Mixture models have become more popular in modelling compared to standard distributions. The mixing distributions play a role in capturing the variability of the random variable in the conditional distribution. Studies have lately focused on finite mixture models as mixing distributions in the mixing mechanism. In the present work, we consider a Normal Variance Mean mix ture model. The mixing distribution is a finite mixture of two special cases of Generalised Inverse Gaussian distribution with indexes -1/2 and -3/2. The parameters of the mixed model are obtained via the Expectation-Maximization (EM) algorithm. The iterative scheme is based on a presentation of the normal equations. An application to some financial data has been done.
Mixture model provides a general framework for the construction new distribution. Often standard distribution has been considered as mixing distribution for the random variable. One way of extending this work is considering (finite) mixture models as mixing distributions. Jorgensen, Seshadri and Whitmore [
Lindley distribution introduced by Lindley [
Generalized Inverse Gaussian Distribution is a three parameter distribution based on the modified Bessel function of the third kind. Let us denote it by
G I G ( λ , δ , γ ) . It has several special and limiting cases. We have for example,
Inverse Gaussian distribution; i.e. G I G ( − 1 2 , δ , γ ) ; reciprocal iverse Gaussian distribution, i.e., G I G ( − 1 2 , δ , γ ) ; Gamma G I G ( λ ,0, γ ) distribution, Inverse Gamma G I G ( λ , δ ,0 ) ; G I G ( − 3 2 , δ , γ ) , G I G ( 3 2 , δ , γ ) distributions, etc.
Fisher [
[
In our present work, we show that a finite mixture of G I G ( − 1 2 , δ , γ ) and
G I G ( 3 2 , δ , γ ) is also weighted Inverse Gaussian distributions. We have used
the model as a mixing distribution in the Normal Variance Mean mixture. The mean and variance of the mixed model has been given. Parameter estimation has been done using the Expectation Maximization (EM) algorithm introduced by Dempster et al. [
The Generalised Inverse Gaussian is a three parameter model presented as
g ( z ) = ( γ δ ) λ z λ − 1 2 K λ ( δ γ ) exp { − 1 2 ( δ 2 z + γ 2 z ) } (1)
z > 0 ; − ∞ < λ < ∞ , δ > 0 , γ > 0
with the raw properties
E ( Z r ) = ( δ γ ) r K λ + r ( δ γ ) K λ ( δ γ )
where r can be positive or negative integers.
Where K λ ( ω ) is the modified Bessel function of the third kind of order λ evaluated at ω with the following properties:
K λ ( ω ) = K − λ ( ω ) (2)
∂ ∂ ω K λ ( ω ) = − 1 2 [ K λ + 1 ( ω ) + K λ − 1 ( ω ) ] (3)
∂ ∂ ω K λ ( ω ) = λ ω K λ ( ω ) − K λ + 1 ( ω ) (4)
2 λ ω K λ ( ω ) = K λ + 1 ( ω ) − K λ − 1 ( ω ) (5)
K λ ( ω ) = π 2 ω e − ω ( 1 + ∑ i = 1 ∞ ∏ i = 1 n 4 γ 2 − ( 2 i − 1 ) 2 n ! ( 8 ω ) n ) (6)
If λ = n + 1 2 , where n is a positive integer, then we have
Corollary 1
