A central role in parametric estimation is played by the likelihood function, though in presence of outliers its estimates are very unstable; for this reason alternative estimators have been proposed in literature, for instance Huber (1981) and Hampel et al. (1986). The approach based on minimizing the Integrated Square Error, or L2 estimate criterion, seems to be particularly helpful in all those situations where, due to large sample size, a careful data preparation is not feasible and hence data may heavily be contaminated by outliers, e.g. Scott (2001). This choice can be motivated by the fact that in the α-family of estimators proposed by Basu et al. (1998), L2 E is the more robust to outliers, even if it is the less efficient. But it must pointed out that the main problem of its use is due to the fact that explicit forms for the solutions are generally not available. However, in many situations, we dispose of feasible computationally closed-forms expressions so that L2 criteria can be performed by any standard non linear optimization code. To this proposal, it is important to remind that, whatever the algorithm, its convergence to optimal solutions strongly depends on its initial configurations. Several random based methods to generate the vector of the initial guesses for the algorithm have been proposed in literature, see for instance McLachlan (1988), Markatou (2000b) and Scott (2005), In our study we investigate the possibility to resort to an heuristic optimization approach, e.g. Winker (2001). More precisely, for a given problem, we exploit the heuristic approach, which does not provide an exact solution, to obtain an initial configuration for classic minimization algorithms. In our paper we shall discuss our procedure with the aim of outliers detection in the case of finite mixtures of normal densities.
The Heuristic Approach in Finding Initial Guesses for L_2 Robust Estimates
ISAIA, Ennio Davide;DURIO, Alessandra
2006-01-01
Abstract
A central role in parametric estimation is played by the likelihood function, though in presence of outliers its estimates are very unstable; for this reason alternative estimators have been proposed in literature, for instance Huber (1981) and Hampel et al. (1986). The approach based on minimizing the Integrated Square Error, or L2 estimate criterion, seems to be particularly helpful in all those situations where, due to large sample size, a careful data preparation is not feasible and hence data may heavily be contaminated by outliers, e.g. Scott (2001). This choice can be motivated by the fact that in the α-family of estimators proposed by Basu et al. (1998), L2 E is the more robust to outliers, even if it is the less efficient. But it must pointed out that the main problem of its use is due to the fact that explicit forms for the solutions are generally not available. However, in many situations, we dispose of feasible computationally closed-forms expressions so that L2 criteria can be performed by any standard non linear optimization code. To this proposal, it is important to remind that, whatever the algorithm, its convergence to optimal solutions strongly depends on its initial configurations. Several random based methods to generate the vector of the initial guesses for the algorithm have been proposed in literature, see for instance McLachlan (1988), Markatou (2000b) and Scott (2005), In our study we investigate the possibility to resort to an heuristic optimization approach, e.g. Winker (2001). More precisely, for a given problem, we exploit the heuristic approach, which does not provide an exact solution, to obtain an initial configuration for classic minimization algorithms. In our paper we shall discuss our procedure with the aim of outliers detection in the case of finite mixtures of normal densities.
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
