Main Article Content


In this paper, we introduce some new goodness-of-fit tests for the Rayleigh distribution based on Jeffreys, Lin-Wong and Renyi divergence measures. Then, the new proposed tests are compared with other tests in the literature. We compare the power of considered tests for some alternative distributions whose powers are calculated by Monte Carlo simulation. Finally, we conclude that entropy-based tests have a good performance in terms of power and among them Jeffreys test is the best one.



Divergence measure Entropy Goodness-of-fit Jeffreys Lin-Wong Monte Carlo Simulation Power Rayleigh distribution Renyi

Article Details

How to Cite
Amir Jahanshahi, S. M., Habibi Rad, A., & Fakoor, V. (2020). Some New Goodness-of-fit Tests for Rayleigh Distribution. Pakistan Journal of Statistics and Operation Research, 16(2), 305-315.


  1. Abbasnejad, M. (2011). Some goodness of fit tests based on Rényi information, Applied Mathematical Sciences, 5, 1921-1934.
  2. Abbasnejad, M., Arghami, N. R. & Tavakoli, M. (2012). A Goodness of Fit Test for Exponentiality Based on Lin-Wong Information, JIRSS, 11(2), 191-202.
  3. Abd-Elfattah, A. M. (2011). Goodness of fit test for the generalized Rayleigh distribution with unknown parameters, Journal of Statistical Computation and Simulation, 81(3), 357-366.
  4. Ahrari, V., Baratpour, S., Habibirad, A. & Fakoor, V. (2019). Goodness of fit tests for Rayleigh distribution based on quantiles, Communications in Statistics - Simulation and Computation, DOI: 10.1080/03610918.2019.1651336.
  5. Alizadeh, R., Alizadeh, H. & Ebrahimi, A. (2012). An entropy test for the Rayleigh distribution and power comparison, Journal of Statistical Computation and Simulation, 151-158.
  6. Alizadeh Noughabi, H. & Arghami, N. R. (2011a). Monte Carlo comparison of five exponentiality tests using different entropy estimates, J. Statist. Comput. Simul., 81, 1579-1592.
  7. Alizadeh Noughabi, H. & Arghami, N. R. (2011). Testing exponentiality based on characterizations of the exponential distribution, Journal of Statistical Computation and Simulation, 81, 1641-1651.
  8. Alizadeh Noughabi, H. & Balakrishnan, N. (2014). Goodness of Fit Using a New Estimate of Kullback-Leibler Information Based on Type II Censored Data, IEEE Transactions on Reliability, 99.
  9. Alizadeh Noughabi, H. (2010). A new estimator of entropy and its application in testing normality, J. Stat. Comput. Simul, 80(10), 1151-1162.
  10. Arellano-Valle, R. B., Galea-Rojas, M. & Iglesias, P. (2000). Bayesian sensitivity analysis in elliptical linear regression models, J. Stat. Plan. Infer, 86, 175-199.
  11. Al-Omari, A. & Zamanzade, E. (2016). Different goodness of fit tests for Rayleigh distribution in ranked set sampling, Pakistan Journal of Statistics and Operation Research, 12(1), 25-39.
  12. Badr, M. M. (2019). Goodness-of-fit tests for the Compound Rayleigh distribution with application to real data, Heliyon 5(8).
  13. Baratpour, S. & Khodadadi, F. (2012). A Cumulative Residual Entropy Characterization of the Rayleigh Distribution and Related Goodness-of-Fit Test, J. Statist. Res. Iran, 9, 115-131.
  14. Baratpour, S. & Habibi, A. (2012). Testing Goodness-of-Fit for Exponential Distribution Based on Cumulative Residual Entropy, Communications in Statistics - Theory and Methods, 41, 1387-1396.
  15. Best, D. J., Rayner, J. C. W. & Thas, O. (2010). Easily applied tests of fit for the Rayleigh distribution, Sankhya B, 72, 254-263.
  16. Correa, J. C. (1995). A new estimator of entropy, Comm. Statist. Theory Methods, 24, 2439-2449.
