 + lu2 * plu12invt) + (lu1^theta) * (lu2^theta) *
(1 + 2 * lu1 * lu2 * (plu12invt^2) - plu12invt *
(-1 + lu1 + lu2 - theta))) * (theta^2))
}
# 2d second derivative wrt theta of the log-density
.galambos.12.S <- function(theta, u1, u2) {
lu1 <- -log(u1)
lu2 <- -log(u2)
plu1mt <- lu1^(-theta)
plu2mt <- lu2^(-theta)
lu1t <- lu1^theta
lu2t <- lu2^theta
plu12invt <- (plu1mt + plu2mt)^(1 / theta)
logplu <- log(plu1mt + plu2mt)
llu1 <- log(lu1)
llu2 <- log(lu2)
t2 <- theta^2
tp12 <- (1 + theta)^2
lu12t <- lu1^(1 + 2 * theta)
lu13t <- lu1^(1 + 3 * theta)
lu22t <- lu2^(1 + 2 * theta)
lu23t <- lu2^(1 + 3 * theta)
aaa <- lu2t * theta * llu1 + lu1t * theta * llu2 + (lu1t + lu2t) * logplu
(-2 * (lu1t + lu2t) * (lu22t * plu12invt * (-1 + lu1 * plu12invt) +
lu12t * plu12invt * (-1 + lu2 * plu12invt) + lu1t * lu2t * (1 + 2 *
lu1 * lu2 * (plu12invt^2) - plu12invt * (-1 + lu1 + lu2 - theta))) *
(lu23t * plu12invt * (-1 + (-1 + lu1) * plu12invt) * (theta * llu1 +
logplu) + lu13t * plu12invt * (-1 + (-1 + lu2) * plu12invt) *
(theta * llu2 + logplu) + (lu1^(2 * theta)) * lu2t * (theta * llu2 +
logplu - plu12invt * (theta * (lu1 + theta) * llu1 + (-3 + lu1 + lu2 -
2 * theta) * theta * llu2 + (-3 + 2 * lu1 + lu2 - theta) * logplu) +
(plu12invt^2) * (theta * (theta + (-1 + lu2 - theta) * (lu1 + theta) *
llu1 + ((1 + theta) * (1 - lu2 + theta) + lu1 * (-1 + 2 * lu2 + theta)) *
llu2) + (1 - lu2 + lu1 * (-2 + 3 * lu2) + theta) * logplu)) + lu1t *
(lu2^(2 * theta)) * (theta * llu1 + logplu + plu12invt * (theta *
(3 - lu1 - lu2 + 2 * theta) * llu1 - theta * (lu2 + theta) * llu2 +
(3 - lu1 - 2 * lu2 + theta) * logplu) + (plu12invt^2) * (theta *
(theta + (lu1 * (-1 + 2 * lu2 - theta) + lu2 * (-1 + theta) + tp12) *
llu1 + (-1 + lu1 - theta) * (lu2 + theta) * llu2) + (1 - 2 * lu2 + lu1 *
(-1 + 3 * lu2) + theta) * logplu))) - (lu22t * plu12invt *
(-1 + lu1 * plu12invt) + lu12t * plu12invt * (-1 + lu2 * plu12invt) +
lu1t * lu2t * (1 + 2 * lu1 * lu2 * (plu12invt^2) - plu12invt *
(-1 + lu1 + lu2 - theta))) * theta * (lu1t * llu1 + lu2t * llu2) *
(lu23t * plu12invt * (-1 + (-1 + lu1) * plu12invt) * (theta * llu1 +
logplu) + lu13t * plu12invt * (-1 + (-1 + lu2) * plu12invt) *
(theta * llu2 + logplu) + (lu1^(2 * theta)) * lu2t * (theta * llu2 +
logplu - plu12invt * (theta * (lu1 + theta) * llu1 + (-3 + lu1 + lu2 - 2 *
theta) * theta * llu2 + (-3 + 2 * lu1 + lu2 - theta) * logplu) +
(plu12invt^2) * (theta * (theta + (-1 + lu2 - theta) *
(lu1 + theta) * llu1 + ((1 + theta) * (1 - lu2 + theta) + lu1 *
(-1 + 2 * lu2 + theta)) * llu2) + (1 - lu2 + lu1 * (-2 + 3 * lu2) +
theta) * logplu)) + lu1t * (lu2^(2 * theta)) * (theta * llu1 + logplu +
plu12invt * (theta * (3 - lu1 - lu2 + 2 * theta) * llu1 - theta *
(lu2 + theta) * llu2 + (3 - lu1 - 2 * lu2 + theta) * logplu) +
