Nothing
R/internal_PIOS_gumbel.R
In gofCopula: Goodness-of-Fit Tests for Copulae
Defines functions
.gumb.V.123
.gumb.S.123
.gumb.123.density
.gumb.12.density
.S.gumb.12
.V.gumb.12
# Internal functions for the derivation of the gumbel copula for the PIOSRn and
# PIOSTn test statistics. Called by .Rn and .Tn, the internal functions for
# their test statistics.
# 2d first derivative wrt theta of the log-density
.V.gumb.12 <- function(theta, u1, u2) {
a <- .Power(.Power(-log(u1), theta) + .Power(-log(u2), theta), 1 / theta)
a2 <- a^2
res.f <- (-(theta * (.Power(-log(u1), theta) * (1 - 3 * a + a2) +
.Power(theta, 2) * (.Power(-log(u1), theta) -
.Power(-log(u2), theta)) + theta * (-1 + a) * (2 *
.Power(-log(u1), theta) - .Power(-log(u2), theta))) *
log(-log(u1))) + (1 + theta * (-1 + a) - 3 * a + a2) *
(.Power(-log(u1), theta) + .Power(-log(u2), theta)) *
log(.Power(-log(u1), theta) + .Power(-log(u2), theta)) +
theta * (theta * (.Power(-log(u1), theta) +
.Power(-log(u2), theta)) - (-(theta * (-1 + a) *
(.Power(-log(u1), theta) - 2 * .Power(-log(u2), theta))) +
.Power(theta, 2) * (-.Power(-log(u1), theta) +
.Power(-log(u2), theta)) + (1 - 3 * a + a2) *
.Power(-log(u2), theta)) * log(-log(u2)))) /
(.Power(theta, 2) * (-1 + theta + a) * (.Power(-log(u1), theta) +
.Power(-log(u2), theta)))
return(res.f)
}
# 2d second derivative wrt theta of the log-density
.S.gumb.12 <- function(theta, u1, u2) {
a <- .Power(.Power(-log(u1), theta) + .Power(-log(u2), theta), 1 / theta)
a2 <- a^2
a3 <- a^3
b <- .Power(log(u1) * log(u2), theta)
d <- log(.Power(-log(u1), theta) + .Power(-log(u2), theta))
res.f <- -((.Power(theta, 2) * (.Power(-log(u1), 2 * theta) *
(2 - 2 * a + a2) * a + 2 * .Power(theta, 4) * b + 5 *
.Power(theta, 3) * (-1 + a) * b + .Power(theta, 2) *
(.Power(-log(u1), 2 * theta) * a + 4 * b - 9 * a * b + 4 *
a2 * b) + theta * (-3 * .Power(-log(u1), 2 * theta) * a + 2 *
.Power(-log(u1), 2 * theta) * a2 - b + 4 * a * b - 4 * a2 *
b + a3 * b)) * .Power(log(-log(u1)), 2) + (2 +
.Power(theta, 2) + theta * (-3 + 2 * a) - 2 * a + a2) * a *
(.Power(-log(u1), 2 * theta) + .Power(-log(u2), 2 * theta) +
2 * b) * .Power(d, 2) + 2 * theta * d * ((-1 +
.Power(theta, 2) * (-1 + a) + 2 * theta * (1 - 3 * a + a2) +
4 * a - 4 * a2 + a3) * (.Power(-log(u1), 2 * theta) +
.Power(-log(u2), 2 * theta) + 2 * b) - (2 + .Power(theta, 2) +
theta * (-3 + 2 * a) - 2 * a + a2) * a *
(.Power(-log(u2), 2 * theta) + b) * log(-log(u2))) +
.Power(theta, 2) * (.Power(theta, 2) * (.Power(-log(u1), 2 *
theta) + .Power(-log(u2), 2 * theta) + 2 * b) - 2 *
(-1 + .Power(theta, 2) * (-1 + a) + 2 * theta * (1 - 3 * a +
a2) + 4 * a - 4 * a2 + a3) * (.Power(-log(u2), 2 * theta) + b) *
log(-log(u2)) + ((2 - 2 * a + a2) * a * .Power(-log(u2), 2 *
theta) + 2 * .Power(theta, 4) * b + 5 * .Power(theta, 3) *
(-1 + a) * b + .Power(theta, 2) * (a * .Power(-log(u2), 2 *
theta) + 4 * b - 9 * a * b + 4 * a2 * b) + theta * (-3 * a *
.Power(-log(u2), 2 * theta) + 2 * a2 * .Power(-log(u2),
2 * theta) - b + 4 * a * b - 4 * a2 * b + a3 * b)) *
.Power(log(-log(u2)), 2)) - 2 * theta * log(-log(u1)) *
((2 + .Power(theta, 2) + theta * (-3 + 2 * a) - 2 * a + a2) *
a * (.Power(-log(u1), 2 * theta) + b) * d + theta *
((-1 + .Power(theta, 2) * (-1 + a) + 2 * theta *
(1 - 3 * a + a2) + 4 * a - 4 * a2 + a3) * (.Power(-log(u1), 2 *
theta) + b) + (-1 + theta) * (2 * .Power(theta, 3) +
.Power(theta, 2) * (-3 + 5 * a) + theta * (1 - 5 * a + 4 * a2) +
(2 - 2 * a + a2) * a) * b * log(-log(u2))))) /
(.Power(theta, 4) * .Power(-1 + theta + a, 2) *
.Power(.Power(-log(u1), theta) + .Power(-log(u2), theta), 2)))
return(res.f)
}
# 2d density
.gumb.12.density <- function(x, 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)
a <- lu1^x + lu2^x
xm1 <- x - 1
res.f <- (lu1^xm1 * (xm1 + a^(1 / x)) * a^(-2 + 1 / x) * lu2^xm1) /
(exp(a^(1 / x)) * u1 * u2)
return(res.f)
}
# 3d density
.gumb.123.density <- function(x, u) {
if (is.null(dim(u))) {
u1 <- u[1]
u2 <- u[2]
u3 <- u[3]
} else {
u1 <- u[, 1]
u2 <- u[, 2]
u3 <- u[, 3]
}
p123 <- (-log(u1))^x + (-log(u2))^x + (-log(u3))^x
p123.x <- p123^(1 / x)
res.f <- ((1 + 2 * (x^2) + 3 * x * (-1 + p123.x) - 3 * p123.x + p123.x^2) *
p123^(-3) * ((-(log(u1) * log(u2) * log(u3)))^(-1 + x))) /
(exp(p123.x) * u1 * u2 * u3)
return(res.f)
}
# 3d second derivative wrt theta of the log-density
.gumb.S.123 <- function(theta, u) {
if (is.null(dim(u))) {
u1 <- u[1]
u2 <- u[2]
u3 <- u[3]
} else {
u1 <- u[, 1]
u2 <- u[, 2]
u3 <- u[, 3]
}
l1 <- log(u1)
l2 <- log(u2)
l3 <- log(u3)
ll1 <- log(-l1)
ll2 <- log(-l2)
ll3 <- log(-l3)
ti <- 1 / theta
pu1t <- (-l1)^theta
pu2t <- (-l2)^theta
pu3t <- (-l3)^theta
pl13t <- (l1 * l3)^theta
pl12t <- (l1 * l2)^theta
pl23t <- (l2 * l3)^theta
pu123t <- pu1t + pu2t + pu3t
lt <- log(pu123t)
b <- pu123t^ti
b2 <- b^2
b3 <- b^3
b4 <- b^4
ll32 <- ll3^2
a <- b * ll32
lpu123t <- lt^2
t2 <- theta^2
t3 <- theta^3
t4 <- theta^4
t5 <- theta^5
t6 <- theta^6
res.f <- (-.Power(-3 * t2 * pu1t + 4 * t3 * pu1t - 3 * t2 * pu2t + 4 * t3 *
pu2t + 3 * t2 * pu1t * b + 3 * t2 * pu2t * b - 3 * t2 *
pu3t + 4 * t3 * pu3t + 3 * t2 * b * pu3t - theta *
(pu1t * (-1 + 7 * b - 6 * b2 + b3) + t2 * (-1 + b) *
(8 * pu1t - 3 * pu2t - 3 * pu3t) + t3 * (4 * pu1t - 2 *
pu2t - 2 * pu3t) + theta * (1 - 3 * b + b2) * (5 *
pu1t - pu2t - pu3t)) * ll1 - theta * (pu2t * (-1 + 7 *
