79
Figure 6.7: Cullen and Frey graph of nonfood expenditure of city households Based on the information from Cullen and Frey graph we fit the Bogor
city data to the suspected distributions and also NIG distribution see Table 6.4, the log-likelihood and AIC values of NIG are the smallest ones then NIG
is the most fit distribution to the data. The histogram of food and non-food expenditures with normal and NIG curves for Bogor city are given in Figure
6.8.
Table 6.4: Fitting result for city households expenditure
Variable Distribution Loglikelihood
AIC Food expenditure
Gamma -8796.591 17597.18
Beta -8765.206 17534.41
NIG -8764.030 17530.06
Nonfood expenditure Gamma -8885.023 17774.05
Log-normal -8845.174 17696.35
NIG -8844.451 17692.90
The same procedure was also applied to Bogor district data, and the sim- ilar result was obtained see Table 6.5 for fitting result of district households
expenditure. Based on the likelihood and AIC values we concluded that NIG is also the most fit distribution to the data.
We predicted the unobserved X
j
using b
µ
j
in 6.1. The predicted X
j
= b
µ
j
and the generalized variance of the two groups city and district are
80
a Food expenditure b Nonfood expenditure
Figure 6.8: Histogram, normal curve - - - and NIG curve —- Table 6.5: Fitting result for district households expenditure
Variable Distribution Loglikelihood
AIC Food expenditure
Gamma -16899.71 33803.42
Lognormal -16755.71 33515.42
NIG -16749.2 33504.41
Nonfood expenditure Gamma -17137.26 34278.51
Log-normal -16988.61 33981.22
NIG -17039.90 34085.80
a City x
1
=food, x
2
=nonfood b District x
1
=food, x
2
=nonfood
Figure 6.9: Bivariate normal inverse-Gaussian distribution for expenditure of Bogor households data
81 presented in Table 6.6. From the generalized variance values in the tables
we obtain information that the expenditure of households in Bogor city is more diverse than the expenditure of households in Bogor district, i.e. the
variability of expenditure in the city is about twice as much as in the district.
Table 6.6: Comparison of bivariate dispersions between city and district City
District b
µ
j
6 .1572 × 10
5
5 .1789 × 10
5
b ψ
14 .37281 × 10
22
7 .193387 × 10
22
6.2.2. The Overall Dispersion of Stock Returns Data
The second data concern the daily stock returns for year 1999 from five petroleum companies 5 dimensional data, i.e Schlumberger Ltd., No-
ble Drilling Corp., Phillips Petroleum, Halliburton Co. and Occidental Petroleum. Total n = 253 were used. Stock return data are known to be
a NIG model. Here we examine the data using histogram with normal and NIG curves. The corresponding histograms for each variable are given in
Figure 6.10. Using the covariance matrix as dispersion measure for five dimensional data can be hard to interpret. Sometimes it is also useful to
have an overall measure of dispersion in the data. Hence we can use the generalized variance and the standardized generalized variance.
In fact the inverse-Gaussian component in the data was an unobserved component and we estimated using 6.1 For these data we obtain the stan-
dardized generalized variance b
µ
j
and the generalized variance using 6.1 and 4.1, the predicted values are presented in Table 6.7 below.
Table 6.7: overall measure of dispersion estimates
b µ
j
b ψ
1 .434353 × 10
−3
8 .708342 × 10
−24
82
a Schlumberger Ltd. b Noble Drilling Corp.
c Phillips Petroleum d Halliburton Co.
e Occidental Petroleum
Figure 6.10: Histograms with normal and NIG curves
7. CONCLUSION AND SUGGESTION
7.1. Conclusion
In this work we have studied the generalized variance of normal stable Tweedie models and its estimations. We proposed the Bayesian estimator
of the generalized variance of normal-Poisson model as a particular case of normal stable Tweedie models and its characterization by variance function
and by generalized variance function.
We introduced the generalization of NST models by replacing X
1
by X
j
for j ∈ {1,...,k}. For the generalized variance estimations of some NST models using ML, UMVU and Bayesian estimators, simulation studies show
that UMVU produces a better estimation than ML estimator. It also points out that Bayesian estimators can be used as an alternative estimator when
UMVU doest not exist in normal Poisson case. However, all methods are consistent estimators and they become more similar when the sample size
increases. The simulation studies also show that when we have zero values in the Poisson component X
j
= 0, the corresponding normal components become the Dirac distribution at zero. This situation does not affect the
generalized variance estimations for reasonable proportions of zeros in the sample.
We successfully proved that the characterization of normal-Poisson
j
models by variance function is obtained through analytical calculations and using some properties of NEF. Also, the characterization of normal Poisson
models by generalized variance which is the solution to a specific Monge- Ampère equation: det K
′′ µ
θ = exp k × θ
⊤
˜θ
c j
on R
k
can be solved using the infinite divisibility property of normal-Poisson.
For the stable Tweedie variance modeling under normality, the simula- tion studies point out that
b µ
j
is a consistent estimator of µ
j
in the context of conditional homoscedasticity from 6.1. The application of this variance
modeling under normality can be used as a standardized scalar measure of dispersion for multivariate data which is also related to the standardized
generalized variance.
7.2. Suggestion
The simulation studies show that the situation when we have zero values in the Poisson component does not affect the generalized variance estima-
tions. However, the problem might arise when the situation can theoretically change the distribution of all normal components into Dirac distribution. To
solve this problem, it could be interesting to improve the model by replacing the variance X
j
of normal components by E[ ηX
j
]. If ηy = y then we have
X
c j
|X
j
following N0,µ
j
; consequently, in practice µ
j
can be approximated 83
