hparetomixt.negloglike.tailpen: Maximum Likelihood Estimation for a Mixture of Hybrid Paretos...
In condmixt: Conditional Density Estimation with Neural Network Conditional Mixtures

Description Usage Arguments Details Value Author(s) References See Also Examples

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.

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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, ...)

`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`

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.

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)

Julie Carreau

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

hparetomixt.init, hparetomixt.negloglike

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)