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Abstract:
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In this talk, we propose a general framework to study the singularity structures of the Fisher information matrix under finite mixtures of skew normal distributions. These models have become popular in recent years due to their flexibility in modeling asymmetric data. However, they appear to contain various singularities under both the exact-fitted setting, i.e when the number of mixing components is known, or the over-fitted setting, i.e when the number of mixing components is bounded but unknown. These singularities happen not only in the vicinity of symmetry but also in the setting of homologous sets, a phenomenon due to the interaction among the parameters of the mixing measure. Apart from these singularities, skew normal density also has the non-linear partial differential equation structure. It leads to two ways of characterizing the singularity levels. One way is based on the solvability of inhomogeneous system of polynomial equations while the another way is based on the combining strength among multiple systems of polynomial equations. The rich spectrum of the singularity structures consequently leads to various degrees of parameter estimation under skew normal mixtures.
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