alik {ExtremalDep}

R Documentation

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.

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.

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.

Author(s)

Simone Padoan, simone.padoan@unibocconi.it, https://mypage.unibocconi.it/simonepadoan/; Boris Beranger, borisberanger@gmail.com https://www.borisberanger.com/;

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

################################################
# 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)

if (interactive()){
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)
}

## Dataset NO2-SO2-NO (NSN)

if (interactive()){
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)
}

## Dataset PM10-NO-NO2 (PNN)

if (interactive()){
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)
}

## Dataset PM10-NO-NO2-SO2 (PNNS)

if (interactive()){
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)
}

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