Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

Further Results on Characteristic Functions Without Contour Integration

  • Received : 2014.07.31
  • Accepted : 2014.08.29
  • Published : 2014.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

Characteristic functions play an important role in probability and statistics; however, a rigorous derivation of these functions requires contour integration, which is unfamiliar to most statistics students. Without resorting to contour integration, Datta and Ghosh (2007) derived the characteristic functions of normal, Cauchy, and double exponential distributions. Here, we derive the characteristic functions of t, truncated normal, skew-normal, and skew-t distributions. The characteristic functions of normal, cauchy distributions are obtained as a byproduct. The derivations are straightforward and can be presented in statistics masters theory classes.

Keywords

References

  1. Azzalini, A. (1985). A class of distributions which includes the normal ones, Scandinavian Journal of Statistics, 12, 171-178.
  2. Azzalini, A. and Capitanio, A. (1999). Statistical applications of the multivariate skew normal distribution, Journal of the Royal Statistical Society, Series B, 61, 579-602. https://doi.org/10.1111/1467-9868.00194
  3. Azzalini, A. and Capitanio, A. (2014). The Skew-Normal and Related Families, Cambridge University Press, Cambridge.
  4. Barndorff-Nielsen, O., Kent, J. and Sorensen, M. (1982). Normal variance-mean mixtures and z distributions, International Statistical Review, 50, 145-159. https://doi.org/10.2307/1402598
  5. Datta, G. S. and Ghosh, M. (2007). Characteristic functions without contour integration, The American Statistician, 61, 67-70. https://doi.org/10.1198/000313007X170422
  6. Durrett, R. (1996). Probability: Theory and Examples, 2nd edition, Duxbury Press, New York.
  7. Henze, N. (1986). A probabilistic representation of the skew-normal distribution, Scandinavian Journal of Statistics, 13, 271-275.
  8. Kim, H. M. and Genton, M. G. (2011). Characteristic function of scale mixtures of multivariate skew-normal distributions, Journal of Multivariate Analysis, 102, 1105-1117. https://doi.org/10.1016/j.jmva.2011.03.004
  9. Kotz, S. and Nadarajah, S. (2004). Multivariate t Distributions and Their Application, Cambridge University Press, Cambridge.
  10. Pewsey, A. (2000). The wrapped skew-normal distribution on the circle, Communications in Statistics-Theory and Methods, 29, 2459-2472. https://doi.org/10.1080/03610920008832616
  11. Watson, G. N. (1966). A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge.