dep2fit: Dependence model fit (stepwise)

Description

The conditional Heffernan--Tawn model is used to fit the dependence in time of a stationary series. A standard 2-stage procedure is used.

Usage

dep2fit(ts, u.mar = 0, u.dep,
        lapl = FALSE, method.mar = c("mle","mom","pwm"),
        nlag = 1, conditions = TRUE)

Value

alpha: parameter controlling the conditional extremal expectation.
beta: parameter controlling the conditional extremal expectation and variance.
res: empirical residual of the model.
pars.se: vector of length 2 giving the estimated standard errors for alpha and beta given by the hessian matrix of the likelihood function used in the first step of the inference procedure.

Arguments

ts: numeric vector; time series to be fitted.
u.mar: marginal threshold; used when transforming the time series to Laplace scale.
u.dep: dependence threshold; level above which the dependence is modelled. u.dep can be lower than u.mar.
lapl: logical; is ts on the Laplace scale already? The default (FALSE) assumes unknown marginal distribution.
method.mar: a character string defining the method used to estimate the marginal GPD; either "mle" for maximum likelihood of "mom" for method of moments. Defaults to "mle".
nlag: integer; number of lags to be considered when modelling the dependence in time.
conditions: logical; should conditions on $\alpha$ and $\beta$ be set? (see Details) Defaults to TRUE.

Details

Consider a stationary time series $(X_t)$ with Laplace marginal distribution; the fitting procedure consists of fitting $$X_t = \alpha_t\times x_0 + x_0^{\beta_t}\times Z_t,\quad t=1,\ldots,m,$$ with $m$ the number of lags considered. A likelihood is maximised assuming $Z_t\sim N(\mu_t, \sigma^2_t)$, then an empirical distribution for the $Z_t$ is derived using the estimates of $\alpha_t$ and $\beta_t$ and the relation $$\hat Z_t = \frac{X_t - \hat\alpha_t\times x_0}{x_0^{\hat\beta_t}}.$$

conditions implements additional conditions suggested by Keef, Papastathopoulos and Tawn (2013) on the ordering of conditional quantiles. These conditions help with getting a consistent fit by shrinking the domain in which $(\alpha,\beta)$ live.

Examples

Run this code

## generate data from an AR(1)
## with Gaussian marginal distribution
n   <- 10000
dep <- 0.5
ar    <- numeric(n)
ar[1] <- rnorm(1)
for(i in 2:n)
  ar[i] <- rnorm(1, mean=dep*ar[i-1], sd=1-dep^2)
plot(ar, type="l")
plot(density(ar))
grid <- seq(-3,3,0.01)
lines(grid, dnorm(grid), col="blue")

## rescale margin
ar <- qlapl(pnorm(ar))

## fit model without constraints...
fit1 <- dep2fit(ts=ar, u.mar = 0.95, u.dep=0.98, conditions=FALSE)
fit1$a; fit1$b

## ...and compare with a fit with constraints
fit2 <- dep2fit(ts=ar, u.mar = 0.95, u.dep=0.98, conditions=TRUE)
fit2$a; fit2$b# should be similar, as true parameters lie well within the constraints