Normal Inverse Gaussian (NIG) distribution introduced by Barndorff-Nielsen [1] is the popular normal variance-mean mixture, with an inverse Gaussian mixing distribution that has been used to handle non-normal distributed data.

Efforts have been done and are still going on to identify alternatives to the NIG distributions: Aas and Haff [2] [3] considered Skew Student’s t distribution as an alternative. Corlu and Corlu [4] used Generalized Lambda distribution, a generalization of Tukey’s Lambda as an alternative.

Finite mixtures are more flexible (robust) than single mixing distributions; a result that has been stated a lot in statistical literature.

Very few studies have used finite mixtures as mixing distributions. Hence an attempt has been done here in this paper of ours. This idea is motivated by the fact that finite mixtures are more flexible than single distributions. Nadarajah, Zhang and Chan [5] have stated that finite mixtures of normal distributions are flexible than single normal distribution; finite mixtures of stable distributions are flexible than a single stable distribution; finite mixtures of student’s t distributions are flexible than single student’s distribution.

The second motivation is that very few studies on continuous mixtures have used finite mixtures as mixing distributions; with the exception of Lindley [6] distribution and its generalization in Poisson mixtures (e.g., Sankaran [7]; Mahmoudi and Zakerzadeh [8] ). Lindley distribution and its generalizations are basically finite gamma mixtures.

GIG is a three parameter distribution denoted by G I G ( λ , δ , γ ) . When λ = − 1 2 , the mixing distribution is G I G ( − 1 2 , δ , γ ) which is Inverse Gaussian and the mixture is Normal Inverse Gaussian (NIG). In our present work, we consider two other special cases: G I G ( 1 2 , δ , γ ) and G I G ( 3 2 , δ , γ ) . A finite mixture of these cases is shown to be a weighted Inverse Gaussian distribution. The concept of weighted distribution was introduced by Fisher [9] and elaborated by Patil and Rao [10]. The reciprocal Inverse Gaussian, i.e., G I G ( 1 2 , δ , γ ) and the finite mixture of G I G ( − 1 2 , δ , γ ) and G I G ( 1 2 , δ , γ ) are shown to be weighted inverse Gaussian distribution by Gupta and Kundu [11].

The finite mixture model is used as a mixing distribution in Normal Variance Mean mixture to construct the proposed model. The maximum likelihood parameter estimates of the mixture are obtained via the Expectation-Maximization (EM) algorithm. For data analysis we have used three datasets: s&p500 index, Range Resource Corporation (RRC) and Shares of Chevron Corporation (CVX). We consider log returns for the period 3/01/2000 to 1/07/2013 with 702 observations for each dataset.

Consider a random variable Z with a probability distribution f ( z ) . Let w ( Z ) be a random function of Z. We can construct a new probability distribution

g ( z ) = w ( z ) E [ w ( Z ) ] f ( z ) , − ∞ < x < ∞ (1)

which is a weighted distribution of f ( z ) . The concept of weighted distribution was introduced by Fisher [9] and elaborated by Patil and Rao [10].

In this work we consider weighted distribution for the Inverse Gaussian (IG) distribution.

Now, suppose Z ∼ I G ( γ , δ ) the Inverse Gaussian distribution with parameters γ and δ and probability density function given by

f ( z ) = δ 2 π exp ( δ γ ) z − 3 2 exp ( − 1 2 ( δ 2 z + γ 2 z ) ) (2)

Let

w ( Z ) = Z + Z 2 1 + δ γ

E [ w ( Z ) ] = δ ( γ 2 + 1 ) γ 3

∴ g ( z ) = γ 3 δ ( γ 2 + 1 ) [ z + z 2 1 + δ γ ] f ( z ) (3)

which is also a finite mixture of G I G ( 1 2 , δ , γ ) and G I G ( 3 2 , δ , γ ) . That is

g ( z ) = p G I G ( 1 2 , δ , γ ) + ( 1 − p ) G I G ( 3 2 , δ , γ )

with

p = γ 2 γ 2 + 1

The mean and variance for the weighted distribution are

E [ Z ] = γ 2 ( 1 + δ γ ) 2 + δ 2 γ 2 + 3 δ γ + 3 γ 2 ( 1 + δ γ ) ( γ 2 + 1 ) (4)

v a r ( Z ) = ( ( δ 2 γ 2 + 3 δ γ + 3 ) ( 1 + δ γ ) γ 2 + ( δ 3 γ 3 + 6 δ 2 γ 2 + 15 δ γ + 15 ) ) ( 1 + δ γ ) ( γ 2 + 1 ) γ 4 ( 1 + δ γ ) 2 ( γ 2 + 1 ) 2                           − ( γ 2 ( 1 + δ γ ) 2 + δ 2 γ 2 + 3 δ γ + 3 ) 2 γ 4 ( 1 + δ γ ) 2 ( γ 2 + 1 ) 2 (5)

A stochastic representation of a Normal Variance-Mean mixture is given by

X = μ + β Z + Z Y

where

Y ∼ N ( 0,1 )

which gives the hierarchical representation

X / Z = z ∼ N ( μ + β z , z ) (6)

being the conditional pdf and g(z) the mixing distribution.