b - 6 * b2 + b3) - theta * (1 - 3 * b + b2) * (pu1t -
5 * pu2t + pu3t) - 2 * t3 * (pu1t - 2 * pu2t + pu3t) -
t2 * (-1 + b) * (3 * pu1t - 8 * pu2t + 3 * pu3t)) * ll2 -
pu1t * lt + 3 * theta * pu1t * lt - 2 * t2 * pu1t * lt -
pu2t * lt + 3 * theta * pu2t * lt - 2 * t2 * pu2t * lt +
7 * pu1t * b * lt - 9 * theta * pu1t * b * lt + 2 * t2 *
pu1t * b * lt + 7 * pu2t * b * lt - 9 * theta * pu2t *
b * lt + 2 * t2 * pu2t * b * lt - 6 * pu1t * b2 * lt +
3 * theta * pu1t * b2 * lt - 6 * pu2t * b2 * lt + 3 *
theta * pu2t * b2 * lt + pu1t * b3 * lt + pu2t * b3 *
lt - pu3t * lt + 3 * theta * pu3t * lt - 2 * t2 * pu3t *
lt + 7 * b * pu3t * lt - 9 * theta * b * pu3t * lt + 2 *
t2 * b * pu3t * lt - 6 * b2 * pu3t * lt + 3 * theta *
b2 * pu3t * lt + b3 * pu3t * lt + t2 * pu1t * ll3 - 3 *
t3 * pu1t * ll3 + 2 * t4 * pu1t * ll3 + t2 * pu2t *
ll3 - 3 * t3 * pu2t * ll3 + 2 * t4 * pu2t * ll3 - 3 *
t2 * pu1t * b * ll3 + 3 * t3 * pu1t * b * ll3 - 3 *
t2 * pu2t * b * ll3 + 3 * t3 * pu2t * b * ll3 + t2 *
pu1t * b2 * ll3 + t2 * pu2t * b2 * ll3 + theta * pu3t *
ll3 - 5 * t2 * pu3t * ll3 + 8 * t3 * pu3t * ll3 - 4 *
t4 * pu3t * ll3 - 7 * theta * b * pu3t * ll3 + 15 *
t2 * b * pu3t * ll3 - 8 * t3 * b * pu3t * ll3 + 6 *
theta * b2 * pu3t * ll3 - 5 * t2 * b2 * pu3t * ll3 -
theta * b3 * pu3t * ll3, 2) + (1 + 2 * t2 + 3 * theta *
(-1 + b) - 3 * b + b2) * (4 * t4 * (pu1t^2) + 4 * t4 *
(pu2t^2) + 8 * t4 * pl12t + 4 * t4 * (pu3t^2) + 8 *
t4 * pl13t + 8 * t4 * pl23t + t2 * ((pu1t^2) * (1 - 15 *
b + 25 * b2 - 10 * b3 + b4) + theta * (-1 + 7 * b - 6 *
b2 + b3) * (7 * (pu1t^2) - 3 * pl12t - 3 * pl13t) +
t2 * (1 - 3 * b + b2) * (18 * (pu1t^2) + (pu2t^2) -
16 * pl12t + (pu3t^2) - 16 * pl13t + 2 * pl23t) + 2 *
t4 * (4 * (pu1t^2) + (pu2t^2) - 7 * pl12t + (pu3t^2) -
7 * pl13t + 2 * pl23t) + t3 * (-1 + b) * (20 *
(pu1t^2) + 3 * (pu2t^2) - 27 * pl12t + 3 * (pu3t^2) -
27 * pl13t + 6 * pl23t)) * (ll1^2) + t2 * ((pu2t^2) *
(1 - 15 * b + 25 * b2 - 10 * b3 + b4) + t3 * (-1 + b) *
(3 * (pu1t^2) + 20 * (pu2t^2) - 27 * pl12t + 3 *
(pu3t^2) + 6 * pl13t - 27 * pl23t) + t2 * (1 - 3 * b +
b2) * ((pu1t^2) + 18 * (pu2t^2) - 16 * pl12t +
(pu3t^2) + 2 * pl13t - 16 * pl23t) + 2 * t4 *
((pu1t^2) + 4 * (pu2t^2) - 7 * pl12t + (pu3t^2) + 2 *
pl13t - 7 * pl23t) + theta * (-1 + 7 * b - 6 * b2 + b3) *
(7 * (pu2t^2) - 3 * pl12t - 3 * pl23t)) * (ll2^2) + 2 *
theta * (pu1t^2) * lt - 4 * t3 * (pu1t^2) * lt + 2 *
theta * (pu2t^2) * lt - 4 * t3 * (pu2t^2) * lt + 4 *
theta * pl12t * lt - 8 * t3 * pl12t * lt - 14 * theta *
(pu1t^2) * b * lt + 4 * t3 * (pu1t^2) * b * lt - 14 *
theta * (pu2t^2) * b * lt + 4 * t3 * (pu2t^2) * b *
lt - 28 * theta * pl12t * b * lt + 8 * t3 * pl12t * b *
lt + 12 * theta * (pu1t^2) * b2 * lt + 12 * theta *
(pu2t^2) * b2 * lt + 24 * theta * pl12t * b2 * lt - 2 *
theta * (pu1t^2) * b3 * lt - 2 * theta * (pu2t^2) * b3 *
lt - 4 * theta * pl12t * b3 * lt + 2 * theta * (pu3t^2) *
lt - 4 * t3 * (pu3t^2) * lt - 14 * theta * b * (pu3t^2) *
lt + 4 * t3 * b * (pu3t^2) * lt + 12 * theta * b2 *
(pu3t^2) * lt - 2 * theta * b3 * (pu3t^2) * lt + 4 *
theta * pl13t * lt - 8 * t3 * pl13t * lt - 28 * theta *
b * pl13t * lt + 8 * t3 * b * pl13t * lt + 24 * theta *
b2 * pl13t * lt - 4 * theta * b3 * pl13t * lt + 4 *
theta * pl23t * lt - 8 * t3 * pl23t * lt - 28 * theta *
b * pl23t * lt + 8 * t3 * b * pl23t * lt + 24 * theta *
b2 * pl23t * lt - 4 * theta * b3 * pl23t * lt +
(pu1t^2) * lpu123t - 3 * theta * (pu1t^2) * lpu123t +
2 * t2 * (pu1t^2) * lpu123t + (pu2t^2) * lpu123t - 3 *
theta * (pu2t^2) * lpu123t + 2 * t2 * (pu2t^2) *
lpu123t + 2 * pl12t * lpu123t - 6 * theta * pl12t *
lpu123t + 4 * t2 * pl12t * lpu123t - 15 * (pu1t^2) *
b * lpu123t + 21 * theta * (pu1t^2) * b * lpu123t -
6 * t2 * (pu1t^2) * b * lpu123t - 15 * (pu2t^2) * b *
lpu123t + 21 * theta * (pu2t^2) * b * lpu123t - 6 * t2 *
(pu2t^2) * b * lpu123t - 30 * pl12t * b * lpu123t + 42 *
theta * pl12t * b * lpu123t - 12 * t2 * pl12t * b *
lpu123t + 25 * (pu1t^2) * b2 * lpu123t - 18 * theta *
(pu1t^2) * b2 * lpu123t + 2 * t2 * (pu1t^2) * b2 *
lpu123t + 25 * (pu2t^2) * b2 * lpu123t - 18 * theta *
(pu2t^2) * b2 * lpu123t + 2 * t2 * (pu2t^2) * b2 *
lpu123t + 50 * pl12t * b2 * lpu123t - 36 * theta *
pl12t * b2 * lpu123t + 4 * t2 * pl12t * b2 * lpu123t -
10 * (pu1t^2) * b3 * lpu123t + 3 * theta * (pu1t^2) *
b3 * lpu123t - 10 * (pu2t^2) * b3 * lpu123t + 3 * theta *
(pu2t^2) * b3 * lpu123t - 20 * pl12t * b3 * lpu123t + 6 *
theta * pl12t * b3 * lpu123t + (pu1t^2) * b4 * lpu123t +
(pu2t^2) * b4 * lpu123t + 2 * pl12t * b4 * lpu123t +
(pu3t^2) * lpu123t - 3 * theta * (pu3t^2) * lpu123t + 2 *
t2 * (pu3t^2) * lpu123t - 15 * b * (pu3t^2) * lpu123t +
21 * theta * b * (pu3t^2) * lpu123t - 6 * t2 * b *
(pu3t^2) * lpu123t + 25 * b2 * (pu3t^2) * lpu123t - 18 *
theta * b2 * (pu3t^2) * lpu123t + 2 * t2 * b2 *
(pu3t^2) * lpu123t - 10 * b3 * (pu3t^2) * lpu123t + 3 *
theta * b3 * (pu3t^2) * lpu123t + b4 * (pu3t^2) *
lpu123t + 2 * pl13t * lpu123t - 6 * theta * pl13t *
lpu123t + 4 * t2 * pl13t * lpu123t - 30 * b * pl13t *
lpu123t + 42 * theta * b * pl13t * lpu123t - 12 * t2 *
b * pl13t * lpu123t + 50 * b2 * pl13t * lpu123t - 36 *
theta * b2 * pl13t * lpu123t + 4 * t2 * b2 * pl13t *
lpu123t - 20 * b3 * pl13t * lpu123t + 6 * theta * b3 *
pl13t * lpu123t + 2 * b4 * pl13t * lpu123t + 2 * pl23t *
lpu123t - 6 * theta * pl23t * lpu123t + 4 * t2 * pl23t *
