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.

1 2 3 | ```
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- |

`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 |

`...` |
optional arguments for |

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`

1 2 3 4 5 6 | ```
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)
``` |

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