  17. Dyer, D. D. & Whisenand, C. W. (1973). Best linear unbiased estimator of the parameter of the Rayleigh distribution, IEEE Transaction on Reliability, 22, 27-34.
  18. Dudewicz, E. J. & Van der Meulen, E. C. (1981). Entropy-based tests of uniformity, J. Amer. Statist. Assoc. 76, 967-974.
  19. Ebrahimi, N., Pflughoeft, K. & Soofi, E. (1994). Two measures of sample entropy, Statist. Probab. Lett. 20, 225-234.
  20. Ebrahimi, N., Habibullah, M. & Soofi, E. S. (1992). Testing exponentiality based on Kullback-Leibler information, J. R. Statist. Soc., 54, 739-748.
  21. Finkelstein, J. & Schafer, R. E. (1971). Imported goodness of fit tests, Biometrika, 58, 641-645.
  22. Garrido, A. (2009). About some properties of the Kullback-Leibler divergence, Advanced Modeling and Optimization, 11(4).
  23. Jahanshahi, S. M. A., Habibirad, A., Fakoor, V. (2016). A Goodness-of-Fit Test for Rayleigh Distribution Based on Hellinger Distance, Annals of Data Science, 3(4), 401-411.
  24. Jeffreys, H. (1946). An invariant form for the prior probability in estimation problems, Proc. Roy. Soc. Lon., Ser. A, 186, 453-461.
  25. Johnson, N. L., Kotz, S. & Balakrishnan, N. (1994). Continuous univariate distributions, Wiley, New York.
  26. Lin, J. & Wong, S. K. M. (1990). A new directed divergence measure and its characterization, Int. J. General Systems, 17, 73-81.
  27. Kapur, J. N. & Kesavan, H. K. (1992). Entropy optimization principles with application, Academic Press.
  28. Meintanis, S. G. and Iliopoulos, G. (2003). Tests of fit for the Rayleigh distribution based on the empirical Laplace transform, Ann. Inst. Statist. Math., 55(1), 137-151.
  29. Morris, K. & Szynal, D. (2008). Some U-statistics in goodness-of-fit tests derived from characterizations via record values, Int. J. of Pure and Appl. Math., 46, 507-582.
  30. Miller, K. & Sackrowttz, H. (1967). Relationships Between Biased and Unbiased Rayleigh Distributions, SIAM journal on applied Mathematics, 15, 1490-1495.
  31. North B., Curtis, D. & Sham, P. (2003). A Note on the Calculation of Empirical p-values from Monte Carlo Procedures, American Journal of Human Genetics, 71(2), 439-441.
  32. Pashardes, S. & Christofides, C. (1995). Statistical analysis of wind speed and direction in Cyprus, Solar Energy, 55(5), 405-414.
  33. Polovko, A. M. (1968). Fundamentals of reliability theory, Academic Press, New York.
  34. Rayleigh, J. W. S. (1880). On the resultant of a large number of vibrations of some pitch and of arbitrary phase, Philosophical Magazine, 5-th Series, 10, 73-78.
  35. Safavinejad, M., Jomhoori, S. & Alizadeh Noughabi, H. (2015). A density-based empirical likelihood ratio goodness-of-fit test for the Rayleigh distribution and power comparison, Journal of Statistical Computation and Simulation, 85, 3322-3334.
  36. Shannon, C. E. (1948). A mathematical theory of communication. Bell System Technical Journal, 27, 379-423.
  37. Siddiqui, M. M. (1962). Some problems connected with Rayleigh distributions, J. Res. Natl. Bur. Stand. 60D, 167-174.
  38. Ullah, A. (1996). Entropy, divergence and distance measures with econometric applications, J. Stat. Plan. Infer., 49, 137-162.
  39. Vasicek, O. (1976). A test for normality based on sample entropy, J. R. Statist. Soc., 38, 54-59.
  40. Van Es, B. (1992). Estimating functional related to a density by a lass of statistic based on spacing, Scand. J. Statist., 19, 61-72.
  41. Zamanzade, E. & Mahdizadeh, M. (2017). Goodness of fit tests for Rayleigh distribution based on Phi-divergence, Revista Colombiana de Estadística 40(2), 279-290.