K n + 1 2 ( ω ) = π 2 ω e − ω [ 1 + ∑ i = 1 n ( n + i ) ! i ! ( n − i ) ! ( 2 ω ) − i ] (7)
For more properties see Abramowitz and Stegun [
The GIG has a number of special cases when the parameter λ take specific values. In particular when λ = − 1 2 we obtain the Inverse Gaussian distribution presented as
f ( z ) = δ 2 π exp ( δ γ ) z − 3 2 exp ( − 1 2 ( δ 2 z + γ 2 z ) ) (8)
and when λ = 3 2 we obtain the special case presented as
f ( z ) = γ 3 e δ γ 2 π ( 1 + δ γ ) z 1 2 exp { − 1 2 ( δ 2 z + γ 2 z ) } (9)
Let Z be a random variable with pdf f ( z ) . A function of Z, w ( Z ) is also a random variable with expectation
E [ w ( Z ) ] = ∫ − ∞ ∞ w ( z ) f ( z ) d z
∴ 1 = ∫ − ∞ ∞ w ( z ) E [ w ( Z ) ] f ( z ) d z
Thus
g ( z ) = w ( z ) E [ w ( Z ) ] f ( z ) , − ∞ < x < ∞ (10)
is a weighted distribution. It was introduced by Fisher [
Now, suppose Z ∼ I G ( γ , δ ) the Inverse Gaussian distribution with parameters γ and δ , Let
w ( Z ) = 1 + Z 2 1 + δ γ (11)
E [ w ( Z ) ] = γ 3 + δ γ 3 (12)
∴ g ( z ) = γ 3 γ 3 + δ ( 1 + z 2 1 + δ γ ) f ( z ) (13)
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 3 γ 3 + δ
The mean and variance for the weighted distribution are
E [ Z ] = γ 3 γ 3 + δ [ δ γ 4 ( 1 + δ γ ) + δ 3 γ 2 + 3 δ 2 γ + 3 δ γ 5 ( 1 + δ γ ) ] (14)
= δ γ 4 ( 1 + δ γ ) + δ 3 γ 2 + 3 δ 2 γ + 3 δ γ 2 ( 1 + δ γ ) ( γ 3 + δ ) (15)
v a r ( Z ) = ( δ γ 6 ( 1 + δ γ ) 2 + δ 5 γ 4 + 10 δ 4 γ 3 + 45 δ 3 γ 2 + 105 δ 2 γ + 105 δ ) ( 1 + δ γ ) ( γ 3 + δ ) γ 6 ( 1 + δ γ ) 2 ( γ 3 + δ ) (16)
− ( δ γ 4 ( 1 + δ γ ) + δ 3 γ 2 + 3 δ 2 γ + 3 δ ) 2 γ 6 ( 1 + δ γ ) 2 ( γ 3 + δ ) (17)
Consider the distribution X | Z ∼ N ( μ + β z , z ) and the random variable Z following a weighted Inverse Gaussian distribution as given in formula (10).
In general the integral formulation for constructing Normal Weighted Inverse Gaussian (NWIG) distributions is presented as
f ( x ) = δ e δ γ e β ( x − μ ) 2 π ∫ 0 ∞ w ( z ) E [ w ( Z ) ] z − 2 e − 1 2 { δ 2 ϕ x z + α 2 z } d z (18)
One of the attractive feature of constructing distributions using mixture approach is that properties of the mixed model can be expressed in terms of the properties of the mixing distribution. In the Normal Variance Mean mixing mechanism we obtain
M X ( t ) = e μ t M Z ( β t + t 2 2 )
E ( X ) = μ + β E ( Z )
V a r ( X ) = E ( Z ) + β 2 V a r ( Z )
μ 3 ( X ) = 3 β v a r ( Z ) + β 3 μ 3 ( Z )
μ 4 ( X ) = β 4 μ 4 ( Z ) + 6 β 2 μ 3 ( Z ) + 6 β 2 E [ Z ] v a r ( Z ) + 3 E [ Z 2 ]
The mixing mechanism has also been used by Barndorff-Nielsen Barndorff-Nielsen [
w ( z ) E [ w ( Z ) ] = γ 3 γ 3 + δ ( 1 + Z 2 1 + δ γ ) (19)
Therefore the mixed model becomes,
f ( x ) = δ γ 3 e δ γ 2 π ( γ 3 + δ ) e β ( x − μ ) ∫ 0 ∞ ( 1 + z 2 1 + δ γ ) z − 2 e − 1 2 ( α 2 z + δ 2 ϕ ( x ) z ) d z = δ γ 3 e δ γ 2 π ( γ 3 + δ ) e β ( x − μ ) ∫ 0 ∞ ( z − 1 − 1 + z − 1 − 1 1 + δ γ ) e − 1 2 ( α 2 z + δ 2 ϕ ( x ) z ) d z
