The Overall Dispersion of Stock Returns Data

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 84 by the empirical mean X j of Poisson component. This improvement can also be applied to other NST models which have zero values in the stable Tweedie component, i.e. normal compound Poisson family with 1 p 2. Investigation and discussion on this proposed solution will become another interesting work. References Anderson DN, Arnold BC. 1993. Linnik distributions and processes. J Appl Probab 30:330–340 Arvanitis LG and Afonja B. 1971. Use of the generalized variance and the gradient projection method in multivariate stratified sampling. Biometrics 27: 119–127 Bar-Lev S, Bschouty D, Enis P, Letac G, Lu I, Richard D. 1994. The diagonal multivariate natural exponential families and their classification. J Theor Probab 7: 883–929 Barndorff-Nielsen OE. 1997. Normal inverse Gaussian distribution and stochastic volatility modelling. Scand J Stat 24: 1–3 Barndorff-Nielsen OE. 1998. Processes of normal inverse Gaussian type. Financ Stoch 2: 41–8 Bandorff-Nielsen OE, Jørgensen B. 1991. Some parametric models on the simplex. J Multivar Anal 39: 106–116 Barndorff-Nielsen OE, Kent J, Sørensen M. 1982. Normal variance-mean mixtures and z distributions. Int Stat Rev 50: 145–159 Behara M, Giri N. 1983. Generalized variance statistic in testing of hypothesis in complex multivariate Gaussian distribution. Archiv Math 40: 538–543 Bernardoff Ph, Kokonendji CC, Puig B. 2008. Generalized variance estima- tors in the multivariate gamma models. Math Meth of Stat 17: 66–73 Bernardo JM, Smith AFM. 1993. Bayesian Theory. Wiley, New York Bertoin J. 1996. Lévy Processes. Cambridge University Press: Cambridge Bobotas P, Kourouklis S. 2013. Improved estimation of the covariance matrix and the generalized variance of a multivariate normal distribution: some unifying results. Sankhya: Indian J Stat 75-A: 26–50 Boubacar Maïnassara Y, Kokonendji CC. 2014. Normal stable Tweedie mod- els and power-generalized variance function of only one component, TEST 23: 585–606 Casalis M. 1996. The 2d + 4 simple quadratic natural exponential families on R d . Ann Stat 24: 1828–1854 Cuenin J, Faivre A, Kokonendji CC. 2017. On generalized variances of prod- ucts of powered components and multiple stable Tweedie models. Com- mun Stat - Theor M, In Press DOI : 10.108003610926.2016.1146770 Cuenin J, Jørgensen B, Kokonendji CC. 2016. Simulations of full multivariate Tweedie with flexible dependence structure. Comput Stat 31 4: 1477-1492 Damsleth E, El-Shaarawi AH. 1989. ARMA models with double exponen- tially distributed noise. J R Stat Soc Series B 511:61–69 Feller W. 1971. An Introduction to Probability Theory and its Applications, Vol. II, Second edition. Wiley: New York Fernandez-Ponce JM, Suarez-Llorens A. 2003. A multivariate dispersion or- dering based on quantiles more widely separated. J Multivar Anal 85: 40–53 85