(plu12invt^2) * (theta * (theta + (lu1 * (-1 + 2 * lu2 - theta) + lu2 *
(-1 + theta) + ((1 + theta)^2)) * llu1 + (-1 + lu1 - theta) *
(lu2 + theta) * llu2) + (1 - 2 * lu2 + lu1 * (-1 + 3 * lu2) + theta) *
logplu))) + ((lu22t * plu12invt * (-1 + lu1 * plu12invt) + lu12t *
plu12invt * (-1 + lu2 * plu12invt) + lu1t * lu2t * (1 + 2 * lu1 * lu2 *
(plu12invt^2) - plu12invt * (-1 + lu1 + lu2 - theta))) * aaa * (lu23t *
plu12invt * (-1 + (-1 + lu1) * plu12invt) * (theta * llu1 + logplu) +
lu13t * plu12invt * (-1 + (-1 + lu2) * plu12invt) * (theta * llu2 +
logplu) + (lu1^(2 * theta)) * lu2t * (theta * llu2 + logplu - plu12invt *
(theta * (lu1 + theta) * llu1 + (-3 + lu1 + lu2 - 2 * theta) * theta *
llu2 + (-3 + 2 * lu1 + lu2 - theta) * logplu) + (plu12invt^2) *
(theta * (theta + (-1 + lu2 - theta) * (lu1 + theta) * llu1 +
((1 + theta) * (1 - lu2 + theta) + lu1 * (-1 + 2 * lu2 + theta)) * llu2) +
(1 - lu2 + lu1 * (-2 + 3 * lu2) + theta) * logplu)) + lu1t *
(lu2^(2 * theta)) * (theta * llu1 + logplu + plu12invt * (theta *
(3 - lu1 - lu2 + 2 * theta) * llu1 - theta * (lu2 + theta) * llu2 +
(3 - lu1 - 2 * lu2 + theta) * logplu) + (plu12invt^2) * (theta *
(theta + (lu1 * (-1 + 2 * lu2 - theta) + lu2 * (-1 + theta) + tp12) *
llu1 + (-1 + lu1 - theta) * (lu2 + theta) * llu2) + (1 - 2 * lu2 + lu1 *
(-1 + 3 * lu2) + theta) * logplu)))) / theta - ((2 * lu12t * plu12invt *
(lu1t + lu2t) * (-1 + lu2 * plu12invt) * t2 * llu1 + lu1t * lu2t *
(lu1t + lu2t) * (1 + 2 * lu1 * lu2 * (plu12invt^2) - plu12invt *
(-1 + lu1 + lu2 - theta)) * t2 * llu1 + 2 * lu22t * plu12invt *
(lu1t + lu2t) * (-1 + lu1 * plu12invt) * t2 * llu2 + lu1t * lu2t *
(lu1t + lu2t) * (1 + 2 * lu1 * lu2 * (plu12invt^2) - plu12invt *
(-1 + lu1 + lu2 - theta)) * t2 * llu2 - lu12t * lu2 * (plu12invt^2) *
(lu2t * theta * llu1 + lu1t * theta * llu2 + (lu1t + lu2t) * logplu) -
lu1 * lu22t * (plu12invt^2) * aaa + lu22t * plu12invt *
(1 - lu1 * plu12invt) * aaa + lu12t * plu12invt * (1 - lu2 * plu12invt) *
aaa + lu1t * lu2t * plu12invt * ((lu1t + lu2t) * t2 - 4 * lu1 * lu2 *
plu12invt * aaa - (1 - lu1 - lu2 + theta) * (lu2t * theta * llu1 + lu1t *
theta * llu2 + (lu1t + lu2t) * logplu))) * (lu23t * plu12invt *
(-1 + (-1 + lu1) * plu12invt) * (theta * llu1 + logplu) + lu13t *
plu12invt * (-1 + (-1 + lu2) * plu12invt) * (theta * llu2 + logplu) +
(lu1^(2 * theta)) * lu2t * (theta * llu2 + logplu - plu12invt *
(theta * (lu1 + theta) * llu1 + (-3 + lu1 + lu2 - 2 * theta) * theta *
llu2 + (-3 + 2 * lu1 + lu2 - theta) * logplu) + (plu12invt^2) *
(theta * (theta + (-1 + lu2 - theta) * (lu1 + theta) * llu1 +
((1 + theta) * (1 - lu2 + theta) + lu1 * (-1 + 2 * lu2 + theta)) * llu2) +
(1 - lu2 + lu1 * (-2 + 3 * lu2) + theta) * logplu)) + lu1t *
(lu2^(2 * theta)) * (theta * llu1 + logplu + plu12invt *