f ( x ) = δ γ 3 e δ γ e β ( x − μ ) π ( γ 3 + δ ) { [ δ ϕ ( x ) α ] − 1 K 1 ( α δ ϕ ( x ) ) + [ δ ϕ ( x ) α ] K 1 ( α δ ϕ ( x ) ) 1 + δ γ } = δ γ 3 e δ γ e β ( x − μ ) π ( γ 3 + δ ) { α δ ϕ ( x ) + δ ϕ ( x ) α ( 1 + δ γ ) } K 1 ( α δ ϕ ( x ) ) = δ γ 3 e δ γ e β ( x − μ ) π ( γ 3 + δ ) α δ ( 1 + δ γ ) ϕ ( x ) { α 2 ( 1 + δ γ ) + δ 2 ϕ ( x ) } K 1 ( α δ ϕ ( x ) ) = γ 3 e δ γ e β ( x − μ ) ( ϕ ( x ) ) − 1 α π ( γ 3 + δ ) ( 1 + δ γ ) { α 2 ( 1 + δ γ ) + δ 2 ϕ ( x ) } K 1 ( α δ ϕ ( x ) ) (20)
With the following properties
E ( X ) = μ + β δ γ 4 ( 1 + δ γ ) + δ 3 γ 2 + 3 δ 2 γ + 3 δ γ 2 ( 1 + δ γ ) ( γ 3 + δ ) (21)
v a r ( X ) = δ γ 4 ( 1 + δ γ ) + δ 3 γ 2 + 3 δ 2 γ + 3 δ γ 2 ( 1 + δ γ ) ( γ 3 + δ ) − β 2 [ ( δ γ 6 ( 1 + δ γ ) 2 + δ 5 γ 4 + 10 δ 4 γ 3 + 45 δ 3 γ 2 + 105 δ 2 γ + 105 δ ) ( 1 + δ γ ) ( γ 3 + δ ) γ 6 ( 1 + δ γ ) 2 ( γ 3 + δ ) − ( δ γ 4 ( 1 + δ γ ) + δ 3 γ 2 + 3 δ 2 γ + 3 δ ) 2 γ 6 ( 1 + δ γ ) 2 ( γ 3 + δ ) ] (22)
The log-likelihood function
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 3 + δ ) ) ( 1 + δ γ ) − 1 2 log ϕ ( x i ) + log [ α 2 ( 1 + δ γ ) + δ 2 ϕ ( x i ) ] + log K 1 ( α δ ϕ ( x i ) ) } = 3 n log γ + n δ γ + β ∑ i = 1 n x i − n β μ − n log ( α π ( γ 3 + δ ) ) ( 1 + δ γ ) − 1 2 ∑ i = 1 n log ϕ ( x i ) + ∑ i = 1 n log [ α 2 ( 1 + δ γ ) + δ 2 ϕ ( x i ) ] + ∑ i = 1 n log K 1 ( α δ ϕ ( x i ) ) (23)
EM algorithm is a powerful technique for maximum likelihood estimation for data containing missing values or data that can be considered as containing missing values. It was introduced by Dempster et al. [
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 [
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 )
[
Since the conditional distribution for the six models is normal distribution as presented in formula (5.2), we have
l 1 = − n 2 log ( 2 π ) − 1 2 ∑ i = 1 n log z i − ∑ i = 1 n ( x i − μ − β z i ) 2 2 z i
Therefore
∂ ∂ β l 1 = 0 ⇒ ∑ i = 1 n ( x i − μ ^ − β ^ z i ) = 0
i.e., ∑ i = 1 n x i − n μ ^ − β ^ ∑ i = 1 n z i = 0
∴ μ ^ = x ¯ − β ^ z ¯
where x ¯ = ∑ i = 1 n x i n and z ¯ = ∑ i = 1 n z i n .
Similarly,
∂ ∂ μ l 1 = 0 ⇒ ∑ i = 1 n x i z i − μ ^ ∑ i = 1 n 1 z i − n β ^ = 0
∴ ∑ i = 1 n x i z i − x ¯ ∑ i = 1 n 1 z i + β ^ z ¯ ∑ i = 1 n 1 z i − n β ^ = 0
∴ β ^ = ∑ i = 1 n x i z i − x ¯ ∑ i = 1 n 1 z i n − z ¯ ∑ i = 1 n 1 z i
M-Step for the Mixing Distributionl 2 = n log δ + 3 n log γ + n δ γ − n 2 log ( 2 π ) − n log ( 1 + δ γ ) − n log ( γ 3 + δ ) + ∑ i = 1 n log ( 1 + δ γ + z i 2 ) − 3 2 ∑ i = 1 n log z i − δ 2 2 ∑ i = 1 n 1 z i − γ 2 2 ∑ i = 1 n z i (24)
Maximizing with respect to δ and γ we have the following representation
( n δ 2 1 + δ γ − ∑ i = 1 n z i ) γ 2 + ∑ i = 1 n δ γ 1 + δ γ + z i 2 + 3 n δ γ 3 + δ = 0 (25)
( n γ 2 1 + δ γ − ∑ i = 1 n 1 z i ) δ 2 + ∑ i = 1 n δ γ 1 + δ γ + z i 2 + n γ 3 γ 3 + δ = 0 (26)
Both equations are quadratic in γ and δ respectively.
