ExtremalDep (version 0.0.3-3)

alik: Approximate likelihood estimation of extremal dependence models.

Description

Estimates the parameters of extremal dependence models. It also provides standard errors and TIC.

Usage

alik(data, model, parastart, c=NULL, trace=0, sig=3)

Arguments

data

A (\(n \times d\)) matrix of angular components, where the rows represent \(n\) independent points in the \(d\)-dimensional unit simplex.

model

A string with the name of the parametric model to be estimated. See Details.

parastart

A vector containing the starting values of the model's parameters for the maximisation of the log-approximate likelihood. See Details.

c

A real value in \([0,1]\), providing the decision rule to allocate a data point to a subset of the simplex. Only required for the Extremal-t, Extremal Skew-t and Asymmetric Logistic models.

trace

Non-negative integer. See the options of the routine optim in R for details. trace=0 is the default.

sig

Non-negative integer. Provides the number of decimal places for the returned object. sig=3 is the default.

Value

Returns a list where par are the estimated parameters, LL is the value of the maximized log-likelihood, TIC is the Takeuchi Information Criterion and SE are the standard errors.

Details

The available parametric extremal dependence models are:

• The Pairwise Beta, called with model="Pairwise". The number of parameters is choose(d,2)+1;

• The Husler-Reiss, called with model="Husler". The number of parameters is choose(d,2);

• The Tilted Dirichlet, called with model="Dirichlet". The number of parameters is \(d\);

• The Extremal-t, called with model="Extremalt". The number of parameters is choose(d,2)+1;

• The Extremal Skew-t, called with model="Skewt". The number of parameters is choose(d,2)+d+1;

• The Asymmetric Logistic, that can be called with model="Asymmetric". The number of dependence parameters is \(2^{d-1}(d+2)-(2d+1)\).

See References and the references therein.

Standard errors are calculated using the sandwich (Godambe) information matrix.

References

Beranger, B. and Padoan, S. A. (2015). Extreme dependence models, chapater of the book Extreme Value Modeling and Risk Analysis: Methods and Applications, Chapman Hall/CRC.

Beranger, B., Padoan, S. A. and Sisson, S. A. (2017). Models for extremal dependence derived from skew-symmetric families. Scandinavian Journal of Statistics, 44(1), 21-45.

Examples

# NOT RUN {
################################################
# The following examples provide the fitting
# results of the air quality data recorded in
# the city center of Leeds, UK, analysed in
# Beranger and Padoan (2015).
################################################

## Load datsets
data(pollution)


## Dataset PM10-NO-SO2 (PNS)

# }
# NOT RUN {
alik(PNS,model="Pairwise",c(1,1,1,1),trace=2,sig=2)
alik(PNS,model="Husler",rep(1,3),trace=2,sig=2)
alik(PNS,model="Dirichlet",rep(0.1,3),trace=2,sig=2)
alik(PNS,model="Extremalt",c(-0.5,-0.4,-0.5,1),c=0.01,trace=2,sig=2)
alik(PNS,model="Asymmetric",c(rep(1.1,4),rep(0.1,9)),c=0.01,trace=2,sig=2)
# }
# NOT RUN {
## Dataset NO2-SO2-NO (NSN)

# }
# NOT RUN {
alik(NSN,model="Pairwise",c(1,1,1,1),trace=2,sig=2)
alik(NSN,model="Husler",rep(1,3),trace=2,sig=2)
alik(NSN,model="Dirichlet",rep(0.1,3),trace=2,sig=2)
alik(NSN,model="Extremalt",c(-0.5,-0.4,-0.5,1),c=0.01,trace=2,sig=2)
alik(NSN,model="Asymmetric",c(rep(1.1,4),rep(0.1,9)),c=0.01,trace=2,sig=2)
# }
# NOT RUN {
## Dataset PM10-NO-NO2 (PNN)

# }
# NOT RUN {
alik(PNN,model="Pairwise",c(1,1,1,1),trace=2,sig=2)
alik(PNN,model="Husler",rep(1,3),trace=2,sig=2)
alik(PNN,model="Dirichlet",rep(0.1,3),trace=2,sig=2)
alik(PNN,model="Extremalt",c(-0.5,-0.4,-0.5,1),c=0.01,trace=2,sig=2)
alik(PNN,model="Asymmetric",c(rep(1.1,4),rep(0.1,9)),c=0.01,trace=2,sig=2)
# }
# NOT RUN {
## Dataset PM10-NO-NO2-SO2 (PNNS)

# }
# NOT RUN {
alik(PNNS,model="Pairwise",rep(1,choose(ncol(PNNS),2)+1),trace=2,sig=2)
alik(PNNS,model="Husler",rep(1,choose(ncol(PNNS),2)),trace=2,sig=2)
alik(PNNS,model="Dirichlet",rep(1,ncol(PNNS)),trace=2,sig=2)
# }
# NOT RUN {
# }