(theta * (3 - lu1 - lu2 + 2 * theta) * llu1 - theta * (lu2 + theta) *
llu2 + (3 - lu1 - 2 * lu2 + theta) * logplu) + (plu12invt^2) *
(theta * (theta + (lu1 * (-1 + 2 * lu2 - theta) + lu2 * (-1 + theta) +
tp12) * llu1 + (-1 + lu1 - theta) * (lu2 + theta) * llu2) +
(1 - 2 * lu2 + lu1 * (-1 + 3 * lu2) + theta) * logplu)))) /
theta + (lu1t + lu2t) * (lu22t * plu12invt * (-1 + lu1 * plu12invt) +
lu12t * plu12invt * (-1 + lu2 * plu12invt) + lu1t * lu2t * (1 + 2 *
lu1 * lu2 * (plu12invt^2) - plu12invt * (-1 + lu1 + lu2 - theta))) *
theta * ((lu1t * lu23t * plu12invt * (-1 - plu12invt + lu1 * plu12invt) *
(llu1 - llu2)) / (lu1t + lu2t) + (lu13t * lu2t *
plu12invt * (-1 - plu12invt + lu2 * plu12invt) * (-llu1 + llu2)) /
(lu1t + lu2t) + 3 * lu23t * plu12invt * (-1 + (-1 + lu1) * plu12invt) *
llu2 * (theta * llu1 + logplu) + 3 * lu13t * plu12invt * (-1 + (-1 + lu2) *
plu12invt) * llu1 * (theta * llu2 + logplu) - ((-1 + lu1) * lu23t *
(plu12invt^2) * (theta * llu1 + logplu) * aaa) /
((lu1t + lu2t) * t2) - (lu23t * plu12invt * (-1 + (-1 + lu1) * plu12invt) *
(theta * llu1 + logplu) * (lu2t * theta * llu1 + lu1t * theta * llu2 +
(lu1t + lu2t) * logplu)) / ((lu1t + lu2t) * t2) - (lu13t * (-1 + lu2) *
(plu12invt^2) * (theta * llu2 + logplu) * (lu2t * theta * llu1 + lu1t *
theta * llu2 + (lu1t + lu2t) * logplu)) / ((lu1t + lu2t) * t2) - (lu13t *
plu12invt * (-1 + (-1 + lu2) * plu12invt) * (theta * llu2 + logplu) *
aaa) / ((lu1t + lu2t) * t2) + 2 * (lu1^(2 * theta)) * lu2t * llu1 *
(theta * llu2 + logplu - plu12invt * (theta * (lu1 + theta) * llu1 +
(-3 + lu1 + lu2 - 2 * theta) * theta * llu2 + (-3 + 2 * lu1 + lu2 -
theta) * logplu) + (plu12invt^2) * (theta * (theta + (-1 + lu2 - theta) *
(lu1 + theta) * llu1 + ((1 + theta) * (1 - lu2 + theta) + lu1 *
(-1 + 2 * lu2 + theta)) * llu2) + (1 - lu2 + lu1 * (-2 + 3 * lu2) +
theta) * logplu)) + (lu1^(2 * theta)) * lu2t * llu2 * (theta * llu2 +
logplu - plu12invt * (theta * (lu1 + theta) * llu1 + (-3 + lu1 + lu2 - 2 *
theta) * theta * llu2 + (-3 + 2 * lu1 + lu2 - theta) * logplu) +
(plu12invt^2) * (theta * (theta + (-1 + lu2 - theta) * (lu1 + theta) *
llu1 + ((1 + theta) * (1 - lu2 + theta) + lu1 * (-1 + 2 * lu2 + theta)) *
llu2) + (1 - lu2 + lu1 * (-2 + 3 * lu2) + theta) * logplu)) +
(lu1^(2 * theta)) * lu2t * (llu2 - (lu2t * llu1 + lu1t * llu2) /
(lu1t + lu2t) + plu12invt * (-(theta * llu1) - (lu1 + theta) * llu1 -
(-3 + lu1 + lu2 - 2 * theta) * llu2 + 2 * theta * llu2 + ((-3 + 2 * lu1 +
lu2 - theta) * (lu2t * llu1 + lu1t * llu2)) /
(lu1t + lu2t) + logplu) + (plu12invt^2) * (theta + (-1 + lu2 - theta) *
(lu1 + theta) * llu1 + ((1 + theta) * (1 - lu2 + theta) + lu1 * (-1 + 2 *
lu2 + theta)) * llu2 - ((1 - lu2 + lu1 * (-2 + 3 * lu2) + theta) *
(lu2t * llu1 + lu1t * llu2)) /
(lu1t + lu2t) + theta * (1 - (1 + lu1 - lu2 + 2 * theta) * llu1 +
