Brazilian Journal of Probability and Statistics

Bayesian analysis based on the Jeffreys prior for the hyperbolic distribution

Thaís C. O. Fonseca, Helio S. Migon, and Marco A. R. Ferreira

Full-text: Open access

Abstract

In this work, we develop Bayesian analysis based on the Jeffreys prior for the hyperbolic family of distributions. It is usually difficult to estimate the four parameters in this class: to be reliable the maximum likelihood estimator typically requires large sample sizes of the order of thousands of observations. Moreover, improper prior distributions may lead to improper posterior distributions, whereas proper prior distributions may dominate the analysis. Here, we show through a simulation study that Bayesian methods based on Jeffreys prior provide reliable point and interval estimators. Moreover, this simulation study shows that for the absolute loss function Bayesian estimators compare favorably to maximum likelihood estimators. Finally, we illustrate with an application to real data that our methodology allows for parameter estimation with remarkable good properties even for a small sample size.

Article information

Source
Braz. J. Probab. Stat., Volume 26, Number 4 (2012), 327-343.

Dates
First available in Project Euclid: 3 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1341320246

Digital Object Identifier
doi:10.1214/11-BJPS142

Mathematical Reviews number (MathSciNet)
MR2949082

Zentralblatt MATH identifier
1319.62060

Keywords
Asymmetry heavy tails normal-mean mixture noninformative prior

Citation

Fonseca, Thaís C. O.; Migon, Helio S.; Ferreira, Marco A. R. Bayesian analysis based on the Jeffreys prior for the hyperbolic distribution. Braz. J. Probab. Stat. 26 (2012), no. 4, 327--343. doi:10.1214/11-BJPS142. https://projecteuclid.org/euclid.bjps/1341320246


Export citation

References

  • Abramowitz, M. and Stegun, A. S. (1972). Handbook of Mathematical Functions. New York: Dover Publications.
  • Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian Journal of Statistics 12, 171–178.
  • Azzalini, A. (2005). The skew-normal distribution and related multivariate families. Scandinavian Journal of Statistics 32, 159–188.
  • Barndorff-Nielsen, O. (1977). Exponentially decreasing distributions for the logarithm of particles size. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 353, 401–419.
  • Barndorff-Nielsen, O. (1979). Models for non-Gaussian variation, with applications to turbulence. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 368, 501–520.
  • Barndorff-Nielsen O. and Blæsild, P. (1981). Hyperbolic distributions and ramifications: contributions to theory and application. In Statistical Distributions in Scientific Work (C. Taillie, G. P. Patil and B. A. Baldessari, eds.) 4, 19–44. Dordrecht: Reidel.
  • Bauer, C. (2000). Value at risk using hyperbolic distributions. Journal of Economics and Business 52, 455–467.
  • Bingham N. H. and Kiesel, R. (2001). Modelling asset returns with hyperbolic distributions. In Asset Return Distributions (J. Knight and S. Satchell, eds.) 1, 1–20. Oxford: Butterworth-Heinemann.
  • Blæsild, P. (1981). The two dimensional hyperbolic distribution and related distributions, with an application to Johannsen’s bean data. Biometrika 68, 251–263.
  • Blæsild, P. and Sørensen, M. (1992). Hyp—A computer program for analysing data by means of the hyperbolic distribution. Research Report 248, Dept. Theoretical Statistics, Aarhus Univ., Aarhus, Denmark.
  • Eberlein E. and Keller, U. (1995). Hyperbolic distributions in finance. Bernoulli 1, 281–299.
  • Eberlein E., Keller, U. and Prause, K. (1998). New insights into smile, mispricing and value at risk: The hyperbolic model. Journal of Business 71, 371–406.
  • Firth, D. (1993). Bias reduction of maximum likelihood estimates. Biometrika 80, 27–38.
  • Fonseca, T. C. O., Ferreira, M. A. R. and Migon, H. S. (2008). Objective Bayesian analysis for the Student-$t$ regression model. Biometrika 95, 325–333.
  • Jørgensen, B. (1982). Statistical Properties of the Generalized Inverse Gaussian Distribution. Lecture Notes in Statistics 9. Heidelberg: Springer.
  • Liseo, B. and Loperfido, N. (2006). A note on the reference priors for the scalar skew-normal distribution. Journal of Statistical Planning and Inference 136, 373–389.
  • Prause, K. (1999). The generalized hyperbolic model: estimation, financial derivatives, and risk measures. Ph.D. thesis, Univ. Freiburg.
  • R Development Core Team (2010). R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. ISBN 3-900051-07-0.
  • Rice, S. and Church, M. (1996). Sampling superficial gravels: The precision of size distribution percentile estimates. Journal of Sedimentary Research 66, 654–665.
  • Silva, R. S., Lopes, H. F. and Migon, H. S. (2006). The extended generalized inverse Gaussian distribution for log-linear and stochastic volatility models. Brazilian Journal of Probability and Statistics 20, 67–91.
  • Zellner, A. (1976). Bayesian and non-Bayesian analysis of the regression model with multivariate Student-$t$ error term. Journal of the American Statistical Association 71, 400–405.