DOI QR코드

DOI QR Code

SOME ALGEBRA FOR PEARSON TYPE VII RANDOM VARIABLES

  • Published : 2008.05.31

Abstract

The distributions of products and ratios of random variables are of interest in many areas of the sciences. In this paper, the exact distributions of the product |XY| and the ratio |X/Y| are derived when X and Y are independent Pearson type VII random variables.

References

  1. M. S. Abu-Salih, Distributions of the product and the quotient of power-function random variables, Arab J. Math. 4 (1983), no. 1-2, 77-90
  2. A. P. Basu and R. H. Lochner, On the distribution of the ratio of two random variables having generalized life distributions, Technometrics 13 (1971), 281-287 https://doi.org/10.2307/1266790
  3. R. P. Bhargava and C. G. Khatri, The distribution of product of independent beta random variables with application to multivariate analysis, Ann. Inst. Statist. Math. 33 (1981), no. 2, 287-296 https://doi.org/10.1007/BF02480942
  4. M. S. Feldstein, The error of forecast in econometric models when the forecast-period exogenous variables are stochastic, Econometrica 39 (1971), no. 1, 55-60 https://doi.org/10.2307/1909139
  5. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Translated from the Russian. Sixth edition. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. Academic Press, Inc., San Diego, CA, 2000
  6. H. G. Grubel, Internationally diversified portfolios : welfare gains capital flows, The American Economic Review 58 (1968), no. 5, 1299-1314
  7. H. L. Harter, On the distribution of Wald's classification statistic, Ann. Math. Statistics 22 (1951), 58-67 https://doi.org/10.1214/aoms/1177729692
  8. D. L. Hawkins and C.-P. Han, Bivariate distributions of some ratios of independent noncentral chi-square random variables, Comm. Statist. A-Theory Methods 15 (1986), no. 1, 261-277 https://doi.org/10.1080/03610928608829120
  9. P. J. Korhonen and S. C. Narula, The probability distribution of the ratio of the absolute values of two normal variables, J. Statist. Comput. Simulation 33 (1989), no. 3, 173-182 https://doi.org/10.1080/00949658908811195
  10. S. Kotz, T. J. Kozubowski, and K. Podgorski, The Laplace Distribution and Generalizations, A revisit with applications to communications, economics, engineering, and finance. Birkhauser Boston, Inc., Boston, MA, 2001
  11. H. J. Malik and R. Trudel, Probability density function of the product and quotient of two correlated exponential random variables, Canad. Math. Bull. 29 (1986), no. 4, 413-418 https://doi.org/10.4153/CMB-1986-065-3
  12. G. Marsaglia, Ratios of normal variables and ratios of sums of uniform variables, J. Amer. Statist. Assoc. 60 (1965), 193-204 https://doi.org/10.2307/2283145
  13. S. Nadarajah and S. Kotz, Skewed distributions generated by the normal kernel, Statist. Probab. Lett. 65 (2003), no. 3, 269-277 https://doi.org/10.1016/j.spl.2003.07.013
  14. T. Pham-Gia, Distributions of the ratios of independent beta variables and applications, Comm. Statist. Theory Methods 29 (2000), no. 12, 2693-2715 https://doi.org/10.1080/03610920008832632
  15. H. Podolski, The distribution of a product of n independent random variables with generalized gamma distribution, Demonstratio Math. 4 (1972), 119-123
  16. S. J. Press, The t-ratio distribution, J. Amer. Statist. Assoc. 64 (1969), 242-252 https://doi.org/10.2307/2283732
  17. S. B. Provost, On the distribution of the ratio of powers of sums of gamma random variables, Pakistan J. Statist. 5 (1989), no. 2, 157-174
  18. A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series, volumes 1, 2 and 3, Gordon & Breach Science Publishers, New York, 1986
  19. P. N. Rathie and H. G. Rohrer, The exact distribution of products of independent random variables, Metron 45 (1987), no. 3-4, 235-245
  20. A. M. Rugman, International Diversification and the Multinational Enterprise, Lexington, 1979
  21. H. Sakamoto, On the distributions of the product and the quotient of the independent and uniformly distributed random variables, Tohoku Math. J. 49 (1943), 243-260
  22. S. M. Shcolnick, On the ratio of independent stable random variables, Stability problems for stochastic models (Uzhgorod, 1984), 349-354, Lecture Notes in Math., 1155, Springer, Berlin, 1985
  23. M. D. Springer and W. E. Thompson, The distribution of products of beta, gamma and Gaussian random variables, SIAM J. Appl. Math. 18 (1970), 721-737 https://doi.org/10.1137/0118065
  24. B. M. Steece, On the exact distribution for the product of two independent beta-distributed random variables, Metron 34 (1976), no. 1-2, 187-190
  25. A. Stuart, Gamma-distributed products of independent random variables, Biometrika 49 (1962), 564-565 https://doi.org/10.1093/biomet/49.3-4.564
  26. J. Tang and A. K. Gupta, On the distribution of the product of independent beta random variables, Statist. Probab. Lett. 2 (1984), no. 3, 165-168 https://doi.org/10.1016/0167-7152(84)90008-7
  27. C. M. Wallgren, The distribution of the product of two correlated t variates, J. Amer. Statist. Assoc. 75 (1980), no. 372, 996-1000 https://doi.org/10.2307/2287194

Cited by

  1. An alpha-power extension for the Birnbaum–Saunders distribution vol.48, pp.4, 2014, https://doi.org/10.1080/02331888.2013.846910