TITLE:
A Special Weight for Inverse Gaussian Mixing Distribution in Normal Variance Mean Mixture with Application
AUTHORS:
Calvin B. Maina, Patrick G. O. Weke, Carolyne A. Ogutu, Joseph A. M. Ottieno
KEYWORDS:
Finite Mixture, Weighted Distribution, Mixed Model, EM-Algorithm
JOURNAL NAME:
Open Journal of Statistics,
Vol.11 No.6,
December
10,
2021
ABSTRACT: Normal
Variance-Mean Mixture (NVMM) provides a general
framework for deriving models with desirable properties for modelling financial
market variables such as exchange rates, equity prices, and interest rates
measured over short time intervals, i.e. daily or weekly. Such data sets are characterized by non-normality and are
usually skewed, fat-tailed and exhibit excess kurtosis. The Generalised Hyperbolic distribution (GHD) introduced by Barndorff-Nielsen (1977) which act as Normal variance-mean mixtures with Generalised Inverse
Gaussian (GIG) mixing distribution nest a number of special and limiting case
distributions. The Normal Inverse Gaussian (NIG) distribution is obtained when
the Inverse Gaussian is the mixing distribution, i.e., the index parameter of the GIG is .
The NIG is very popular because of its analytical tractability. In the mixing
mechanism, the mixing distribution characterizes the prior information of the
random variable of the conditional distribution. Therefore, considering finite
mixture models is one way of extending the work. The GIG is a three parameter
distribution denoted by and nest several
special and limiting cases. When , we have which is called an Inverse Gaussian (IG)
distribution. When , , , we have , and distributions respectively. These distributions
are related to and are called weighted inverse Gaussian
distributions. In this work, we consider a finite mixture of and and show that the mixture is also a weighted Inverse Gaussian
distribution and use it to construct a NVMM. Due to the complexity of the
likelihood, direct maximization is difficult. An EM type algorithm is provided
for the Maximum Likelihood estimation of the parameters of the proposed model.
We adopt an iterative scheme which is not based on explicit solution to the
normal equations. This subtle approach reduces the computational difficulty of
solving the complicated quantities involved directly to designing an iterative
scheme based on a representation of the normal equation. The algorithm is
easily programmable and we obtained a monotonic convergence for the data sets
used.