condmixt (version 1.1)

hparetomixt.negloglike.tailpen: Maximum Likelihood Estimation for a Mixture of Hybrid Paretos with Tail Penalty

Description

In order to drive the tail index estimation, a penalty is introduced in the log-likelihood. The goal of the penalty is to include a priori information which in our case is that only a few mixture components have a heavy tail index which should approximate the tail of the underlying distribution while most other mixture components have a light tail and aim at modelling the central part of the underlying distribution.

Usage

hparetomixt.negloglike.tailpen(params, lambda, w, beta, mu, sigma, x)
hparetomixt.fit.tailpen(params, lambda, w, beta, mu, sigma, x, ...)
hparetomixt.cvtrain.tailpen(m, lambda, w, beta, mu, sigma, x, nfold=5, nstart=1, ...)

Arguments

params

matrix of dimension 4 by m, where m is the number of components, each column of the matrix contains the mixture parameters of one component (pi, xi, mu, sigma)

m

number of mixture components

lambda

penalty parameter which controls the trade-off between the penalty and the negative log-likelihood, takes on positive values

w

penalty parameter in [0,1] which is the proportion of components with light tails, 1-w being the proportion of components with heavy tails

beta

positive penalty parameter which indicates the importance of the light tail components (it is the parameter of an exponential which represents the prior over the light tail components)

mu

penalty parameter in (0,1) which represents the a priori value for the heavy tail index of the underlying distribution

sigma

positive penalty parameter which controls the spread around the a priori value for the heavy tail index of the underlying distribution

x

a vector of length n of observations assumed to be sampled from a mixture of hybrid Paretos

nfold

number of fold for cross-validation estimate, default is 5

nstart

number of re-starts for the optimizer nlm with different initial parameters, default is 1

…

optional arguments for nlm

Value

hparetomixt.negloglike.tailpen returns a single value (the negative log-likelihood for given parameters and sample) and a vector, the gradient, which is passed as an attribute, while hparetomixt.fit.tailpen returns a 4 by m matrix of MLE for the hybrid Pareto mixture parameters and hparetomixt.cvtrain.tailpen returns a cross-validation estimate of the out-of-sample negative log-likelihood for the given model (number of components and penalty parameters)

Details

The penalty term is given by the logarithm of the following two-component mixture, as a function of a tail index parameter xi : w beta exp(-beta xi) + (1-w) exp(-(xi-mu)^2/(2 sigma^2))/(sqrt(2 pi) sigma) where the first term is the prior on the light tail component and the second term is the prior on the heavy tail component.

References

Carreau, J.,Naveau, P. and Sauquet, E. (2009), A statistical rainfall-runoff mixture model with heavy-tailed components, 45, Water Resources Research

See Also

hparetomixt.init, hparetomixt.negloglike

Examples

# NOT RUN {
r <- rfrechet(500,loc=5,scale=5,shape=5)
m <- 2
param.init <- hparetomixt.init(m,r)
hparetomixt.negloglike.tailpen(param.init,10,0.5,20,0.1,0.2,r)
hparetomixt.fit.tailpen(param.init,10,0.5,20,0.1,0.2,r)
hparetomixt.cvtrain.tailpen(2,10,0.5,20,0.1,0.2,r)
# }