Posterior Expectation
E ( Z / X ) = ∫ 0 ∞ z ( 1 + z 2 1 + δ γ ) z − 2 e − 1 2 ( α 2 z + δ 2 ϕ ( x ) z ) d z ∫ 0 ∞ ( 1 + z 2 1 + δ γ ) z − 2 e − 1 2 ( α 2 z + δ 2 ϕ ( x ) z ) d z = 1 2 ∫ 0 ∞ ( z 0 − 1 + z 2 − 1 1 + δ γ ) e − 1 2 ( α 2 z + δ 2 ϕ ( x ) z ) d z 1 2 ∫ 0 ∞ ( z − 1 − 1 + z 1 − 1 1 + δ γ ) e − 1 2 ( α 2 z + δ 2 ϕ ( x ) z ) d z
= K 0 ( α δ ϕ ( x ) ) + [ δ ϕ ( x ) α ] 2 K 2 ( α δ ϕ ( x ) ) 1 + δ γ [ δ ϕ ( x ) α ] − 1 K 1 ( α δ ϕ ( x ) ) + δ ϕ ( x ) α ( 1 + δ γ ) K 1 ( α δ ϕ ( x ) ) = K 0 ( α δ ϕ ( x ) ) + δ 2 ϕ ( x ) α 2 K 2 ( α δ ϕ ( x ) ) 1 + δ γ [ α δ ϕ ( x ) + δ ϕ ( x ) α ( 1 + δ γ ) ] K 1 ( α δ ϕ ( x ) ) = α δ ( 1 + δ γ ) ϕ ( x ) K 0 ( α δ ϕ ( x ) ) + δ 3 ( ϕ ( x ) ) 3 2 α K 2 ( α δ ϕ ( x ) ) [ α 2 ( 1 + δ γ ) + δ 2 ϕ ( x ) ] K 1 ( α δ ϕ ( x ) ) (27)
Similarly,
E ( 1 Z / X ) = ∫ 0 ∞ z − 1 ( 1 + z 2 1 + δ γ ) z − 2 e − 1 2 ( α 2 z + δ 2 ϕ ( x ) z ) d z ∫ 0 ∞ ( 1 + z 2 1 + δ γ ) z − 2 e − 1 2 ( α 2 z + δ 2 ϕ ( x ) z ) d z = 1 2 ∫ 0 ∞ ( z − 2 − 1 + z 0 − 1 1 + δ γ ) e − 1 2 ( α 2 z + δ 2 ϕ ( x ) z ) d z 1 2 ∫ 0 ∞ ( z − 1 − 1 + z 1 − 1 1 + δ γ ) e − 1 2 ( α 2 z + δ 2 ϕ ( x ) z ) d z = [ δ ϕ ( x ) α ] − 2 K 2 ( α δ ϕ ( x ) ) + K 0 ( α δ ϕ ( x ) ) 1 + δ γ [ ( δ ϕ ( x ) α ) − 1 + ( δ ϕ ( x ) α ) 1 1 + δ γ ] K 1 ( α δ ϕ ( x ) )
= α 2 δ 2 ϕ ( x ) K 2 ( α δ ϕ ( x ) ) + K 0 ( α δ ϕ ( x ) ) 1 + δ γ [ α δ ϕ ( x ) + δ ϕ ( x ) α ( 1 + δ γ ) ] K 1 ( α δ ϕ ( x ) ) = α 2 K 2 ( α δ ϕ ( x ) ) + δ 2 ϕ ( x ) 1 + δ γ K 0 ( α δ ϕ ( x ) ) [ α δ ϕ ( x ) + δ 3 ( ϕ ( x ) ) 3 2 α ( 1 + δ γ ) ] K 1 ( α δ ϕ ( x ) ) = α 3 ( 1 + δ γ ) K 2 ( α δ ϕ ( x ) ) + α δ 2 ϕ ( x ) K 0 ( α δ ϕ ( x ) ) [ α 2 δ ( 1 + δ γ ) ϕ ( x ) + δ 3 ( ϕ ( x ) ) 3 2 ] K 1 ( α δ ϕ ( x ) ) (28)
Similarly
E ( z 2 / X ) = ( 1 + δ γ ) α 2 δ 2 ϕ ( x ) K 1 ( α δ ϕ ( x ) ) + δ 4 ( ϕ ( x ) ) 2 K 3 ( α δ ϕ ( x ) ) [ ( 1 + δ γ ) α 4 + α 2 δ 2 ϕ ( x ) ] K 1 ( α δ ϕ ( x ) ) (29)
The posterior expectations for the k-th iteration are:
s i ( k ) = α ( k ) δ ( k ) ( 1 + δ ( k ) γ ( k ) ) ϕ ( k ) ( x ) K 0 ( α ( k ) δ ( k ) ϕ ( k ) ( x ) ) + ( δ ( k ) ) 3 ( ϕ ( k ) ( x ) ) 3 2 α ( k ) K 2 ( α ( k ) δ ( k ) ϕ ( k ) ( x ) ) [ ( α ( k ) ) 2 ( 1 + δ ( k ) γ ( k ) ) + ( δ ( k ) ) 2 ϕ ( k ) ( x ) ] K 1 ( α ( k ) δ ( k ) ϕ ( k ) ( x ) )
