Distributional Nikulin-Rao-Robson validity under a novel gamma extension with characterizations and risk assessment

Document Type : Special Issue Dedicated to memory of Prof. Mahbanoo Tata

Authors

1 Department of Mathematical and Statistical Sciences, Marquette University, USA

2 Department of Quantitative Methods, School of Business, King Faisal University, Al-Ahsa 31982, Saudi Arabia

3 Department of Applied Statistics and Insurance, Faculty of Commerce, Mansoura University, Mansoura 35516, Mansoura, Egypt

4 Laboratory of probability and statistics LaPS, University Badji Mokhtar, Annaba, Algeria

5 Department of Statistics, American University, Cairo, Egypt

6 Department of Statistics, Cornell University, Ithaca, NY, USA

7 Department of Statistics, Mathematics and Insurance, Benha University, Benha, Egypt

8 Department of Applied, Mathematical and Actuarial Statistics, Faculty of Commerce, Damietta University, Damietta, Egypt

Abstract

In this work, a novel probability distribution is introduced and studied. Some characterizations are presented. Several financial risk indicators, such as the value-at-risk, tail-valueat-risk, tail variance, tail Mean-Variance, and mean excess loss function are considered under the maximum likelihood estimation, the ordinary least squares, the weighted least squares, and the Anderson Darling estimation methods. These four methods were applied for the actuarial evaluation under a simulation study and under an application to insurance claims data. For distributional validation under the complete data, the well-known Nikulin-Rao-Robson statistic is considered. The Nikulin-Rao-Robson test statistic is assessed under a simulation study and under three complete real data sets. For censored distributional validation, a new version of the Nikulin-Rao-Robson statistic is considered. The new Nikulin-Rao-Robson test statistic is assessed under a comprehensive simulation study and under three censored real data sets.