Suppose the mixing distribution follows formula (3) and noting that

w ( z ) E [ w ( Z ) ] = γ 3 δ ( γ 2 + 1 ) ( Z + Z 2 1 + δ γ ) (7)

the mixed model becomes

f ( x ) = δ e δ γ 2 π e β ( x − μ ) ∫ 0 ∞ w ( z ) E [ w ( Z ) ] z − 2 e − 1 2 ( α 2 z + δ 2 ϕ ( x ) z ) d z = δ e δ γ 2 π e β ( x − μ ) γ 3 δ ( γ 2 + 1 ) ∫ 0 ∞ ( z + z 2 1 + δ γ ) z − 2 e − 1 2 ( α 2 z + δ 2 ϕ ( x ) z ) d z = γ 3 e δ γ 2 π ( γ 2 + 1 ) e β ( x − μ ) ∫ 0 ∞ ( z 0 − 1 + z 1 − 1 1 + δ γ ) e − α 2 2 ( z + δ 2 ϕ ( x ) α 2 z ) d z

f ( x ) = γ 3 e δ γ e β ( x − μ ) π ( γ 2 + 1 ) { K 0 ( α δ ϕ ( x ) ) + δ ϕ ( x ) α ( 1 + δ γ ) K 1 ( α δ ϕ ( x ) ) } = γ 3 e δ γ e β ( x − μ ) α π ( 1 + δ γ ) ( γ 2 + 1 ) { α ( 1 + δ γ ) K 0 ( α δ ϕ ( x ) ) + δ ϕ ( x ) K 1 ( α δ ϕ ( x ) ) } (8)

With the following properties

E ( X ) = μ + β γ 2 ( 1 + δ γ ) 2 + δ 2 γ 2 + 3 δ γ + 3 γ 2 ( 1 + δ γ ) ( γ 2 + 1 ) (9)

v a r ( X ) = γ 2 ( 1 + δ γ ) 2 + δ 2 γ 2 + 3 δ γ + 3 γ 2 ( 1 + δ γ ) ( γ 2 + 1 )       − β 2 ( ( δ 2 γ 2 + 3 δ γ + 3 ) ( 1 + δ γ ) γ 2 + ( δ 3 γ 3 + 6 δ 2 γ 2 + 15 δ γ + 15 ) ) ( 1 + δ γ ) ( γ 2 + 1 ) γ 4 ( 1 + δ γ ) 2 ( γ 2 + 1 ) 2       − ( γ 2 ( 1 + δ γ ) 2 + δ 2 γ 2 + 3 δ γ + 3 ) 2 γ 4 ( 1 + δ γ ) 2 ( γ 2 + 1 ) 2 (10)

Note

K λ ( ω ) denotes modified Bessel function of the third kind with index λ evaluated at ω .

K λ ( ω ) = 1 2 ∫ 0 ∞     x λ − 1 e − 1 2 ( x + 1 x ) d x (11)

with the following properties

a) K 1 2 ( ω ) = K − 1 2 ( ω ) = π 2 ω e − ω (12)

b) K 3 2 ( ω ) = K − 3 2 ( ω ) = π 2 ω e − ω ( 1 + 1 ω ) (13)

c) K 5 2 ( ω ) = K − 5 2 ( ω ) = π 2 ω e − ω ( 1 + 3 ω + 3 ω 2 ) (14)

d) K 7 2 ( ω ) = K − 7 2 ( ω ) = π 2 ω e − ω ( 1 + 6 ω + 15 ω 2 + 15 ω 3 ) (15)

e) K 9 2 ( ω ) = K − 9 2 ( ω ) = π 2 ω e − ω ( 1 + 10 ω + 45 ω 2 + 105 ω 3 + 105 ω 4 ) (16)

f) K 11 2 ( ω ) = K − 11 2 ( ω ) = π 2 ω e − ω ( 1 + 15 ω + 105 ω 2 + 420 ω 3 + 945 ω 4 + 945 ω 5 ) (17)

which are necessary in deriving the properties and estimates of the proposed models. For more definition and properties (see Abramowitz and Stegun [12] ).