w i ( k ) = ( α ( k ) ) 3 ( 1 + δ ( k ) γ ( k ) ) K 2 ( α ( k ) δ ( k ) ϕ ( k ) ( x ) ) + α ( k ) ( δ ( k ) ) 2 ϕ ( k ) ( x ) K 0 ( α ( k ) δ ( k ) ϕ ( k ) ( x ) ) [ ( α ( k ) ) 2 δ ( k ) ( 1 + δ ( k ) γ ( k ) ) ϕ ( k ) ( x ) + ( δ ( k ) ) 3 ( ϕ ( k ) ( x ) ) 3 2 ] K 1 ( α ( k ) δ ( k ) ϕ ( k ) ( x ) )
v i ( k ) = ( 1 + δ ( k ) γ ( k ) ) ( α ( k ) ) 2 ( δ ( k ) ) 2 ϕ ( k ) ( x i ) K 1 ( α ( k ) δ ( k ) ϕ ( k ) ( x ) ) + ( δ ) 4 ( ϕ ( k ) ( x i ) ) 2 K 3 ( α ( k ) δ ( k ) ϕ ( k ) ( x ) ) [ 1 + δ ( k ) γ ( k ) ( α ( k ) ) 4 + ( α ( k ) ) 2 ( δ ( k ) ) 2 ϕ ( k ) ( x i ) ] K 1 ( α ( k ) δ ( k ) ϕ ( k ) ( x ) )
Now, define the iterative scheme as follows:
let
a 1 ( k + 1 ) = n ( δ ( k ) ) 2 1 + δ ( k ) γ ( k ) − ∑ i = 1 n s i ( k )
b 1 ( k + 1 ) = ∑ i = 1 n δ ( k ) 1 + δ ( k ) γ ( k ) + v i ( k )
c 1 ( k + 1 ) = 3 n δ ( k ) ( γ ( k ) ) 3 + δ ( k )
let
t ( k + 1 ) = − b 1 ( k + 1 ) − ( b 1 ( k + 1 ) ) 2 − 4 a 1 ( k + 1 ) c 1 ( k + 1 ) 2 a 1 ( k + 1 ) (30)
using the square root transformation, we have
γ k + 1 = t ( k + 1 ) (31)
Similarly, define
a 2 ( k + 1 ) = n ( γ ( k + 1 ) ) 2 1 + δ ( k ) γ ( k + 1 ) − ∑ i = 1 n w i ( k ) (32)
b 2 ( k + 1 ) = ∑ i = 1 n δ ( k ) γ ( k + 1 ) 1 + δ ( k ) γ ( k + 1 ) + v i ( k ) (33)
c 2 ( k + 1 ) = n ( γ ( k + 1 ) ) 3 ( γ ( k + 1 ) ) 3 + δ ( k ) (34)
let
s ( k + 1 ) = − b 2 ( k + 1 ) − ( b 2 ( k + 1 ) ) 2 − 4 a 2 ( k + 1 ) c 2 ( k + 1 ) 2 a 2 ( k + 1 ) (35)
using the square root transformation, we have
δ k + 1 = s ( k + 1 ) (36)
and the k-th iteration for the loglikelihood is given by
l ( k ) = 3 n log γ ( k ) + n δ ( k ) γ ( k ) + β ( k ) ∑ i = 1 n ( x i − μ ( k ) ) − n log α ( k ) π − n log ( 1 + δ ( k ) γ ( k ) ) − n log ( ( γ ( k ) ) 3 + δ ( k ) ) − 1 2 ∑ i = 1 n log ϕ ( k ) ( x i ) + ∑ i = 1 n log [ ( α ( k ) ) 2 ( 1 + δ ( k ) γ ( k ) ) + ( δ ( k ) ) 2 ϕ ( k ) ( x i ) ] + ∑ i = 1 n log K 1 ( α ( k ) δ ( k ) ϕ ( k ) ( x i ) ) (37)
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
The data used in this research is for the s & p 500 weekly returns for the period 3/01/2000 to 1/07/2013 with 702 observations. The histogram for the weekly log-returns shows 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 (
