Abstract
We develop new goodness of fit tests for Rayleigh distribution based on fixed point characterization. We use U-Statistic theory to derive the test statistics. First we develop a test for complete data and then discuss, how the right censored observations can be incorporated in the testing procedure. The asymptotic properties of the test statistic in both uncensored and censored cases are studied in detail. Extensive Monte Carlo simulation studies are carried out to validate the performance of the proposed tests. We illustrate the procedures using real data sets. We also provide, a goodness of fit test for the standard Rayleigh distribution based on jackknife empirical likelihood.
Similar content being viewed by others
Data availability
The source of all data sets used for illustration purpose are mentioned in the manuscript.
References
Ahrari, V., Baratpour, S., Habibirad, A., & Fakoor, V. (2022). Goodness of fit tests for Rayleigh distribution based on quantiles. Communications in Statistics-Simulation and Computation, 51(2), 341–357.
Ahsanullah, M. & Shakil, M. (2013). Characterizations of Rayleigh distribution based on order statistics and record values. The Bulletin of the Malaysian Mathematical Society Series 2, 36(3), 625–635.
Andrews, D. F., & Herzberg, A. M. (1985). Data: A Collection of Problems from Many Fields for the Student and Research Worker. New York: Springer Series in Statistics.
Baratpour, S., & Khodadadi, F. (2013). A cumulative residual entropy characterization of the Rayleigh distribution and related goodness-of-fit test. Journal of Statistical Research of Iran, 9(2), 115–131.
Best, D. J., Rayner, J. C., & Thas, O. (2010). Easily applied tests of fit for the Rayleigh distribution. Sankhya B, 72(2), 254–263.
Bovaird, T., & Lineweaver, C. H. (2017). A flat inner disc model as an alternative to the Kepler dichotomy in the Q1–Q16 planet population. Monthly Notices of the Royal Astronomical Society, 468(2), 1493–1504.
Caroni, C. (2002). The correct “ball bearings’’ data. Lifetime Data Analysis, 8(4), 395–399.
Celik, A. N. (2004). A statistical analysis of wind power density based on the Weibull and Rayleigh models at the southern region of Turkey. Renewable Energy, 29(4), 593–604.
Datta, S., Bandyopadhyay, D., & Satten, G. A. (2010). Inverse Probability of Censoring Weighted U-statistics for Right-Censored Data with an Application to Testing Hypotheses. Scandinavian Journal of Statistics, 37(4), 680–700.
Dey, S., & Dey, T. (2014). Statistical inference for the Rayleigh distribution under progressively Type-II censoring with binomial removal. Applied Mathematical Modelling, 38(3), 974–982.
Fang, J., & Margot, J. L. (2012). Architecture of planetary systems based on Kepler data: Number of planets and coplanarity. The Astrophysical Journal, 761(2), 92.
Jahanshahi, S. M. A., Rad, A. H., & Fakoor, V. (2016). A goodness-of-fit test for Rayleigh distribution based on Hellinger distance. Annals of Data Science, 3(4), 401–411.
Jing, B. Y., Yuan, J., & Zhou, W. (2009). Jackknife empirical likelihood. Journal of the American Statistical Association, 104(487), 1224–1232.
Kim, C., & Han, K. (2009). Estimation of the scale parameter of the Rayleigh distribution with multiply type-II censored sample. Journal of Statistical Computation and Simulation, 79(8), 965–976.
Lawless, J. F. (2011). Statistical models and methods for lifetime data. New York: John Wiley & Sons.
Lee, A. J. (2019). U-Statistics: Theory and Practice. New York: Marcel Dekker Inc.
Liebenberg, S. C., Ngatchou-Wandji, J., & Allison, J. S. (2022). On a new goodness-of-fit test for the Rayleigh distribution based on a conditional expectation characterization. Communications in Statistics-Theory and Methods, 51(15), 5226–5240.
Lieblein, J., & Zelen, M. (1956). Statistical investigation of the fatigue life of deep-groove ball bearings. Journal of Research of the National Bureau of Standards, 57(5), 273–316.
Meintanis, S., & Iliopoulos, G. (2003). Tests of fit for the Rayleigh distribution based on the empirical Laplace transform. Annals of the Institute of Statistical Mathematics, 55(1), 137–151.
