The class of multivariate Erlang mixtures with common scale parameter has many desirable properties and has widely been used in insurance loss modeling. The parameters of a multivariate Erlang mixture are normally estimated using an expectation–maximization (EM) algorithm as shown in Lee and Lin (2012) and Verbelen et al. (2016). However, when fitting the mixture to data of high dimension, the fitted density surface is often not smooth (with deep peaks and valleys) and the tail fitting may also be rather unsatisfactory. In this paper, we propose a generalized expectation conditional maximization (GECM) algorithm that maximizes a penalized likelihood with a proposed roughness penalty. The roughness penalty is based on integrated squared second derivative of the density function of aggregate data, which is used in functional data analysis. We illustrate the performance of the proposed method through some numerical experiments and real data applications.

具有通用比例参数的多元Erlang混合物类别具有许多理想的属性，并已广泛用于保险损失建模中。通常使用期望最大化（EM）算法估算多元Erlang混合物的参数，如Lee和Lin（2012）和Verbelen等人所示。（2016）。但是，当将混合物拟合为高维数据时，拟合的密度表面通常不平滑（具有深的峰和谷），并且尾部拟合也可能不太令人满意。在本文中，我们提出了一种广义期望条件条件最大化（GECM）算法，该算法以拟议的粗糙度代价最大化了被惩罚的可能性。粗糙度损失基于集合数据的密度函数的平方二阶平方平方，该平方二阶导数用于函数数据分析。