Keywords

Main Subjects

References

[1] Abouelmagd, T. H. M., Hamed, M. S., Hamedani, G. G., Ali, M. M., Goual, H., Korkmaz, M. C. and Yousof, H. M. (2019a). The zero truncated Poisson Burr X family of distributions with properties, characterizations, applications, and validation test. Journal of Nonlinear Sciences and Applications (JNSA), 12(5), 314-336.
[2] Aidi, K., Butt, N. S., Ali, M. M., Ibrahim, M., Yousof, H. M. and Shehata, W. A. M. (2021). A Modi ed Chi-square Type Test Statistic for the Double Burr X Model with Applications to Right Censored Medical and Reliability Data. Pakistan Journal of
Statistics and Operation Research, 17(3), 615-623.
[3] Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (1999). Coherent measures of risk. Mathematical  nance, 9(3), 203-228.
[4] Bagdonavicius, V., & Nikulin, M. (2011). Chi-squared goodness-of- t test for right censored data. International Journal of Applied Mathematics and Statistics, 24(1), 1-11.
[5] Cabarbaye, A., Faure, J., Laulheret, R., & Cabarbaye, A. (2009). Interest of a global optimization tool for reliability models adjustment and systems optimization. In Reliability, Risk, and Safety, Three Volume Set (pp. 617-624). CRC Press.
[6] Castillo, E. and Hadi, A. S. (1995). A Method for Estimating Parameters and Quantiles of Continuous Distributions of Random Variables, Computational Statistics and Data Analysis, 20, 421{439.
[7] Emam, W.; Tashkandy, Y.; Hamedani, G.G.; Shehab, M.A.; Ibrahim, M.; Yousof, H.M. (2023). A Novel Discrete Generator with Modeling Engineering, Agricultural and Medical Count and Zero-In ated Real Data with Bayesian, and Non-Bayesian Inference. Mathematics 2023, 11, 1125. https://doi.org/10.3390/math11051125
[8] Emam, W.; Tashkandy, Y.; Goual, H.; Hamida, T.; Hiba, A.; Ali, M.M.; Yousof, H.M.; Ibrahim, M. (2023). A New One-Parameter Distribution for Right Censored Bayesian and Non-Bayesian Distributional Validation under Various Estimation Methods. Mathematics 2023, 11, 897. https://doi.org/10.3390/math11040897
[9] Goual, H. and Yousof, H. M. (2020). Validation of Burr XII inverse Rayleigh model via a modi ed chi-squared goodness-of- t test. Journal of Applied Statistics, 47(3), 393-423.
[10] Goual, H., Yousof, H. M. and Ali, M. M. (2020). Lomax inverse Weibull model: properties, applications, and a modi ed Chi-squared goodness-of- t test for validation. Journal of Nonlinear Sciences & Applications (JNSA), 13(6), 330-353.
[11] Goual, H., Yousof, H. M. and Ali, M. M. (2019). Validation of the odd Lindley exponentiated exponential by a modi ed goodness of  t test with applications to censored and complete data. Pakistan Journal of Statistics and Operation Research, 15(3), 745-771.
[12] Gupta, R. C., Gupta, P. L. and Gupta, R. D. (1998). Modeling failure time data by Lehman alternatives. Communications in Statistics-Theory and methods, 27(4), 887-904.
[13] Glanzel, W. (1987). A characterization theorem based on truncated moments and its application to some distribution families, Mathematical Statistics and Probability Theory (Bad Tatzmannsdorf, 1986), Vol. B, Reidel, Dordrecht, 1987, 75{84.
[14] Glanzel, W. (1990). Some consequences of a characterization theorem based on truncated moments, Statistics: A Journal of Theoretical and Applied Statistics, 21(4), 613{618.
[15] Hamedani, G.G. (2013). On certain generalized gamma convolution distributions II, Technical Report, No. 484, MSCS, Marquette University, 2013.
[16] Hamed, M. S., Cordeiro, G. M. and Yousof, H. M. (2022). A New Compound Lomax Model: Properties, Copulas, Modeling and Risk Analysis Utilizing the Negatively Skewed Insurance Claims Data. Pakistan Journal of Statistics and Operation Research, 18(3), 601-631. https://doi.org/10.18187/pjsor.v18i3.3652
[17] Ibrahim, M., Aidi, K., Ali, M. M. and Yousof, H. M. (2023). A Novel Test Statistic for Right Censored Validity under a new Chen extension with Applications in Reliability and Medicine. Annals of Data Science, 10(5), 1285{1299.
[18] Ibrahim, M., Altun, E., Goual, H., and Yousof, H. M. (2020). Modi ed goodness-of- t type test for censored validation under a new Burr type XII distribution with di erent methods of estimation and regression modeling. Eurasian Bulletin of Mathematics, 3(3), 162-182.
[19] Ibrahim, M.; Emam, W.; Tashkandy, Y.; Ali, M.M.; Yousof, H.M. Bayesian and Non-Bayesian Risk Analysis and Assessment under Left-Skewed Insurance Data and a Novel Compound Reciprocal Rayleigh Extension. Mathematics 2023, 11, 1593. https://doi.org/10.3390/math11071593
