A Normal Weighted Inverse Gaussian Distribution for Skewed and Heavy-Tailed Data

Calvin B Maina,Joseph A M Ottieno,Patrick G O Weke,Carolyne A Ogutu

doi:10.4236/am.2022.132013

Abstract

High frequency financial data is characterized by non-normality: asymmetric, leptokurtic and fat-tailed behaviour. The normal distribution is therefore inadequate in capturing these characteristics. To this end, various flexible distributions have been proposed. It is well known that mixture distributions produce flexible models with good statistical and probabilistic properties. In this work, a finite mixture of two special cases of Generalized Inverse Gaussian distribution has been constructed. Using this finite mixture as a mixing distribution to the Normal Variance Mean Mixture we get a Normal Weighted Inverse Gaussian (NWIG) distribution. The second objective, therefore, is to construct and obtain properties of the NWIG distribution. The maximum likelihood parameter estimates of the proposed model are estimated via EM algorithm and three data sets are used for application. The result shows that the proposed model is flexible and fits the data well.

Highlights

It is well known that mixture distributions produce flexible models with good statistical and probabilistic properties
The maximum likelihood parameter estimates of the proposed model are estimated via EM algorithm and three data sets are used for application
We show that two special cases of Generalised Inverse Gaussian (GIG) distribution can be expressed as Weighted Inverse Gaussian (WIG) distribution

Summary

Introduction

It is well known that mixture distributions produce flexible models with good statistical and probabilistic properties. Our first objective is to construct and obtain properties of a finite mixture of two special cases of Generalized Inverse Gaussian distribution. Nielsen [1] as a Normal Variance-Mean Mixture is obtained when the Generalized Inverse Gaussian (GIG) distribution is the mixing distribution. 2. The two special cases and their finite mixture are weighted Inverse Gaussian distributions. The two special cases and their finite mixture are weighted Inverse Gaussian distributions Using this finite mixture as a mixing distribution to the Normal. Generalized Hyperbolic Distribution (GHD) is a normal variance mean mixture with GIG mixing distribution. It is a five parameter distribution denoted by GH (λ,α , β ,δ , μ ).

Proposed Mixing Distribution

Proposed Model

E-Step

Application

Conclusions