# R/weibull.info.R In texmex: Statistical Modelling of Extreme Values

```weibull.info <-
# phi=log(beta) and gamma are both linear in their covariates.
# If penalization is used, the calculation accounts for this, but the resulting
# estimates of variance will be too low and bias might dominate MSE
function(o, method="observed"){
if (class(o) != "evmOpt"){ stop("object must be of class 'evmOpt'") }
if (method != "observed"){ stop("only 'observed' information is implemented") }

x <- o\$data\$D\$phi
z <- o\$data\$D\$gamma
ns <- ncol(x)
nk <- ncol(z)
phi <- coef(o)[1:ns]
gamma <- coef(o)[(ns+1):(ns + nk)]

phi.i <- colSums(phi * t(x))
gamma.i <- colSums(gamma * t(z))
w.i <- (o\$data\$y - o\$threshold)

# Second derivatives of penalties
p <- matrix(0, nrow=ns+nk, ncol=ns+nk)
if (o\$penalty %in% c("gaussian", "quadratic")){ # note if Lasso penalty used then 2nd deriv is zero hence no term for this
Si <- solve(o\$priorParameters[[2]])
for (i in 1:(ns+nk)){
for (j in 1:(ns + nk)){
p[i,j] <- 2*Si[i,j]
}
}
}

# Second and mixed derivatives of log-lik wrt coefficients of linear predictors

# linear predictors in beta
#		d2li.dbeta2 <- gamma.i / beta.i^2 * (1 - (1 + gamma.i) * (w.i/beta.i)^gamma.i )
#		d2li.dbetadgamma <- 1/beta.i  * ( (w.i/beta.i)^gamma.i - 1) + gamma.i/beta.i * (w.i/beta.i)^gamma.i / w.i * log(w.i/beta.i)
#		d2li.dgamma2 <- - gamma.i^(-2) - (w.i/beta.i)^gamma.i * (log(w.i/beta.i))^2

d2li.dphi2 <- -gamma.i  * (w.i^gamma.i ) * exp(-gamma.i*phi.i) * (1 + gamma.i)
d2li.dphidgamma <- w.i^gamma.i * exp( - gamma.i * phi.i ) *  (gamma.i * (log(w.i) - phi.i) + 1 ) - 1
d2li.dgamma2 <- - gamma.i^(-2) - (w.i/exp(phi.i))^gamma.i * (log(w.i) - phi.i)^2

# Matrix has 4 blocks, 2 of which are transposes of each other. Need block for beta parameters,
# block for gamma parameters and block for the cross of them.

Ip <- matrix(0, ncol=ns, nrow=ns)
for (u in 1:ns){
for (v in 1:ns){
Ip[u,v] <- -sum(x[,u] * x[,v] * d2li.dphi2)
}
}

Ix <- matrix(0, ncol=nk, nrow=nk)
for (s in 1:nk){
for (t in 1:nk){
Ix[s,t] <- -sum(z[,s] * z[,t] * d2li.dgamma2)
}
}

Ipx <- matrix(0, ncol=nk, nrow=ns)
for (u in 1:ns){
for (s in 1:nk){
Ipx[u,s] <- -sum(z[,s] * x[,u] * d2li.dphidgamma )
}
}

i <- rbind( cbind(Ip, Ipx), cbind(t(Ipx), Ix))

# return observed Information matrix.   Note that an estimate of the covariance matrix is given by the inverse of this matrix.
i - p
}
```

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texmex documentation built on May 2, 2019, 5:41 a.m.