lpu123t - 30 * b * pl23t * lpu123t + 42 * theta * b *
pl23t * lpu123t - 12 * t2 * b * pl23t * lpu123t + 50 *
b2 * pl23t * lpu123t - 36 * theta * b2 * pl23t *
lpu123t + 4 * t2 * b2 * pl23t * lpu123t - 20 * b3 *
pl23t * lpu123t + 6 * theta * b3 * pl23t * lpu123t + 2 *
b4 * pl23t * lpu123t - 6 * t4 * (pu1t^2) * ll3 + 8 * t5 *
(pu1t^2) * ll3 - 6 * t4 * (pu2t^2) * ll3 + 8 * t5 *
(pu2t^2) * ll3 - 12 * t4 * pl12t * ll3 + 16 * t5 *
pl12t * ll3 + 6 * t4 * (pu1t^2) * b * ll3 + 6 * t4 *
(pu2t^2) * b * ll3 + 12 * t4 * pl12t * b * ll3 - 2 * t2 *
(pu3t^2) * ll3 + 16 * t4 * (pu3t^2) * ll3 - 16 * t5 *
(pu3t^2) * ll3 + 14 * t2 * b * (pu3t^2) * ll3 - 16 * t4 *
b * (pu3t^2) * ll3 - 12 * t2 * b2 * (pu3t^2) * ll3 + 2 *
t2 * b3 * (pu3t^2) * ll3 - 2 * t2 * pl13t * ll3 + 10 *
t4 * pl13t * ll3 - 8 * t5 * pl13t * ll3 + 14 * t2 * b *
pl13t * ll3 - 10 * t4 * b * pl13t * ll3 - 12 * t2 * b2 *
pl13t * ll3 + 2 * t2 * b3 * pl13t * ll3 - 2 * t2 *
pl23t * ll3 + 10 * t4 * pl23t * ll3 - 8 * t5 * pl23t *
ll3 + 14 * t2 * b * pl23t * ll3 - 10 * t4 * b * pl23t *
ll3 - 12 * t2 * b2 * pl23t * ll3 + 2 * t2 * b3 * pl23t *
ll3 - 2 * t2 * (pu1t^2) * lt * ll3 + 6 * t3 * (pu1t^2) *
lt * ll3 - 4 * t4 * (pu1t^2) * lt * ll3 - 2 * t2 *
(pu2t^2) * lt * ll3 + 6 * t3 * (pu2t^2) * lt * ll3 - 4 *
t4 * (pu2t^2) * lt * ll3 - 4 * t2 * pl12t * lt * ll3 +
12 * t3 * pl12t * lt * ll3 - 8 * t4 * pl12t * lt * ll3 +
14 * t2 * (pu1t^2) * b * lt * ll3 - 18 * t3 * (pu1t^2) *
b * lt * ll3 + 4 * t4 * (pu1t^2) * b * lt * ll3 + 14 *
t2 * (pu2t^2) * b * lt * ll3 - 18 * t3 * (pu2t^2) * b *
lt * ll3 + 4 * t4 * (pu2t^2) * b * lt * ll3 + 28 * t2 *
pl12t * b * lt * ll3 - 36 * t3 * pl12t * b * lt * ll3 +
8 * t4 * pl12t * b * lt * ll3 - 12 * t2 * (pu1t^2) * b2 *
lt * ll3 + 6 * t3 * (pu1t^2) * b2 * lt * ll3 - 12 * t2 *
(pu2t^2) * b2 * lt * ll3 + 6 * t3 * (pu2t^2) * b2 * lt *
ll3 - 24 * t2 * pl12t * b2 * lt * ll3 + 12 * t3 * pl12t *
b2 * lt * ll3 + 2 * t2 * (pu1t^2) * b3 * lt * ll3 + 2 *
t2 * (pu2t^2) * b3 * lt * ll3 + 4 * t2 * pl12t * b3 *
lt * ll3 - 2 * theta * (pu3t^2) * lt * ll3 + 10 * t2 *
(pu3t^2) * lt * ll3 - 16 * t3 * (pu3t^2) * lt * ll3 +
8 * t4 * (pu3t^2) * lt * ll3 + 30 * theta * b *
(pu3t^2) * lt * ll3 - 70 * t2 * b * (pu3t^2) * lt *
ll3 + 48 * t3 * b * (pu3t^2) * lt * ll3 - 8 * t4 * b *
(pu3t^2) * lt * ll3 - 50 * theta * b2 * (pu3t^2) * lt *
ll3 + 60 * t2 * b2 * (pu3t^2) * lt * ll3 - 16 * t3 * b2 *
(pu3t^2) * lt * ll3 + 20 * theta * b3 * (pu3t^2) * lt *
ll3 - 10 * t2 * b3 * (pu3t^2) * lt * ll3 - 2 * theta *
b4 * (pu3t^2) * lt * ll3 - 2 * theta * pl13t * lt * ll3 +
8 * t2 * pl13t * lt * ll3 - 10 * t3 * pl13t * lt * ll3 +
4 * t4 * pl13t * lt * ll3 + 30 * theta * b * pl13t * lt *
ll3 - 56 * t2 * b * pl13t * lt * ll3 + 30 * t3 * b *
pl13t * lt * ll3 - 4 * t4 * b * pl13t * lt * ll3 - 50 *
theta * b2 * pl13t * lt * ll3 + 48 * t2 * b2 * pl13t *
lt * ll3 - 10 * t3 * b2 * pl13t * lt * ll3 + 20 * theta *
b3 * pl13t * lt * ll3 - 8 * t2 * b3 * pl13t * lt * ll3 -
2 * theta * b4 * pl13t * lt * ll3 - 2 * theta * pl23t *
lt * ll3 + 8 * t2 * pl23t * lt * ll3 - 10 * t3 * pl23t *
lt * ll3 + 4 * t4 * pl23t * lt * ll3 + 30 * theta * b *
pl23t * lt * ll3 - 56 * t2 * b * pl23t * lt * ll3 + 30 *
t3 * b * pl23t * lt * ll3 - 4 * t4 * b * pl23t * lt *
ll3 - 50 * theta * b2 * pl23t * lt * ll3 + 48 * t2 *
b2 * pl23t * lt * ll3 - 10 * t3 * b2 * pl23t * lt *
ll3 + 20 * theta * b3 * pl23t * lt * ll3 - 8 * t2 *
b3 * pl23t * lt * ll3 - 2 * theta * b4 * pl23t * lt *
ll3 + t4 * (pu1t^2) * ll32 - 3 * t5 * (pu1t^2) * ll32 +
2 * t6 * (pu1t^2) * ll32 + t4 * (pu2t^2) * ll32 - 3 *
t5 * (pu2t^2) * ll32 + 2 * t6 * (pu2t^2) * ll32 + 2 *
t4 * pl12t * ll32 - 6 * t5 * pl12t * ll32 + 4 * t6 *
pl12t * ll32 - 3 * t4 * (pu1t^2) * a + 3 * t5 *
(pu1t^2) * a - 3 * t4 * (pu2t^2) * a + 3 * t5 *
(pu2t^2) * a - 6 * t4 * pl12t * a + 6 * t5 * pl12t * a +
t4 * (pu1t^2) * b2 * ll32 + t4 * (pu2t^2) * b2 * ll32 +
2 * t4 * pl12t * b2 * ll32 + t2 * (pu3t^2) * ll32 - 7 *
t3 * (pu3t^2) * ll32 + 18 * t4 * (pu3t^2) * ll32 - 20 *
t5 * (pu3t^2) * ll32 + 8 * t6 * (pu3t^2) * ll32 - 15 *
t2 * b * (pu3t^2) * ll32 + 49 * t3 * b * (pu3t^2) *
ll32 - 54 * t4 * b * (pu3t^2) * ll32 + 20 * t5 * b *
(pu3t^2) * ll32 + 25 * t2 * b2 * (pu3t^2) * ll32 - 42 *
t3 * b2 * (pu3t^2) * ll32 + 18 * t4 * b2 * (pu3t^2) *
ll32 - 10 * t2 * b3 * (pu3t^2) * ll32 + 7 * t3 * b3 *
(pu3t^2) * ll32 + t2 * b4 * (pu3t^2) * ll32 + 3 * t3 *
pl13t * ll32 - 16 * t4 * pl13t * ll32 + 27 * t5 * pl13t *
ll32 - 14 * t6 * pl13t * ll32 - 21 * t3 * b * pl13t *
ll32 + 48 * t4 * b * pl13t * ll32 - 27 * t5 * b * pl13t *
ll32 + 18 * t3 * b2 * pl13t * ll32 - 16 * t4 * b2 *
pl13t * ll32 - 3 * t3 * b3 * pl13t * ll32 + 3 * t3 *
pl23t * ll32 - 16 * t4 * pl23t * ll32 + 27 * t5 * pl23t *
ll32 - 14 * t6 * pl23t * ll32 - 21 * t3 * b * pl23t *
ll32 + 48 * t4 * b * pl23t * ll32 - 27 * t5 * b * pl23t *
ll32 + 18 * t3 * b2 * pl23t * ll32 - 16 * t4 * b2 *
pl23t * ll32 - 3 * t3 * b3 * pl23t * ll32 - 2 * theta *
ll1 * (-(theta * (pl12t * (1 - 15 * b + 25 * b2 - 10 *
b3 + b4) - theta * (-1 + 7 * b - 6 * b2 + b3) *
((pu1t^2) + (pu2t^2) - 8 * pl12t + pl13t + pl23t) - 2 *
t4 * (2 * (pu1t^2) + 2 * (pu2t^2) - 8 * pl12t -
(pu3t^2) + pl13t + pl23t) - t2 * (1 - 3 * b + b2) * (5 *
(pu1t^2) + 5 * (pu2t^2) - 25 * pl12t - (pu3t^2) + 4 *
pl13t + 4 * pl23t) - t3 * (-1 + b) * (8 * (pu1t^2) + 8 *
