The estimation of the parameters of a mixture of Gaussian densities is considered, within the framework of maximum likelihood. Due to unboundedness of the likelihood function, the maximum likelihood estimator fails to exist. We adopt a solution to likelihood function degeneracy which consists in penalizing the likelihood function. The resulting penalized likelihood function is then bounded over the parameter space and the existence of the penalized maximum likelihood estimator is granted. As original contribution we provide asymptotic properties, and in particular a consistency proof, for the penalized maximum likelihood estimator. Numerical examples are provided in the finite data case, showing the performances of the penalized estimator compared to the standard one.
To cite this version:Gabriela Ciuperca, Cécile Mercadier. Semi-parametric estimation for heavy tailed distributions. Extremes, Springer Verlag (Germany), 2010, 13 (1) In this paper, we generalize several works in the extreme value theory for the estimation of the extreme value index and the second order parameter. Weak consistency and asymptotic normality are proven under classical assumptions. Some numerical simulations and computations are also performed to illustrate the finite-sample and the limiting behavior of the estimators.
We propose a general adaptive LASSO method for a quantile regression model.
Our method is very interesting when we know nothing about the first two moments
of the model error. We first prove that the obtained estimators satisfy the
oracle properties, which involves the relevant variable selection without using
hypothesis test. Next, we study the proposed method when the (multiphase) model
changes to unknown observations called change-points. Convergence rates of the
change-points and of the regression parameters estimators in each phase are
found. The sparsity of the adaptive LASSO quantile estimators of the regression
parameters is not affected by the change-points estimation. If the phases
number is unknown, a consistent criterion is proposed. Numerical studies by
Monte Carlo simulations show the performance of the proposed method, compared
to other existing methods in the literature, for models with a single phase or
for multiphase models. The adaptive LASSO quantile method performs better than
known variable selection methods, as the least squared method with adaptive
LASSO penalty, $L_1$-method with LASSO-type penalty and quantile method with
SCAD penalty
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