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

```gpd.info <-
# Compute the observed information matrix from a gpd object.
# The expressions are given in Appendix A of Davison & Smith 1990.
# Note we are using a simpler parameterisation in which phi = log(sigma)
# and xi are both linear in their covariates. Xi is -k used in Davison and Smith.
# 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\$xi
ns <- ncol(x); nk <- ncol(z)
phi <- coef(o)[1:ns]
xi <- coef(o)[(ns+1):(ns + nk)]

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

if (any(xi.i < -.50)){ message("Fitted values of xi < -0.5") }

# 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

d2li.dphi2 <- -(1 + 1/xi.i) * xi.i * w.i / (1 + xi.i*w.i)^2
d2li.dphidxi <- 1/xi.i^2 * (1/(1 + xi.i*w.i) - 1) + (1+1/xi.i)*w.i/(1 + xi.i*w.i)^2
d2li.dxi2 <- -2/xi.i^3 * log(1 + xi.i*w.i) + 2*w.i/(xi.i^2 * (1 + xi.i*w.i)) + (1 + 1/xi.i)*w.i^2/(1 + xi.i*w.i)^2

# Matrix has 4 blocks, 2 of which are transposes of each other. Need block for phi parameters,
# block for xi 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.dxi2)
}
}

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.dphidxi )
}
}

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.