(pu2t^2) - 34 * pl12t - 3 * (pu3t^2) + 5 * pl13t + 5 *
pl23t)) * ll2) + ((1 - 15 * b + 25 * b2 - 10 * b3 + b4) *
((pu1t^2) + pl12t + pl13t) + t2 * (1 - 3 * b + b2) * (8 *
(pu1t^2) - 3 * (pu2t^2) + 5 * pl12t - 3 * (pu3t^2) + 5 *
pl13t - 6 * pl23t) + 2 * t3 * (-1 + b) * (2 * (pu1t^2) -
(pu2t^2) + pl12t - (pu3t^2) + pl13t - 2 * pl23t) +
theta * (-1 + 7 * b - 6 * b2 + b3) * (5 * (pu1t^2) -
(pu2t^2) + 4 * pl12t - (pu3t^2) + 4 * pl13t - 2 *
pl23t)) * lt + theta * ((pu1t^2) - 8 * t2 * (pu1t^2) +
8 * t3 * (pu1t^2) + 3 * t2 * (pu2t^2) - 4 * t3 *
(pu2t^2) + pl12t - 5 * t2 * pl12t + 4 * t3 * pl12t - 7 *
(pu1t^2) * b + 8 * t2 * (pu1t^2) * b - 3 * t2 *
(pu2t^2) * b - 7 * pl12t * b + 5 * t2 * pl12t * b + 6 *
(pu1t^2) * b2 + 6 * pl12t * b2 - (pu1t^2) * b3 - pl12t *
b3 + 3 * t2 * (pu3t^2) - 4 * t3 * (pu3t^2) - 3 * t2 * b *
(pu3t^2) + pl13t - 5 * t2 * pl13t + 4 * t3 * pl13t - 7 *
b * pl13t + 5 * t2 * b * pl13t + 6 * b2 * pl13t - b3 *
pl13t + 6 * t2 * pl23t - 8 * t3 * pl23t - 6 * t2 * b *
pl23t - ((1 - 15 * b + 25 * b2 - 10 * b3 + b4) *
pl13t - theta * (-1 + 7 * b - 6 * b2 + b3) * ((pu1t^2) +
pl12t + (pu3t^2) - 8 * pl13t + pl23t) - 2 * t4 * (2 *
(pu1t^2) - (pu2t^2) + pl12t + 2 * (pu3t^2) - 8 * pl13t +
pl23t) - t2 * (1 - 3 * b + b2) * (5 * (pu1t^2) -
(pu2t^2) + 4 * pl12t + 5 * (pu3t^2) - 25 * pl13t + 4 *
pl23t) - t3 * (-1 + b) * (8 * (pu1t^2) - 3 * (pu2t^2) +
5 * pl12t + 8 * (pu3t^2) - 34 * pl13t + 5 * pl23t)) *
ll3)) - 2 * theta * ll2 * ((-(t2 * (1 - 3 * b + b2) *
(3 * (pu1t^2) - 8 * (pu2t^2) - 5 * pl12t + 3 * (pu3t^2) +
6 * pl13t - 5 * pl23t)) - theta * (-1 + 7 * b - 6 * b2 +
b3) * ((pu1t^2) - 5 * (pu2t^2) - 4 * pl12t + (pu3t^2) +
2 * pl13t - 4 * pl23t) - 2 * t3 * (-1 + b) * ((pu1t^2) -
2 * (pu2t^2) - pl12t + (pu3t^2) + 2 * pl13t - pl23t) +
(1 - 15 * b + 25 * b2 - 10 * b3 + b4) * ((pu2t^2) +
pl12t + pl23t)) * lt + theta * (3 * t2 * (pu1t^2) - 4 *
t3 * (pu1t^2) + (pu2t^2) - 8 * t2 * (pu2t^2) + 8 * t3 *
(pu2t^2) + pl12t - 5 * t2 * pl12t + 4 * t3 * pl12t - 3 *
t2 * (pu1t^2) * b - 7 * (pu2t^2) * b + 8 * t2 *
(pu2t^2) * b - 7 * pl12t * b + 5 * t2 * pl12t * b +
6 * (pu2t^2) * b2 + 6 * pl12t * b2 - (pu2t^2) * b3 -
pl12t * b3 + 3 * t2 * (pu3t^2) - 4 * t3 * (pu3t^2) - 3 *
t2 * b * (pu3t^2) + 6 * t2 * pl13t - 8 * t3 * pl13t - 6 *
t2 * b * pl13t + pl23t - 5 * t2 * pl23t + 4 * t3 *
pl23t - 7 * b * pl23t + 5 * t2 * b * pl23t + 6 * b2 *
pl23t - b3 * pl23t - ((1 - 15 * b + 25 * b2 - 10 * b3 +
b4) * pl23t - theta * (-1 + 7 * b - 6 * b2 + b3) *
((pu2t^2) + pl12t + (pu3t^2) + pl13t - 8 * pl23t) + 2 *
t4 * ((pu1t^2) - 2 * (pu2t^2) - pl12t - 2 * (pu3t^2) -
pl13t + 8 * pl23t) + t2 * (1 - 3 * b + b2) * ((pu1t^2) -
5 * (pu2t^2) - 4 * pl12t - 5 * (pu3t^2) - 4 * pl13t +
25 * pl23t) + t3 * (-1 + b) * (3 * (pu1t^2) - 8 *
(pu2t^2) - 5 * pl12t - 8 * (pu3t^2) - 5 * pl13t + 34 *
pl23t)) * ll3)))) / (t4 * ((1 + 2 * t2 + 3 * theta *
(-1 + b) - 3 * b + b2)^(2)) * ((pu123t)^(2)))
return(res.f)
}
# 3d first derivative wrt theta of the log-density
.gumb.V.123 <- function(theta, u) {
if (is.null(dim(u))) {
u1 <- u[1]
u2 <- u[2]
u3 <- u[3]
} else {
u1 <- u[, 1]
u2 <- u[, 2]
u3 <- u[, 3]
}
l1 <- log(u1)
l2 <- log(u2)
l3 <- log(u3)
ll1 <- log(-l1)
ll2 <- log(-l2)
ll3 <- log(-l3)
ti <- 1 / theta
pu1t <- (-l1)^theta
pu2t <- (-l2)^theta
pu3t <- (-l3)^theta
pu123t <- pu1t + pu2t + pu3t
lt <- log(pu123t)
b <- .Power(pu123t, ti)
b2 <- b^2
b3 <- b^3
t2 <- theta^2
t3 <- theta^3
t4 <- theta^4
t5 <- theta^5
res.f <- (-3 * t2 * pu1t + 4 * t3 * pu1t - 3 * t2 * pu2t + 4 * t3 * pu2t +
3 * t2 * pu1t * b + 3 * t2 * pu2t * b - 3 * t2 * pu3t + 4 * t3 *
pu3t + 3 * t2 * b * pu3t - theta * (pu1t * (-1 + 7 * b - 6 * b2 +
b3) + t2 * (-1 + b) * (8 * pu1t - 3 * pu2t - 3 * pu3t) + t3 *
(4 * pu1t - 2 * pu2t - 2 * pu3t) + theta * (1 - 3 * b + b2) * (5 *
pu1t - pu2t - pu3t)) * ll1 - theta * (pu2t * (-1 + 7 * b - 6 *
b2 + b3) - theta * (1 - 3 * b + b2) * (pu1t - 5 * pu2t + pu3t) -
2 * t3 * (pu1t - 2 * pu2t + pu3t) - t2 * (-1 + b) * (3 * pu1t -
8 * pu2t + 3 * pu3t)) * ll2 - pu1t * lt + 3 * theta * pu1t * lt -
2 * t2 * pu1t * lt - pu2t * lt + 3 * theta * pu2t * lt - 2 * t2 *
pu2t * lt + 7 * pu1t * b * lt - 9 * theta * pu1t * b * lt + 2 *
t2 * pu1t * b * lt + 7 * pu2t * b * lt - 9 * theta * pu2t * b *
lt + 2 * t2 * pu2t * b * lt - 6 * pu1t * b2 * lt + 3 * theta *
pu1t * b2 * lt - 6 * pu2t * b2 * lt + 3 * theta * pu2t * b2 * lt +
pu1t * b3 * lt + pu2t * b3 * lt - pu3t * lt + 3 * theta *
pu3t * lt - 2 * t2 * pu3t * lt + 7 * b * pu3t * lt - 9 * theta *
b * pu3t * lt + 2 * t2 * b * pu3t * lt - 6 * b2 * pu3t * lt + 3 *
theta * b2 * pu3t * lt + b3 * pu3t * lt + t2 * pu1t * ll3 - 3 *
t3 * pu1t * ll3 + 2 * t4 * pu1t * ll3 + t2 * pu2t * ll3 - 3 * t3 *
pu2t * ll3 + 2 * t4 * pu2t * ll3 - 3 * t2 * pu1t * b * ll3 + 3 *
t3 * pu1t * b * ll3 - 3 * t2 * pu2t * b * ll3 + 3 * t3 * pu2t * b *
ll3 + t2 * pu1t * b2 * ll3 + t2 * pu2t * b2 * ll3 + theta * pu3t *
ll3 - 5 * t2 * pu3t * ll3 + 8 * t3 * pu3t * ll3 - 4 * t4 * pu3t *
ll3 - 7 * theta * b * pu3t * ll3 + 15 * t2 * b * pu3t * ll3 - 8 *
t3 * b * pu3t * ll3 + 6 * theta * b2 * pu3t * ll3 - 5 * t2 * b2 *
pu3t * ll3 - theta * b3 * pu3t * ll3) / (t2 * (1 + 2 * t2 + 3 *
theta * (-1 + b) - 3 * b + b2) * (pu123t))
return(res.f)
}
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Defines functions .gumb.V.123 .gumb.S.123 .gumb.123.density .gumb.12.density .S.gumb.12 .V.gumb.12