Morgan, E. C., Lackner, M., Vogel, R. M., & Baise, L. G. (2011). Probability distributions for offshore wind speeds. Energy Conversion and Management, 52(1), 15–26.
Nanda, A. K. (2010). Characterization of distributions through failure rate and mean residual life functions. Statistics & Probability Letters, 80(9–10), 752–755.
Noughabi, R. A., Noughabi, H. A., & Behabadi, E. M. A. (2014). An entropy test for the Rayleigh distribution and power comparison. Journal of Statistical Computation and Simulation, 84(1), 151–158.
Owen, A. B. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika, 75(2), 237–249.
Owen, A. (1990). Empirical likelihood ratio confidence regions. The Annals of Statistics, 18(1), 90–120.
R Core Team (2021). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/.
Rayleigh, L. (1880). XII. On the resultant of a large number of vibrations of the same pitch and of arbitrary phase. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 10(60), 73-78.
Ross, N. (2011). Fundamentals of Stein’s method. Probability Surveys, 8, 210–293.
Safavinejad, M., Jomhoori, S., & Noughabi, H. A. (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(16), 3322–3334.
Shi, X. (1984). The approximate independence of jackknife pseudo-values and the bootstrap methods. Journal of Wuhan Institute Hydra-Electric Engineering, 2, 83–90.
Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Berkeley symposium on mathematical statistics and probability, 6(2), 583–603.
Sudheesh, K. K. (2009). On Stein’s identity and its applications. Statistics & Probability Letters, 79(12), 1444–1449.
Sudheesh, K. K., & Tibiletti, L. (2012). Moment identity for discrete random variable and its applications. Statistics, 46(6), 767–775.
Sudheesh, K. K., & Dewan, I. (2016). On generalized moment identity and its applications: a unified approach. Statistics, 50(5), 1149–1160.
Thomas, D. R., & Grunkemeier, G. L. (1975). Confidence interval estimation of survival probabilities for censored data. Journal of the American Statistical Association, 70(352), 865–871.
Vaisakh K. M., Sreedevi E.P., & Sudheesh K.K. (2021). A new goodness of fit test for gamma distribution with censored observations. arXiv preprint arXiv:2108.00503.
Zamanzade, E. & Mahdizadeh, M. (2017). Goodness of fit tests for Rayleigh distribution based on Phi-divergence. Revista Colombiana de Estadística, 40(2), 279-290.
Acknowledgements
We would like to thank the anonymous reviewers for their suggestions to improve the quality of the paper substantially. Vaisakh K.M. and Sreedevi E.P. would like to thank Kerala State Council for Science, Technology and Environment for the financial support to carry out this research work. Thomas Xavier would like to thank Dr. Isha Dewan and Dr. Sudheesh K. K. for introducing him to this problem.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix
Simplification of \(\Delta (F)\)
Consider
Now, changing the order of integration we have
where \(\bar{F}(x) = 1 - F(x)\). The last identity follows from the fact that \(2\bar{F}(x)dF(x)\) is the density function of \(\min (X_1,X_2)\). Again,
Substituting (15) and (16) in (14), we obtain
Proof of Theorem 3:
Define
Since \({{\widehat{\sigma }}}^2\) is a consistent estimators of \(\sigma ^2\), by Slutsky’s theorem, the asymptotic distribution of \(\sqrt{n}({\widehat{\Delta }}-\Delta (F))\) and \(\sqrt{n}({\widehat{\Delta }}^{*}-E({\widehat{\Delta }}^{*}))\) are same. Now we observe that \({\widehat{\Delta }}^*\) is a U statistic with symmetric kernel,
Hence using the central limit theorem for U-statistics we have the asymptotic normality of \({\widehat{\Delta }}^*\). The asymptotic variance is \(4\sigma _1^2\) where \(\sigma _1^2\) is given by Lee (2019)
Consider
Also
Substituting equations (19) and (20) in equation (18) we obtain the variance expression as specified in the theorem. \(\square\)
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Vaisakh, K.M., Xavier, T. & Sreedevi, E.P. Goodness of fit test for Rayleigh distribution with censored observations. J. Korean Stat. Soc. 52, 794–815 (2023). https://doi.org/10.1007/s42952-023-00222-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s42952-023-00222-7