Given a random sample of size n from our proposed model the log-likelihood function is given by

l = log L = ∑ i = 1 n log f ( x i ) = ∑ i = 1 n { 3 log γ + δ γ + β ( x i − μ ) − log ( α π ( γ 2 + 1 ) ( 1 + δ γ ) )   + log [ α ( 1 + δ γ ) K 0 ( α δ ϕ ( x ) ) + δ ϕ ( x ) K 1 ( α δ ϕ ( x ) ) ] } = 3 n log γ + n δ γ + β ∑ i = 1 n     x i − n β μ − n log ( α π ( γ 2 + 1 ) ( 1 + δ γ ) )   + ∑ i = 1 n log [ α ( 1 + δ γ ) K 0 ( α δ ϕ ( x ) ) + δ ϕ ( x ) K 1 ( α δ ϕ ( x ) ) ] (18)

The derivatives of the log-likelihood function involve the Bessel function and the parameters are difficult to separate. Direct maximization is not an easy. In this work we adopt the Expectation-Maximization (EM) algorithm. The EM-algorithm, introduced by Dempster et al. [13], is a powerful technique for maximum likelihood estimation for data containing missing values or data that can be considered as containing missing values.

Karlis [14] considered the mixing operation responsible for producing missing values. He applied the algorithm in estimating the parameters for the Normal Inverse Gaussian. The iterative scheme was based on explicit solution to the normal equations. In our present work the iterative scheme is based on a representation of the normal equation. This subtle approach overcomes the computational difficulty of solving the complicated quantities numerically.

Assume that the true data are made of an observed part X and unobserved part Z. This then ensures the log likelihood of the complete data ( x i , z i ) for i = 1 , 2 , 3 , ⋯ , n factorizes into two parts, i.e.,

log L = log ∏ i = 1 n     f ( x i / z i ) + log ∏ i = 1 n     g ( z i ) = ∑ i = 1 n log f ( x i / z i ) + ∑ i = 1 n log g ( z i )

where

l 1 = ∑ i = 1 n log f ( x i / z i )

and

l 2 = ∑ i = 1 n log g ( z i )

Let ( P t ) 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

X t = log P t − log P t − 1

In this section we consider three datasets for data analysis: Range Resource Corporation (RRC), Shares of Chevron Corporation (CVX) and s&p500 index. The period 3/01/2000 to 1/07/2013 with 702 observations for each data set is considered. The histogram for the weekly log-returns in Figures 1-3 for s&p 500, RRC and CVX show that the data is negatively skewed and exhibiting 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 typical the other data sets.

Table 1 provides descriptive statistics of the data sets for the return series in consideration. We observe that the excess kurtosis that indicates the leptokurtic behaviour of the returns. The log-returns has a distributions with relatively heavier

tails than the normal distribution. We observe skewness for the data sets which indicates that the two tails of the returns behave slightly differently.

We use Karlis [14] formulation to obtain the initial values for the EM algorithm. For this dataset, the values obtained are in Table 2.

The stopping criterion is when

l ( k ) − l ( k − 1 ) l ( k ) < t o l (31)

where tol is the tolerance level. Table 3 below shows the NIG maximum likelihood parameter estimates using the EM-algorithm at t o l = 10 − 3 .

We now wish to obtain the maximum likelihood parameter estimates for the proposed model via the EM algorithm. Tables 3-5 illustrate monotonic convergence

at different levels. The log-likelihood and AIC for each data sets are also provided.

Figures 4-6 show how the proposed models fit the data sets. It is clear that the proposed model is a good fit compared to the normal distribution and hence a good alternative to the Normal Inverse Gaussian distribution.

Remark:

Expressing the proposed model in terms of its components we have

f ( x ) = γ 2 γ 2 + 1 × G H D ( 1 2 , α , δ , β , μ ) + 1 γ 2 + 1 × G H D ( 3 2 , α , δ , β , μ ) (32)

The finite mixture therefore is flexible in determining between G H D ( 1 2 , α , δ , β , μ ) and G H D ( 3 2 , α , δ , β , μ ) depending on the nature of the data. Table 6 below gives different values of weights for the three data sets considered based on the maximum likelihood estimates.

Two special cases of the Generalized Inverse Gaussian with indexes 1 2 and 3 2

have been used to construct a finite mixture model. The model has been used as a mixing distribution for Normal Variance-Mean mixture to a Normal Weighted Inverse Gaussian Model. The mean and variance of the proposed model have been obtained.

Three data sets: Range Resource Corporation (RRC), Shares of Chevron Corporation (CVX) and s&p500 index for the period 3/01/2000 to 1/07/2013 with 702 observations have been used for data analysis. An iterative scheme has been presented for parameter estimation by the EM algorithm. The iterative scheme demonstrates a monotonic convergence. The method of moment estimates for NIG worked well for the three data sets. The model fits the data sets well. Hence the model is a good alternative to NIG.

The authors gratefully acknowledge the financial support from Kisii University.

The authors declare no conflicts of interest regarding the publication of this paper.

Maina, C.B., Weke, P.G.O., Ogutu, C.A. and Ottieno, J.A.M. (2021) A Special Weight for Inverse Gaussian Mixing Distribution in Normal Variance Mean Mixture with Application. Open Journal of Statistics, 11, 977-992. https://doi.org/10.4236/ojs.2021.116057