# Internal functions for the derivation of the gumbel copula for the PIOSRn and
# PIOSTn test statistics. Called by .Rn and .Tn, the internal functions for
# their test statistics.
# 2d first derivative wrt theta of the log-density
.V.gumb.12 <- function(theta, u1, u2) {
a <- .Power(.Power(-log(u1), theta) + .Power(-log(u2), theta), 1 / theta)
a2 <- a^2
res.f <- (-(theta * (.Power(-log(u1), theta) * (1 - 3 * a + a2) +
.Power(theta, 2) * (.Power(-log(u1), theta) -
.Power(-log(u2), theta)) + theta * (-1 + a) * (2 *
.Power(-log(u1), theta) - .Power(-log(u2), theta))) *
log(-log(u1))) + (1 + theta * (-1 + a) - 3 * a + a2) *
(.Power(-log(u1), theta) + .Power(-log(u2), theta)) *
log(.Power(-log(u1), theta) + .Power(-log(u2), theta)) +
theta * (theta * (.Power(-log(u1), theta) +
.Power(-log(u2), theta)) - (-(theta * (-1 + a) *
(.Power(-log(u1), theta) - 2 * .Power(-log(u2), theta))) +
.Power(theta, 2) * (-.Power(-log(u1), theta) +
.Power(-log(u2), theta)) + (1 - 3 * a + a2) *
.Power(-log(u2), theta)) * log(-log(u2)))) /
(.Power(theta, 2) * (-1 + theta + a) * (.Power(-log(u1), theta) +
.Power(-log(u2), theta)))
return(res.f)
}
# 2d second derivative wrt theta of the log-density
.S.gumb.12 <- function(theta, u1, u2) {
a <- .Power(.Power(-log(u1), theta) + .Power(-log(u2), theta), 1 / theta)
a2 <- a^2
a3 <- a^3
b <- .Power(log(u1) * log(u2), theta)
d <- log(.Power(-log(u1), theta) + .Power(-log(u2), theta))
res.f <- -((.Power(theta, 2) * (.Power(-log(u1), 2 * theta) *
(2 - 2 * a + a2) * a + 2 * .Power(theta, 4) * b + 5 *
.Power(theta, 3) * (-1 + a) * b + .Power(theta, 2) *
(.Power(-log(u1), 2 * theta) * a + 4 * b - 9 * a * b + 4 *
a2 * b) + theta * (-3 * .Power(-log(u1), 2 * theta) * a + 2 *
.Power(-log(u1), 2 * theta) * a2 - b + 4 * a * b - 4 * a2 *
b + a3 * b)) * .Power(log(-log(u1)), 2) + (2 +
.Power(theta, 2) + theta * (-3 + 2 * a) - 2 * a + a2) * a *
(.Power(-log(u1), 2 * theta) + .Power(-log(u2), 2 * theta) +
2 * b) * .Power(d, 2) + 2 * theta * d * ((-1 +
.Power(theta, 2) * (-1 + a) + 2 * theta * (1 - 3 * a + a2) +
4 * a - 4 * a2 + a3) * (.Power(-log(u1), 2 * theta) +
.Power(-log(u2), 2 * theta) + 2 * b) - (2 + .Power(theta, 2) +
theta * (-3 + 2 * a) - 2 * a + a2) * a *
(.Power(-log(u2), 2 * theta) + b) * log(-log(u2))) +
.Power(theta, 2) * (.Power(theta, 2) * (.Power(-log(u1), 2 *
theta) + .Power(-log(u2), 2 * theta) + 2 * b) - 2 *
(-1 + .Power(theta, 2) * (-1 + a) + 2 * theta * (1 - 3 * a +
a2) + 4 * a - 4 * a2 + a3) * (.Power(-log(u2), 2 * theta) + b) *
log(-log(u2)) + ((2 - 2 * a + a2) * a * .Power(-log(u2), 2 *
theta) + 2 * .Power(theta, 4) * b + 5 * .Power(theta, 3) *
(-1 + a) * b + .Power(theta, 2) * (a * .Power(-log(u2), 2 *
theta) + 4 * b - 9 * a * b + 4 * a2 * b) + theta * (-3 * a *
.Power(-log(u2), 2 * theta) + 2 * a2 * .Power(-log(u2),
2 * theta) - b + 4 * a * b - 4 * a2 * b + a3 * b)) *
.Power(log(-log(u2)), 2)) - 2 * theta * log(-log(u1)) *
((2 + .Power(theta, 2) + theta * (-3 + 2 * a) - 2 * a + a2) *
a * (.Power(-log(u1), 2 * theta) + b) * d + theta *
((-1 + .Power(theta, 2) * (-1 + a) + 2 * theta *
(1 - 3 * a + a2) + 4 * a - 4 * a2 + a3) * (.Power(-log(u1), 2 *
theta) + b) + (-1 + theta) * (2 * .Power(theta, 3) +
.Power(theta, 2) * (-3 + 5 * a) + theta * (1 - 5 * a + 4 * a2) +
(2 - 2 * a + a2) * a) * b * log(-log(u2))))) /
(.Power(theta, 4) * .Power(-1 + theta + a, 2) *
.Power(.Power(-log(u1), theta) + .Power(-log(u2), theta), 2)))
return(res.f)
}
# 2d density
.gumb.12.density <- function(x, 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)
a <- lu1^x + lu2^x
xm1 <- x - 1
res.f <- (lu1^xm1 * (xm1 + a^(1 / x)) * a^(-2 + 1 / x) * lu2^xm1) /
(exp(a^(1 / x)) * u1 * u2)
return(res.f)
}
# 3d density
.gumb.123.density <- function(x, u) {
if (is.null(dim(u))) {
u1 <- u[1]
u2 <- u[2]
u3 <- u[3]
} else {
u1 <- u[, 1]
u2 <- u[, 2]
u3 <- u[, 3]
}
p123 <- (-log(u1))^x + (-log(u2))^x + (-log(u3))^x
p123.x <- p123^(1 / x)
res.f <- ((1 + 2 * (x^2) + 3 * x * (-1 + p123.x) - 3 * p123.x + p123.x^2) *
p123^(-3) * ((-(log(u1) * log(u2) * log(u3)))^(-1 + x))) /
(exp(p123.x) * u1 * u2 * u3)
return(res.f)
}
# 3d second derivative wrt theta of the log-density
.gumb.S.123 <- function(theta, u) {
if (is.null(dim(u))) {
u1 <- u[1]
u2 <- u[2]
u3 <- u[3]
} else {
u1 <- u[, 1]
u2 <- u[, 2]
u3 <- u[, 3]
}
l1 <- log(u1)
l2 <- log(u2)
l3 <- log(u3)
ll1 <- log(-l1)
ll2 <- log(-l2)
ll3 <- log(-l3)
ti <- 1 / theta
pu1t <- (-l1)^theta
pu2t <- (-l2)^theta
pu3t <- (-l3)^theta
pl13t <- (l1 * l3)^theta
pl12t <- (l1 * l2)^theta
pl23t <- (l2 * l3)^theta
pu123t <- pu1t + pu2t + pu3t
lt <- log(pu123t)
b <- pu123t^ti
b2 <- b^2
b3 <- b^3
b4 <- b^4
ll32 <- ll3^2
a <- b * ll32
lpu123t <- lt^2
t2 <- theta^2
t3 <- theta^3
t4 <- theta^4
t5 <- theta^5
t6 <- theta^6
res.f <- (-.Power(-3 * t2 * pu1t + 4 * t3 * pu1t - 3 * t2 * pu2t + 4 * t3 *
pu2t + 3 * t2 * pu1t * b + 3 * t2 * pu2t * b - 3 * t2 *
pu3t + 4 * t3 * pu3t + 3 * t2 * b * pu3t - theta *
(pu1t * (-1 + 7 * b - 6 * b2 + b3) + t2 * (-1 + b) *
(8 * pu1t - 3 * pu2t - 3 * pu3t) + t3 * (4 * pu1t - 2 *
pu2t - 2 * pu3t) + theta * (1 - 3 * b + b2) * (5 *
pu1t - pu2t - pu3t)) * ll1 - theta * (pu2t * (-1 + 7 *
b - 6 * b2 + b3) - theta * (1 - 3 * b + b2) * (pu1t -
5 * pu2t + pu3t) - 2 * t3 * (pu1t - 2 * pu2t + pu3t) -
