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

High frequency financial data is characterized by non-normality: asymmetric, leptokurtic and fat-tailed behaviour. The normal distribution is therefore in-adequate 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.


Introduction
It is well known that mixture distributions produce flexible models with good statistical and probabilistic properties. Our first objective, therefore, is to construct and obtain properties of a finite mixture of two special cases of Generalized Inverse Gaussian distribution. These two special cases are related to the inverse Gaussian distribution which is also a special case of Generalised Inverse Gaussian distribution.
The Generalized Hyperbolic Distribution (GHD) introduced by Barndorff- The two special cases and their finite mixture are weighted Inverse Gaussian distributions. 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.
In literature, the Normal Inverse Gaussian (NIG) distribution has been used repeatedly for financial data which are skewed, leptokurtic and heavy-tailed because they are collected over short-time intervals, such as daily or weekly. Our third objective is to compare the log-likelihood functions of NWIG and NIG distributions.

Proposed Mixing Distribution
We show that two special cases of Generalised Inverse Gaussian (GIG) distribution can be expressed as Weighted Inverse Gaussian (WIG) distribution. A finite mixture of these cases can also be expressed as WIG distribution. The Generalized Inverse Gaussian (GIG) distribution is given by The moments around the origin of the ( ) , , GIG λ δ γ distribution are given by ( ) Remark: This expectation formula works when r is also a negative integer.

Special Cases
This is an Inverse Gaussian (IG) distribution.
This is a Reciprocal Inverse Gaussian (RIG) distribution.
Using the concept of weighted distribution introduced by Fisher (1934) it can be shown that the two special cases are weighted inverse Gaussian distribution.
More specifically, we express 2 g and 3 g in terms of 1 g as follows: and ( ) ( ) A finite mixture of the two cases is given by

Construction of the Mixed Model
Suppose the conditional of x given z is   The log-likelihood function log log e e log 1 1 Similarly,    [6].
Karlis [7] considers the mixing operation responsible for producing missing data.
Assume that the true data are made of an observed part X and unobserved part Z. Kosta [8] observes the log likelihood of the complete data ( ) factorizes into two parts. This implies that the joint density of X and Z is given by The likelihood function is

M-Step for the Mixing Distribution
Differentiating w.r.t γ we obtain

E-Step
The k-th iterations are as follows For the log-likelihood, the k-th iteration is given as

Iterative Scheme
From Equations (19) and (20), we obtain the following iterative scheme

Application
Let ( t P ) denote the price process of a security at time t, in particular of a stock.
In order to allow comparison of investments in different securities we shall investigate the rates of return defined by 1 log log In this section, we consider three data sets for data analysis. They include: Range Resource Corporation (RRC), Shares of Chevron Corporation (CVX) and s&p500 index. The histogram for the weekly log-returns in Figure 1 for RRC illustrates that the data is negatively skewed and exhibits heavy tails. The Q-Q plot shows that the normal distribution is not a good fit for the data, especially at the tails. This is also similar for the other data sets. Table 1 provides descriptive statistics for the return series in consideration.
We observe that the data sets experience excess kurtosis indicates the leptokurtic behaviour of the returns. The log-returns have distributions with relatively heavier tails than the normal distribution. The skewness indicates that the two tails of the returns behave differently. Table 2 below gives the method of moment estimates of NIG for the three data sets. The estimates will be used as initial values for the EM-algorithm.
The stopping criterion is when where tol is the tolerance level chosen; e.g 10 −6 and ( ) k l as given in Equation (11). We now wish to obtain the maximum likelihood parameter estimates of the data sets for the proposed model via the EM algorithm. Tables 3-5 illustrate monotonic convergence at different levels. The loglikelihood and AIC for each data set are also provided.   Using the estimates we obtain the estimates of p for the data sets as shown in Table 6 below: The finite mixture for these data sets is more weighted to the NRIG than the other special case of the GHD when 3 2 λ = − .