  In this article, we present the maximum weighted likelihood estimator (MWLE)
for robust estimations of heavy-tail finite mixture models (FMM). This is
motivated by the complex distributional phenomena of insurance claim severity
data, where flexible density estimation tools such as FMM are needed but MLE
often produces unstable tail estimates under FMM. Under some regularity
conditions, MWLE is proved to be consistent and asymptotically normal. We
further prove that the tail index obtained by MWLE is consistent even if the
model is misspecified, justifying the robustness of MWLE in estimating the tail
part of FMM. With a probabilistic interpretation for MWLE, Generalized
Expectation-Maximization (GEM) algorithm is still applicable for efficient
parameter estimations. We therefore present and compare two distinctive
constructions of complete data to implement the GEM algorithm. By exemplifying
our approach on two simulation studies and a real motor insurance data set, we
show that comparing to MLE, MWLE produces more appropriate estimations on the
tail part of FMM, without much sacrificing the flexibility of FMM in capturing
the body part.