t2 * (-1 + b) * (3 * pu1t - 8 * pu2t + 3 * pu3t)) * ll2 -
pu1t * lt + 3 * theta * pu1t * lt - 2 * t2 * pu1t * lt -
pu2t * lt + 3 * theta * pu2t * lt - 2 * t2 * pu2t * lt +
7 * pu1t * b * lt - 9 * theta * pu1t * b * lt + 2 * t2 *
pu1t * b * lt + 7 * pu2t * b * lt - 9 * theta * pu2t *
b * lt + 2 * t2 * pu2t * b * lt - 6 * pu1t * b2 * lt +
3 * theta * pu1t * b2 * lt - 6 * pu2t * b2 * lt + 3 *
theta * pu2t * b2 * lt + pu1t * b3 * lt + pu2t * b3 *
lt - pu3t * lt + 3 * theta * pu3t * lt - 2 * t2 * pu3t *
lt + 7 * b * pu3t * lt - 9 * theta * b * pu3t * lt + 2 *
t2 * b * pu3t * lt - 6 * b2 * pu3t * lt + 3 * theta *
b2 * pu3t * lt + b3 * pu3t * lt + t2 * pu1t * ll3 - 3 *
t3 * pu1t * ll3 + 2 * t4 * pu1t * ll3 + t2 * pu2t *
ll3 - 3 * t3 * pu2t * ll3 + 2 * t4 * pu2t * ll3 - 3 *
t2 * pu1t * b * ll3 + 3 * t3 * pu1t * b * ll3 - 3 *
t2 * pu2t * b * ll3 + 3 * t3 * pu2t * b * ll3 + t2 *
pu1t * b2 * ll3 + t2 * pu2t * b2 * ll3 + theta * pu3t *
ll3 - 5 * t2 * pu3t * ll3 + 8 * t3 * pu3t * ll3 - 4 *
t4 * pu3t * ll3 - 7 * theta * b * pu3t * ll3 + 15 *
t2 * b * pu3t * ll3 - 8 * t3 * b * pu3t * ll3 + 6 *
theta * b2 * pu3t * ll3 - 5 * t2 * b2 * pu3t * ll3 -
theta * b3 * pu3t * ll3, 2) + (1 + 2 * t2 + 3 * theta *
(-1 + b) - 3 * b + b2) * (4 * t4 * (pu1t^2) + 4 * t4 *
(pu2t^2) + 8 * t4 * pl12t + 4 * t4 * (pu3t^2) + 8 *
t4 * pl13t + 8 * t4 * pl23t + t2 * ((pu1t^2) * (1 - 15 *
b + 25 * b2 - 10 * b3 + b4) + theta * (-1 + 7 * b - 6 *
b2 + b3) * (7 * (pu1t^2) - 3 * pl12t - 3 * pl13t) +
t2 * (1 - 3 * b + b2) * (18 * (pu1t^2) + (pu2t^2) -
16 * pl12t + (pu3t^2) - 16 * pl13t + 2 * pl23t) + 2 *
t4 * (4 * (pu1t^2) + (pu2t^2) - 7 * pl12t + (pu3t^2) -
7 * pl13t + 2 * pl23t) + t3 * (-1 + b) * (20 *
(pu1t^2) + 3 * (pu2t^2) - 27 * pl12t + 3 * (pu3t^2) -
27 * pl13t + 6 * pl23t)) * (ll1^2) + t2 * ((pu2t^2) *
(1 - 15 * b + 25 * b2 - 10 * b3 + b4) + t3 * (-1 + b) *
(3 * (pu1t^2) + 20 * (pu2t^2) - 27 * pl12t + 3 *
(pu3t^2) + 6 * pl13t - 27 * pl23t) + t2 * (1 - 3 * b +
b2) * ((pu1t^2) + 18 * (pu2t^2) - 16 * pl12t +
(pu3t^2) + 2 * pl13t - 16 * pl23t) + 2 * t4 *
((pu1t^2) + 4 * (pu2t^2) - 7 * pl12t + (pu3t^2) + 2 *
pl13t - 7 * pl23t) + theta * (-1 + 7 * b - 6 * b2 + b3) *
(7 * (pu2t^2) - 3 * pl12t - 3 * pl23t)) * (ll2^2) + 2 *
theta * (pu1t^2) * lt - 4 * t3 * (pu1t^2) * lt + 2 *
theta * (pu2t^2) * lt - 4 * t3 * (pu2t^2) * lt + 4 *
theta * pl12t * lt - 8 * t3 * pl12t * lt - 14 * theta *
(pu1t^2) * b * lt + 4 * t3 * (pu1t^2) * b * lt - 14 *
theta * (pu2t^2) * b * lt + 4 * t3 * (pu2t^2) * b *
lt - 28 * theta * pl12t * b * lt + 8 * t3 * pl12t * b *
lt + 12 * theta * (pu1t^2) * b2 * lt + 12 * theta *
(pu2t^2) * b2 * lt + 24 * theta * pl12t * b2 * lt - 2 *
theta * (pu1t^2) * b3 * lt - 2 * theta * (pu2t^2) * b3 *
lt - 4 * theta * pl12t * b3 * lt + 2 * theta * (pu3t^2) *
lt - 4 * t3 * (pu3t^2) * lt - 14 * theta * b * (pu3t^2) *
lt + 4 * t3 * b * (pu3t^2) * lt + 12 * theta * b2 *
(pu3t^2) * lt - 2 * theta * b3 * (pu3t^2) * lt + 4 *
theta * pl13t * lt - 8 * t3 * pl13t * lt - 28 * theta *
b * pl13t * lt + 8 * t3 * b * pl13t * lt + 24 * theta *
b2 * pl13t * lt - 4 * theta * b3 * pl13t * lt + 4 *
theta * pl23t * lt - 8 * t3 * pl23t * lt - 28 * theta *
b * pl23t * lt + 8 * t3 * b * pl23t * lt + 24 * theta *
b2 * pl23t * lt - 4 * theta * b3 * pl23t * lt +
(pu1t^2) * lpu123t - 3 * theta * (pu1t^2) * lpu123t +
2 * t2 * (pu1t^2) * lpu123t + (pu2t^2) * lpu123t - 3 *
theta * (pu2t^2) * lpu123t + 2 * t2 * (pu2t^2) *
lpu123t + 2 * pl12t * lpu123t - 6 * theta * pl12t *
lpu123t + 4 * t2 * pl12t * lpu123t - 15 * (pu1t^2) *
b * lpu123t + 21 * theta * (pu1t^2) * b * lpu123t -
6 * t2 * (pu1t^2) * b * lpu123t - 15 * (pu2t^2) * b *
lpu123t + 21 * theta * (pu2t^2) * b * lpu123t - 6 * t2 *
(pu2t^2) * b * lpu123t - 30 * pl12t * b * lpu123t + 42 *
theta * pl12t * b * lpu123t - 12 * t2 * pl12t * b *
lpu123t + 25 * (pu1t^2) * b2 * lpu123t - 18 * theta *
(pu1t^2) * b2 * lpu123t + 2 * t2 * (pu1t^2) * b2 *
lpu123t + 25 * (pu2t^2) * b2 * lpu123t - 18 * theta *
(pu2t^2) * b2 * lpu123t + 2 * t2 * (pu2t^2) * b2 *
lpu123t + 50 * pl12t * b2 * lpu123t - 36 * theta *
pl12t * b2 * lpu123t + 4 * t2 * pl12t * b2 * lpu123t -
10 * (pu1t^2) * b3 * lpu123t + 3 * theta * (pu1t^2) *
b3 * lpu123t - 10 * (pu2t^2) * b3 * lpu123t + 3 * theta *
(pu2t^2) * b3 * lpu123t - 20 * pl12t * b3 * lpu123t + 6 *
theta * pl12t * b3 * lpu123t + (pu1t^2) * b4 * lpu123t +
(pu2t^2) * b4 * lpu123t + 2 * pl12t * b4 * lpu123t +
(pu3t^2) * lpu123t - 3 * theta * (pu3t^2) * lpu123t + 2 *
t2 * (pu3t^2) * lpu123t - 15 * b * (pu3t^2) * lpu123t +
21 * theta * b * (pu3t^2) * lpu123t - 6 * t2 * b *
(pu3t^2) * lpu123t + 25 * b2 * (pu3t^2) * lpu123t - 18 *
theta * b2 * (pu3t^2) * lpu123t + 2 * t2 * b2 *
(pu3t^2) * lpu123t - 10 * b3 * (pu3t^2) * lpu123t + 3 *
theta * b3 * (pu3t^2) * lpu123t + b4 * (pu3t^2) *
lpu123t + 2 * pl13t * lpu123t - 6 * theta * pl13t *
lpu123t + 4 * t2 * pl13t * lpu123t - 30 * b * pl13t *
lpu123t + 42 * theta * b * pl13t * lpu123t - 12 * t2 *
b * pl13t * lpu123t + 50 * b2 * pl13t * lpu123t - 36 *
theta * b2 * pl13t * lpu123t + 4 * t2 * b2 * pl13t *
lpu123t - 20 * b3 * pl13t * lpu123t + 6 * theta * b3 *
pl13t * lpu123t + 2 * b4 * pl13t * lpu123t + 2 * pl23t *
lpu123t - 6 * theta * pl23t * lpu123t + 4 * t2 * pl23t *
lpu123t - 30 * b * pl23t * lpu123t + 42 * theta * b *