[20] Ibrahim, M., Yadav, A. S., Yousof, H. M., Goual, H. and Hamedani, G. G. (2019). A new extension of Lindley distribution: modi ed validation test, characterizations and di erent methods of estimation. Communications for Statistical Applications and Methods, 26(5), 473-495.
[21] Khedr, A. M., Nofal, Z. M., El Gebaly, Y. M., & Yousof, H. M. A Novel Family of Compound Probability Distributions: Properties, Copulas, Risk Analysis and Assessment under a Reinsurance Revenues Data Set. Thailand Statistician, forthcoming.
[22] Khalil, M. G., Yousof, H. M., Aidi, K., Ali, M. M., Butt, N. S. and Ibrahim. M. (2023). Modi ed Bagdonavicius-Nikulin Goodness-of- t Test Statistic for the Compound Topp Leone Burr XII Model with Various Censored Applications. Statistics, Optimization & Information Computing, forthcoming.
[23] Klein, J. P. and Moeschberger, M. L. (2003). Survival analysis: techniques for censored and truncated data (Vol. 1230). New York: Springer.
[24] Mohamed, H. S., Cordeiro, G. M., Minkah, R., Yousof, H. M., & Ibrahim, M. (2024). A size-of-loss model for the negatively skewed insurance claims data: applications, risk analysis using di erent methods and statistical forecasting. Journal of Applied Statistics, 51(2), 348-369.
[25] Murthy, D. P., Xie, M., & Jiang, R. (2004). Weibull models. John Wiley & Sons.
[26] Nikulin, M. S., (1973a). Chi-square test for normality. In: Proceedings of the International Vilnius Conference on Probability Theory and Mathematical Statistics, Vol. 2, 199-122.
[27] Nikulin, M. S., (1973b), chi-squared test for continuous distribution with shift and scale parameters, Theory of Probability and its Applications, 18(3), 559-568.
[28] Nikulin. M.S, (1973c). On a Chi-squared test for continuous distributions, Theory of Probability and its Applications. 19, 638-639.
[29] Rao, K. C., Robson, D. S., (1974). A chi-square statistic for goodness-of- t tests within the exponential family. Communication in Statistics, 3, 1139-1153.
[30] Ravi, V. and Gilbert, P. D. (2009). BB: An R package for solving a large system of nonlinear equations and for optimizing a high-dimensional nonlinear objective function, J. Statist. Software, 32(4), 1-26.
[31] Smith, R. L. and Naylor, J. (1987). A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution. Journal of the Royal Statistical Society: Series C (Applied Statistics), 36(3), 358-369.
[32] Voinov, V., Nikulin, M., and Balakrishnan, N. (2013), chi-squared Goodness of Fit Tests with Applications, Academic Press, Elsevier.
[33] Yadav, A. S., Goual, H., Alotaibi, R. M., Rezk, H., Ali, M. M. and Yousof, H. M. (2020). Validation of the Topp-Leone-Lomax model via a modi ed Nikulin-Rao-Robson goodness-of- t test with di erent methods of estimation. Symmetry, 12(1), 57.
[34] Yadav, A. S., Shukla, S., Goual, H., Saha, M. and Yousof, H. M. (2022). Validation of xgamma exponential model via Nikulin-Rao-Robson goodness-of-  t test under complete and censored sample with di erent methods of estimation. Statistics, Optimization & Information Computing, 10(2), 457-483.
[35] Yousof, H. M., Ali, M. M., Aidi, K., Ibrahim, M. (2023a). The modi ed Bagdonavicius-Nikulin goodness-of- t test statistic for the right censored distributional validation with applications in medicine and reliability. Statistics in Transition New Series, 24(4), 1-18.
[36] Yousof, H. M., Ali, M. M., Goual, H. and Ibrahim. M. (2021). A new reciprocal Rayleigh extension: properties, copulas, di erent methods of estimation and modi ed right censored test for validation, Statistics in Transition New Series, 23(3), 1-23.
[37] Yousof, H. M., A fy, A. Z., Hamedani, G. G. and Aryal, G. (2017). The Burr X generator of distributions for lifetime data. Journal of Statistical Theory and Applications, 16(3), 288-305.
[38] Yousof, H. M., Aidi, K., Hamedani, G. G., & Ibrahim, M. (2023b). Chi-squared type test for distributional censored and uncensored validity with numerical assessments and real data applications. Japanese Journal of Statistics and Data Science, 6(2), 729-758.
Journal of Mahani Mathematical Research

Articles in Press, Accepted Manuscript
Available Online from 13 April 2024

Files

History

  • Receive Date: 15 February 2024
  • Accept Date: 16 March 2024

Share

How to cite

Statistics