(2 + lu1 - lu2 + 2 * theta) * llu2) + logplu) + (plu12invt *
aaa * (theta * (lu1 + theta) * llu1 + (-3 + lu1 + lu2 - 2 * theta) *
theta * llu2 + (-3 + 2 * lu1 + lu2 - theta) * logplu)) /
((lu1t + lu2t) * t2) - (2 * (plu12invt^2) * aaa * (theta * (theta +
(-1 + lu2 - theta) * (lu1 + theta) * llu1 + ((1 + theta) *
(1 - lu2 + theta) + lu1 * (-1 + 2 * lu2 + theta)) * llu2) +
(1 - lu2 + lu1 * (-2 + 3 * lu2) + theta) * logplu)) /
((lu1t + lu2t) * t2)) + lu1t * (lu2^(2 * theta)) * llu1 *
(theta * llu1 + logplu + plu12invt * (theta * (3 - lu1 - lu2 + 2 * theta) *
llu1 - theta * (lu2 + theta) * llu2 + (3 - lu1 - 2 * lu2 + theta) *
logplu) + (plu12invt^2) * (theta * (theta + (lu1 * (-1 + 2 * lu2 - theta) +
lu2 * (-1 + theta) + tp12) * llu1 + (-1 + lu1 - theta) *
(lu2 + theta) * llu2) + (1 - 2 * lu2 + lu1 * (-1 + 3 * lu2) + theta) *
logplu)) + 2 * lu1t * (lu2^(2 * theta)) * llu2 * (theta * llu1 +
logplu + plu12invt * (theta * (3 - lu1 - lu2 + 2 * theta) * llu1 - theta *
(lu2 + theta) * llu2 + (3 - lu1 - 2 * lu2 + theta) * logplu) +
(plu12invt^2) * (theta * (theta + (lu1 * (-1 + 2 * lu2 - theta) + lu2 *
(-1 + theta) + tp12) * llu1 + (-1 + lu1 - theta) * (lu2 + theta) * llu2) +
(1 - 2 * lu2 + lu1 * (-1 + 3 * lu2) + theta) * logplu)) + lu1t *
(lu2^(2 * theta)) * (llu1 - (lu2t * llu1 + lu1t * llu2) / (lu1t + lu2t) +
(plu12invt^2) * (theta + (lu1 * (-1 + 2 * lu2 - theta) + lu2 *
(-1 + theta) + tp12) * llu1 + (-1 + lu1 - theta) * (lu2 + theta) * llu2 -
((1 - 2 * lu2 + lu1 * (-1 + 3 * lu2) + theta) *
(lu2t * llu1 + lu1t * llu2)) / (lu1t + lu2t) +
theta * (1 + (2 - lu1 + lu2 + 2 * theta) * llu1 + (-1 + lu1 - lu2 -
2 * theta) * llu2) + logplu) + (plu12invt * ((-(lu1^(1 + theta)) + lu2t *
(lu2 + 3 * theta) + lu1t * (3 - lu2 + 4 * theta)) * llu1 +
((lu1^(1 + theta)) + lu1t * (-3 + lu2 - 3 * theta) - lu2t * (lu2 + 2 *
theta)) * llu2 + (lu1t + lu2t) * logplu)) /
(lu1t + lu2t) - (plu12invt * aaa * (theta * (3 - lu1 - lu2 + 2 * theta) *
llu1 - theta * (lu2 + theta) * llu2 + (3 - lu1 - 2 * lu2 + theta) *
logplu)) / ((lu1t + lu2t) * t2) - (2 * (plu12invt^2) *
aaa * (theta * (theta + (lu1 * (-1 + 2 * lu2 - theta) + lu2 *
(-1 + theta) + ((1 + theta)^2)) * llu1 + (-1 + lu1 - theta) *
(lu2 + theta) * llu2) + (1 - 2 * lu2 + lu1 * (-1 + 3 * lu2) + theta) *
logplu)) / ((lu1t + lu2t) * t2)))) / (plu12invt *
((lu1t + lu2t)^2) * ((lu22t * plu12invt * (-1 + lu1 * plu12invt) +
lu12t * plu12invt * (-1 + lu2 * plu12invt) + lu1t * lu2t *
(1 + 2 * lu1 * lu2 * (plu12invt^2) - plu12invt *
(-1 + lu1 + lu2 - theta)))^2) * (theta^3))
}
Try the gofCopula package in your browser
Any scripts or data that you put into this service are public.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Embedding an R snippet on your website
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.