pl23t * lpu123t - 12 * t2 * b * pl23t * lpu123t + 50 *
b2 * pl23t * lpu123t - 36 * theta * b2 * pl23t *
lpu123t + 4 * t2 * b2 * pl23t * lpu123t - 20 * b3 *
pl23t * lpu123t + 6 * theta * b3 * pl23t * lpu123t + 2 *
b4 * pl23t * lpu123t - 6 * t4 * (pu1t^2) * ll3 + 8 * t5 *
(pu1t^2) * ll3 - 6 * t4 * (pu2t^2) * ll3 + 8 * t5 *
(pu2t^2) * ll3 - 12 * t4 * pl12t * ll3 + 16 * t5 *
pl12t * ll3 + 6 * t4 * (pu1t^2) * b * ll3 + 6 * t4 *
(pu2t^2) * b * ll3 + 12 * t4 * pl12t * b * ll3 - 2 * t2 *
(pu3t^2) * ll3 + 16 * t4 * (pu3t^2) * ll3 - 16 * t5 *
(pu3t^2) * ll3 + 14 * t2 * b * (pu3t^2) * ll3 - 16 * t4 *
b * (pu3t^2) * ll3 - 12 * t2 * b2 * (pu3t^2) * ll3 + 2 *
t2 * b3 * (pu3t^2) * ll3 - 2 * t2 * pl13t * ll3 + 10 *
t4 * pl13t * ll3 - 8 * t5 * pl13t * ll3 + 14 * t2 * b *
pl13t * ll3 - 10 * t4 * b * pl13t * ll3 - 12 * t2 * b2 *
pl13t * ll3 + 2 * t2 * b3 * pl13t * ll3 - 2 * t2 *
pl23t * ll3 + 10 * t4 * pl23t * ll3 - 8 * t5 * pl23t *
ll3 + 14 * t2 * b * pl23t * ll3 - 10 * t4 * b * pl23t *
ll3 - 12 * t2 * b2 * pl23t * ll3 + 2 * t2 * b3 * pl23t *
ll3 - 2 * t2 * (pu1t^2) * lt * ll3 + 6 * t3 * (pu1t^2) *
lt * ll3 - 4 * t4 * (pu1t^2) * lt * ll3 - 2 * t2 *
(pu2t^2) * lt * ll3 + 6 * t3 * (pu2t^2) * lt * ll3 - 4 *
t4 * (pu2t^2) * lt * ll3 - 4 * t2 * pl12t * lt * ll3 +
12 * t3 * pl12t * lt * ll3 - 8 * t4 * pl12t * lt * ll3 +
14 * t2 * (pu1t^2) * b * lt * ll3 - 18 * t3 * (pu1t^2) *
b * lt * ll3 + 4 * t4 * (pu1t^2) * b * lt * ll3 + 14 *
t2 * (pu2t^2) * b * lt * ll3 - 18 * t3 * (pu2t^2) * b *
lt * ll3 + 4 * t4 * (pu2t^2) * b * lt * ll3 + 28 * t2 *
pl12t * b * lt * ll3 - 36 * t3 * pl12t * b * lt * ll3 +
8 * t4 * pl12t * b * lt * ll3 - 12 * t2 * (pu1t^2) * b2 *
lt * ll3 + 6 * t3 * (pu1t^2) * b2 * lt * ll3 - 12 * t2 *
(pu2t^2) * b2 * lt * ll3 + 6 * t3 * (pu2t^2) * b2 * lt *
ll3 - 24 * t2 * pl12t * b2 * lt * ll3 + 12 * t3 * pl12t *
b2 * lt * ll3 + 2 * t2 * (pu1t^2) * b3 * lt * ll3 + 2 *
t2 * (pu2t^2) * b3 * lt * ll3 + 4 * t2 * pl12t * b3 *
lt * ll3 - 2 * theta * (pu3t^2) * lt * ll3 + 10 * t2 *
(pu3t^2) * lt * ll3 - 16 * t3 * (pu3t^2) * lt * ll3 +
8 * t4 * (pu3t^2) * lt * ll3 + 30 * theta * b *
(pu3t^2) * lt * ll3 - 70 * t2 * b * (pu3t^2) * lt *
ll3 + 48 * t3 * b * (pu3t^2) * lt * ll3 - 8 * t4 * b *
(pu3t^2) * lt * ll3 - 50 * theta * b2 * (pu3t^2) * lt *
ll3 + 60 * t2 * b2 * (pu3t^2) * lt * ll3 - 16 * t3 * b2 *
(pu3t^2) * lt * ll3 + 20 * theta * b3 * (pu3t^2) * lt *
ll3 - 10 * t2 * b3 * (pu3t^2) * lt * ll3 - 2 * theta *
b4 * (pu3t^2) * lt * ll3 - 2 * theta * pl13t * lt * ll3 +
8 * t2 * pl13t * lt * ll3 - 10 * t3 * pl13t * lt * ll3 +
4 * t4 * pl13t * lt * ll3 + 30 * theta * b * pl13t * lt *
ll3 - 56 * t2 * b * pl13t * lt * ll3 + 30 * t3 * b *
pl13t * lt * ll3 - 4 * t4 * b * pl13t * lt * ll3 - 50 *
theta * b2 * pl13t * lt * ll3 + 48 * t2 * b2 * pl13t *
lt * ll3 - 10 * t3 * b2 * pl13t * lt * ll3 + 20 * theta *
b3 * pl13t * lt * ll3 - 8 * t2 * b3 * pl13t * lt * ll3 -
2 * theta * b4 * pl13t * lt * ll3 - 2 * theta * pl23t *
lt * ll3 + 8 * t2 * pl23t * lt * ll3 - 10 * t3 * pl23t *
lt * ll3 + 4 * t4 * pl23t * lt * ll3 + 30 * theta * b *
pl23t * lt * ll3 - 56 * t2 * b * pl23t * lt * ll3 + 30 *
t3 * b * pl23t * lt * ll3 - 4 * t4 * b * pl23t * lt *
ll3 - 50 * theta * b2 * pl23t * lt * ll3 + 48 * t2 *
b2 * pl23t * lt * ll3 - 10 * t3 * b2 * pl23t * lt *
ll3 + 20 * theta * b3 * pl23t * lt * ll3 - 8 * t2 *
b3 * pl23t * lt * ll3 - 2 * theta * b4 * pl23t * lt *
ll3 + t4 * (pu1t^2) * ll32 - 3 * t5 * (pu1t^2) * ll32 +
2 * t6 * (pu1t^2) * ll32 + t4 * (pu2t^2) * ll32 - 3 *
t5 * (pu2t^2) * ll32 + 2 * t6 * (pu2t^2) * ll32 + 2 *
t4 * pl12t * ll32 - 6 * t5 * pl12t * ll32 + 4 * t6 *
pl12t * ll32 - 3 * t4 * (pu1t^2) * a + 3 * t5 *
(pu1t^2) * a - 3 * t4 * (pu2t^2) * a + 3 * t5 *
(pu2t^2) * a - 6 * t4 * pl12t * a + 6 * t5 * pl12t * a +
t4 * (pu1t^2) * b2 * ll32 + t4 * (pu2t^2) * b2 * ll32 +
2 * t4 * pl12t * b2 * ll32 + t2 * (pu3t^2) * ll32 - 7 *
t3 * (pu3t^2) * ll32 + 18 * t4 * (pu3t^2) * ll32 - 20 *
t5 * (pu3t^2) * ll32 + 8 * t6 * (pu3t^2) * ll32 - 15 *
t2 * b * (pu3t^2) * ll32 + 49 * t3 * b * (pu3t^2) *
ll32 - 54 * t4 * b * (pu3t^2) * ll32 + 20 * t5 * b *
(pu3t^2) * ll32 + 25 * t2 * b2 * (pu3t^2) * ll32 - 42 *
t3 * b2 * (pu3t^2) * ll32 + 18 * t4 * b2 * (pu3t^2) *
ll32 - 10 * t2 * b3 * (pu3t^2) * ll32 + 7 * t3 * b3 *
(pu3t^2) * ll32 + t2 * b4 * (pu3t^2) * ll32 + 3 * t3 *
pl13t * ll32 - 16 * t4 * pl13t * ll32 + 27 * t5 * pl13t *
ll32 - 14 * t6 * pl13t * ll32 - 21 * t3 * b * pl13t *
ll32 + 48 * t4 * b * pl13t * ll32 - 27 * t5 * b * pl13t *
ll32 + 18 * t3 * b2 * pl13t * ll32 - 16 * t4 * b2 *
pl13t * ll32 - 3 * t3 * b3 * pl13t * ll32 + 3 * t3 *
pl23t * ll32 - 16 * t4 * pl23t * ll32 + 27 * t5 * pl23t *
ll32 - 14 * t6 * pl23t * ll32 - 21 * t3 * b * pl23t *
ll32 + 48 * t4 * b * pl23t * ll32 - 27 * t5 * b * pl23t *
ll32 + 18 * t3 * b2 * pl23t * ll32 - 16 * t4 * b2 *
pl23t * ll32 - 3 * t3 * b3 * pl23t * ll32 - 2 * theta *
ll1 * (-(theta * (pl12t * (1 - 15 * b + 25 * b2 - 10 *
b3 + b4) - theta * (-1 + 7 * b - 6 * b2 + b3) *
((pu1t^2) + (pu2t^2) - 8 * pl12t + pl13t + pl23t) - 2 *
t4 * (2 * (pu1t^2) + 2 * (pu2t^2) - 8 * pl12t -
(pu3t^2) + pl13t + pl23t) - t2 * (1 - 3 * b + b2) * (5 *
(pu1t^2) + 5 * (pu2t^2) - 25 * pl12t - (pu3t^2) + 4 *
pl13t + 4 * pl23t) - t3 * (-1 + b) * (8 * (pu1t^2) + 8 *
