DOI QR코드

DOI QR Code

Comparison of parameter estimation methods for normal inverse Gaussian distribution

  • Received : 2019.09.19
  • Accepted : 2019.11.12
  • Published : 2020.01.31

Abstract

This paper compares several methods for estimating parameters of normal inverse Gaussian distribution. Ordinary maximum likelihood estimation and the method of moment estimation often do not work properly due to restrictions on parameters. We examine the performance of adjusted estimation methods along with the ordinary maximum likelihood estimation and the method of moment estimation by simulation and real data application. We also see the effect of the initial value in estimation methods. The simulation results show that the ordinary maximum likelihood estimator is significantly affected by the initial value; in addition, the adjusted estimators have smaller root mean square error than ordinary estimators as well as less impact on the initial value. With real datasets, we obtain similar results to what we see in simulation studies. Based on the results of simulation and real data application, we suggest using adjusted maximum likelihood estimates with adjusted method of moment estimates as initial values to estimate the parameters of normal inverse Gaussian distribution.

Keywords

References

  1. Barndorff-Nielsen OE (1977). Exponentially decreasing distributions for the logarithm of particle size, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 353, 401-419. https://doi.org/10.1098/rspa.1977.0041
  2. Barndorff-Nielsen OE (1997). Processes of normal inverse Gaussian type, Finance and Stochastics, 2, 41-68. https://doi.org/10.1007/s007800050032
  3. Behr A and Potter U (2009). Alternatives to the normal model of stock returns: Gaussian mixture, generalised log F and generalised hyperbolic models, Annals of Finance, 5, 49-68. https://doi.org/10.1007/s10436-007-0089-8
  4. Bolance C, Guillen M, Pelican E, and Vernic R (2008). Skewed bivariate models and nonparametric estimation for the CTE risk measure, Insurance: Mathematics and Economics, 43, 386-393. https://doi.org/10.1016/j.insmatheco.2008.07.005
  5. Eberlein E and Keller U (1995). Hyperbolic distributions in finance, Bernoulli, 1, 281-299. https://doi.org/10.2307/3318481
  6. Eriksson A, Ghysels E, and Wang F (2009). The normal inverse Gaussian distribution and the pricing of derivatives, The Journal of Derivatives, 16, 23-37. https://doi.org/10.3905/JOD.2009.16.3.023
  7. Fajardo J and Farias A (2004). Generalized hyperbolic distributions and Brazilian data, Brazilian Review of Econometrics, 24, 249-271. https://doi.org/10.12660/bre.v24n22004.2712
  8. Figueroa-Lopez JE, Lancette SR, Lee K, and Mi Y (2011). Handbook of Modeling High-Frequency Data in Finance, John Wiley & Sons, Hoboken, 3-26.
  9. Frees E and Valdez E (1998). Understanding relationships using copulas, North American Actuarial Journal, 2, 1-25. https://doi.org/10.1080/10920277.1998.10595667
  10. Ghysels E and Wang F (2014). Moment-implied densities: properties and applications, Journal of Business & Economic Statistics, 32, 88-111. https://doi.org/10.1080/07350015.2013.847842
  11. Hansen B (1994). Autoregressive conditional density estimation, International Economic Review, 35, 705-730. https://doi.org/10.2307/2527081
  12. Karlis D (2002). An EM type algorithm for maximum likelihood estimation of the normal-inverse Gaussian distribution, Statistics & Probability Letters, 57, 43-52. https://doi.org/10.1016/S0167-7152(02)00040-8
  13. Karlis D and Lillestol J (2004). Bayesian estimation of NIG models via Markov chain Monte Carlo methods, Applied Stochastic Models in Business and Industry, 20, 323-338. https://doi.org/10.1002/asmb.544
  14. Kim J (2019). A study on the estimation of spliced distributions using exponential-estimation method (Master's thesis), Korea University, Seoul.
  15. Madan D and Seneta E (1990). The VG model for share market returns, Journal of Financial Economics, 63, 511-524.
  16. McNeil A (1997). Estimating the tails of loss severity distributions using extreme value theory, ASTIN Bulletin, 27, 117-137. https://doi.org/10.2143/AST.27.1.563210
  17. Prause K (1999). The generalized hyperbolic model: estimation, financial derivatives, and risk measures (Ph.D. thesis), University of Freiburg, Germany.
  18. Rydberg TH (1997). The normal inverse Gaussian Levy process: simulation and approximation, Communications in Statistics: Stochastic Models, 13, 887-910. https://doi.org/10.1080/15326349708807456
  19. Yoon J and Song S (2016). A numerical study of adjusted parameter estimation in normal inverse Gaussian distribution, The Korean Journal of Applied Statistics, 29, 741-752. https://doi.org/10.5351/KJAS.2016.29.4.741