We now fit the proposed model to s & p500 weekly log-returns. Using the sample estimates and the NIG estimators we obtain the following estimates as initial values for the EM algorithm (
α ^ = 0.6556607 , β ^ = − 0.1257455 , δ ^ = 0.8310044 , μ ^ = 0.1690855 .
Stopping Criterion
The stopping criterion is when
l ( k ) − l ( k − 1 ) l ( k ) < t o l (38)
| Minimum | Standard.dev | skewness | exc.kurtosis | Maximum | Mean | N |
|---|---|---|---|---|---|---|
| −8.722261488 | 1.157893 | −0.7851156 | 6.408709 | 4.931805023 | 0.006697 | 702 |
| Parameter | Starting Values | EM ( t o l = 10 − 3 ) | EM ( t o l = 10 − 5 ) | EM ( t o l = 10 − 10 ) |
|---|---|---|---|---|
| α ^ | 0.6556607 | 1.025172 | 1.025056 | 1.025061 |
| β ^ | −0.1257455 | −0.082968 | −0.082969 | −0.082988 |
| δ ^ | 0.8310044 | 0.8571405 | 0.8570473 | 0.8570354 |
| μ ^ | 0.1690855 | 0.1506045 | 0.150692 | 0.1507229 |
| Loglikelihood | −1048.305 | −1048.288 | −1048.301 | |
| No. iteration | 9 | 12 | 32 | |
| AIC | 2104.61 | 2104.576 | 2104.602 |
where tol is the tolerance level chosen; e.g. 10−6.
The proposed model fits the data set well as illustrated in
f ( x ) = γ 3 γ 3 + δ × N I G + δ γ 3 + δ × G H D ( 3 2 , α , δ , β , μ ) (39)
Using the parameter estimates p = 0.242793 . Therefore, the finite mixture for these data sets is more weighted to the GHD when λ = 3 2 than to the NIG.
In this work, we have considered a normal variance mean mixture when the mixing distribution is a weighted inverse Gaussian distribution. In specific, we have shown that a finite mixture of two special cases of generalised inverse Gaussian
distribution of indexes − 1 2 and 3 2 is itself a weighted inverse Gaussian distribution.
Further, we have constructed a Normal Weighted Inverse Gaussian distribution, studied its properties and estimated the parameters using the Expectation Maximization (EM) algorithm. The initial values were based on Karlis [
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 Finite Mixture of Generalised Inverse Gaussian with Indexes as Mixing Distribution for Normal Variance Mean Mixture with Application. Open Journal of Statistics, 11, 963-976. https://doi.org/10.4236/ojs.2021.116056