(pu2t^2) - 34 * pl12t - 3 * (pu3t^2) + 5 * pl13t + 5 *
pl23t)) * ll2) + ((1 - 15 * b + 25 * b2 - 10 * b3 + b4) *
((pu1t^2) + pl12t + pl13t) + t2 * (1 - 3 * b + b2) * (8 *
(pu1t^2) - 3 * (pu2t^2) + 5 * pl12t - 3 * (pu3t^2) + 5 *
pl13t - 6 * pl23t) + 2 * t3 * (-1 + b) * (2 * (pu1t^2) -
(pu2t^2) + pl12t - (pu3t^2) + pl13t - 2 * pl23t) +
theta * (-1 + 7 * b - 6 * b2 + b3) * (5 * (pu1t^2) -
(pu2t^2) + 4 * pl12t - (pu3t^2) + 4 * pl13t - 2 *
pl23t)) * lt + theta * ((pu1t^2) - 8 * t2 * (pu1t^2) +
8 * t3 * (pu1t^2) + 3 * t2 * (pu2t^2) - 4 * t3 *
(pu2t^2) + pl12t - 5 * t2 * pl12t + 4 * t3 * pl12t - 7 *
(pu1t^2) * b + 8 * t2 * (pu1t^2) * b - 3 * t2 *
(pu2t^2) * b - 7 * pl12t * b + 5 * t2 * pl12t * b + 6 *
(pu1t^2) * b2 + 6 * pl12t * b2 - (pu1t^2) * b3 - pl12t *
b3 + 3 * t2 * (pu3t^2) - 4 * t3 * (pu3t^2) - 3 * t2 * b *
(pu3t^2) + pl13t - 5 * t2 * pl13t + 4 * t3 * pl13t - 7 *
b * pl13t + 5 * t2 * b * pl13t + 6 * b2 * pl13t - b3 *
pl13t + 6 * t2 * pl23t - 8 * t3 * pl23t - 6 * t2 * b *
pl23t - ((1 - 15 * b + 25 * b2 - 10 * b3 + b4) *
pl13t - theta * (-1 + 7 * b - 6 * b2 + b3) * ((pu1t^2) +
pl12t + (pu3t^2) - 8 * pl13t + pl23t) - 2 * t4 * (2 *
(pu1t^2) - (pu2t^2) + pl12t + 2 * (pu3t^2) - 8 * pl13t +
pl23t) - t2 * (1 - 3 * b + b2) * (5 * (pu1t^2) -
(pu2t^2) + 4 * pl12t + 5 * (pu3t^2) - 25 * pl13t + 4 *
pl23t) - t3 * (-1 + b) * (8 * (pu1t^2) - 3 * (pu2t^2) +
5 * pl12t + 8 * (pu3t^2) - 34 * pl13t + 5 * pl23t)) *
ll3)) - 2 * theta * ll2 * ((-(t2 * (1 - 3 * b + b2) *
(3 * (pu1t^2) - 8 * (pu2t^2) - 5 * pl12t + 3 * (pu3t^2) +
6 * pl13t - 5 * pl23t)) - theta * (-1 + 7 * b - 6 * b2 +
b3) * ((pu1t^2) - 5 * (pu2t^2) - 4 * pl12t + (pu3t^2) +
2 * pl13t - 4 * pl23t) - 2 * t3 * (-1 + b) * ((pu1t^2) -
2 * (pu2t^2) - pl12t + (pu3t^2) + 2 * pl13t - pl23t) +
(1 - 15 * b + 25 * b2 - 10 * b3 + b4) * ((pu2t^2) +
pl12t + pl23t)) * lt + theta * (3 * t2 * (pu1t^2) - 4 *
t3 * (pu1t^2) + (pu2t^2) - 8 * t2 * (pu2t^2) + 8 * t3 *
(pu2t^2) + pl12t - 5 * t2 * pl12t + 4 * t3 * pl12t - 3 *
t2 * (pu1t^2) * b - 7 * (pu2t^2) * b + 8 * t2 *
(pu2t^2) * b - 7 * pl12t * b + 5 * t2 * pl12t * b +
6 * (pu2t^2) * b2 + 6 * pl12t * b2 - (pu2t^2) * b3 -
pl12t * b3 + 3 * t2 * (pu3t^2) - 4 * t3 * (pu3t^2) - 3 *
t2 * b * (pu3t^2) + 6 * t2 * pl13t - 8 * t3 * pl13t - 6 *
t2 * b * pl13t + pl23t - 5 * t2 * pl23t + 4 * t3 *
pl23t - 7 * b * pl23t + 5 * t2 * b * pl23t + 6 * b2 *
pl23t - b3 * pl23t - ((1 - 15 * b + 25 * b2 - 10 * b3 +
b4) * pl23t - theta * (-1 + 7 * b - 6 * b2 + b3) *
((pu2t^2) + pl12t + (pu3t^2) + pl13t - 8 * pl23t) + 2 *
t4 * ((pu1t^2) - 2 * (pu2t^2) - pl12t - 2 * (pu3t^2) -
pl13t + 8 * pl23t) + t2 * (1 - 3 * b + b2) * ((pu1t^2) -
5 * (pu2t^2) - 4 * pl12t - 5 * (pu3t^2) - 4 * pl13t +
25 * pl23t) + t3 * (-1 + b) * (3 * (pu1t^2) - 8 *
(pu2t^2) - 5 * pl12t - 8 * (pu3t^2) - 5 * pl13t + 34 *
pl23t)) * ll3)))) / (t4 * ((1 + 2 * t2 + 3 * theta *
(-1 + b) - 3 * b + b2)^(2)) * ((pu123t)^(2)))
return(res.f)
}
# 3d first derivative wrt theta of the log-density
.gumb.V.123 <- function(theta, u) {
if (is.null(dim(u))) {
u1 <- u[1]
u2 <- u[2]
u3 <- u[3]
} else {
u1 <- u[, 1]
u2 <- u[, 2]
u3 <- u[, 3]
}
l1 <- log(u1)
l2 <- log(u2)
l3 <- log(u3)
ll1 <- log(-l1)
ll2 <- log(-l2)
ll3 <- log(-l3)
ti <- 1 / theta
pu1t <- (-l1)^theta
pu2t <- (-l2)^theta
pu3t <- (-l3)^theta
pu123t <- pu1t + pu2t + pu3t
lt <- log(pu123t)
b <- .Power(pu123t, ti)
b2 <- b^2
b3 <- b^3
t2 <- theta^2
t3 <- theta^3
t4 <- theta^4
t5 <- theta^5
res.f <- (-3 * t2 * pu1t + 4 * t3 * pu1t - 3 * t2 * pu2t + 4 * t3 * pu2t +
3 * t2 * pu1t * b + 3 * t2 * pu2t * b - 3 * t2 * pu3t + 4 * t3 *
pu3t + 3 * t2 * b * pu3t - theta * (pu1t * (-1 + 7 * b - 6 * b2 +
b3) + t2 * (-1 + b) * (8 * pu1t - 3 * pu2t - 3 * pu3t) + t3 *
(4 * pu1t - 2 * pu2t - 2 * pu3t) + theta * (1 - 3 * b + b2) * (5 *
pu1t - pu2t - pu3t)) * ll1 - theta * (pu2t * (-1 + 7 * b - 6 *
b2 + b3) - theta * (1 - 3 * b + b2) * (pu1t - 5 * pu2t + pu3t) -
2 * t3 * (pu1t - 2 * pu2t + pu3t) - t2 * (-1 + b) * (3 * pu1t -
8 * pu2t + 3 * pu3t)) * ll2 - pu1t * lt + 3 * theta * pu1t * lt -
2 * t2 * pu1t * lt - pu2t * lt + 3 * theta * pu2t * lt - 2 * t2 *
pu2t * lt + 7 * pu1t * b * lt - 9 * theta * pu1t * b * lt + 2 *
t2 * pu1t * b * lt + 7 * pu2t * b * lt - 9 * theta * pu2t * b *
lt + 2 * t2 * pu2t * b * lt - 6 * pu1t * b2 * lt + 3 * theta *
pu1t * b2 * lt - 6 * pu2t * b2 * lt + 3 * theta * pu2t * b2 * lt +
pu1t * b3 * lt + pu2t * b3 * lt - pu3t * lt + 3 * theta *
pu3t * lt - 2 * t2 * pu3t * lt + 7 * b * pu3t * lt - 9 * theta *
b * pu3t * lt + 2 * t2 * b * pu3t * lt - 6 * b2 * pu3t * lt + 3 *
theta * b2 * pu3t * lt + b3 * pu3t * lt + t2 * pu1t * ll3 - 3 *
t3 * pu1t * ll3 + 2 * t4 * pu1t * ll3 + t2 * pu2t * ll3 - 3 * t3 *
pu2t * ll3 + 2 * t4 * pu2t * ll3 - 3 * t2 * pu1t * b * ll3 + 3 *
t3 * pu1t * b * ll3 - 3 * t2 * pu2t * b * ll3 + 3 * t3 * pu2t * b *
ll3 + t2 * pu1t * b2 * ll3 + t2 * pu2t * b2 * ll3 + theta * pu3t *
ll3 - 5 * t2 * pu3t * ll3 + 8 * t3 * pu3t * ll3 - 4 * t4 * pu3t *
ll3 - 7 * theta * b * pu3t * ll3 + 15 * t2 * b * pu3t * ll3 - 8 *
t3 * b * pu3t * ll3 + 6 * theta * b2 * pu3t * ll3 - 5 * t2 * b2 *
pu3t * ll3 - theta * b3 * pu3t * ll3) / (t2 * (1 + 2 * t2 + 3 *
theta * (-1 + b) - 3 * b + b2) * (pu123t))
return(res.f)
}
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