A New Three-Parameter Mixed Poisson Transmuted Weighted Exponential Distribution with Applications to Insurance Data
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Keywords:
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Weighted Exponential Distribution, Quadratic Transmutation Map, Dispersed Count Observations, Parameter Estimation
Abstract
The classical Poisson distribution assumes equidispersion and this is hardly the case when given count observations. To improve efficiency and general applicability for dispersed data, this paper obtains a new mixing probability distribution using quadratic transmutation map on the weighed exponential distribution. The mixing distribution is used to obtain a new three parameter mixed Poisson distribution. The new distribution has unimodal shape with positive skewness and tends to zero speedily. Basic properties of the new distribution like the distribution function, moments, and dispersion index are obtained. Maximum likelihood estimates of the parameters of the distribution are obtained. Applications are made using four claims frequencies from different insurance companies. Results show that the new distribution gives a good fit in comparisons with some referred discrete distributions.
References
Adcock, C., M. Eling, and N. Loperfido (2015). Skewed Distributions in Finance and Actuarial Science: A Review. The European Journal of Finance, 21(13-14); 1253–1281
Adetunji, A. A. and S. R. M. Sabri (2021). Modelling Claim Frequency in Insurance Using Count Models. Asian Journal of Probability and Statistics, 14(4); 14–20
AL Kadim, K. A. (2018). Proposed Generalized Formula for Transmuted Distribution. Journal of University of Babylon for Pure and Applied Sciences, 26(4); 66–74
Altun, E. (2019). Weighted Exponential Regression Model: an Alternative to the Gamma Regression Model. International Journal of Modeling, Simulation, and Scientific Computing, 10(06); 1950035
Alzaatreh, A., C. Lee, and F. Famoye (2013). A New Method for Generating Families of Continuous Distributions. Metron, 71(1); 63–79
Aslam, M., Z. Hussain, and Z. Asghar (2018). Cubic Transmuted-g Family of Distributions and its Properties. Stochastics and Quality Control, 33(2); 103–112
Azzalini, A. (1985). A Class of Distributions Which Includes The Normal Ones. Scandinavian Journal of Statistics; 171–178
Bhati, D., D. Sastry, and P. M. Qadri (2015). A New Generalized Poisson-lindley Distribution: Applications and Properties. Austrian Journal of Statistics, 44(4); 35–51
Bourguignon, M., R. B. Silva, and G. M. Cordeiro (2014). The Weibull-G Family of Probability Distributions. Journal of Data Science, 12(1); 53–68
Cordeiro, G. M., E. Ortega, and G. Silva (2012). The Beta Extended Weibull Family. Journal of Probability and Statistical Science, 10(10); 15–40
Cordeiro, G. M., E. M. Ortega, and D. C. da Cunha (2013). The Exponentiated Generalized Class of Distributions. Journal of Data Science, 11(1); 1–27
Denuit, M. (1997). A New Distribution of Poisson Type for the Number of Claims. ASTIN Bulletin: The Journal of the IAA, 27(2); 229–242
Granzotto, D., F. Louzada, and N. Balakrishnan (2017). Cubic Rank Transmuted Distributions: Inferential Issues and Applications. Journal of statistical Computation and Simulation, 87(14); 2760–2778
Greenwood, M. and G. U. Yule (1920). An Inquiry Into the Nature of Frequency Distributions Representative of Multiple Happenings With Particular Reference to the Occurrence of Multiple Attacks of Disease or of Repeated Accidents. Journal of the Royal Statistical Society, 83(2); 255-279
Gupta, R. D. and D. Kundu (2009). A New Class of Weighted Exponential Distributions. Statistics, 43(6); 621–634
Karlis, D. and E. Xekalaki (2005). Mixed Poisson Distributions. International Statistical Review/Revue Internationale de Statistique; 35–58
Mahdavi, A. and D. Kundu (2017). A New Method for Generating Distributions with an Application to Exponential Distribution. Communications in Statistics-Theory and Methods, 46(13); 6543–6557
Meytrianti, A., S. Nurrohmah, and M. Novita (2019). An Alternative Distribution for Modelling Overdispersion Count Data: Poisson Shanker Distribution. ICSA - International Conference on Statistics and Analytics, 1; 108–120
Ohlsson, E. and B. Johansson (2010). Non-life insurance pricing with generalized linear models, volume 174. Springer
Omari, C. O., S. G. Nyambura, and J. M. W. Mwangi (2018). Modeling the Frequency and Severity of Auto Insurance Claims Using Statistical Distributions. Journal of Mathematical Finance, 8(1); 137–160
R Core Team, R. and R. C. Team (2021). R: a language and environment for statistical computing. R Foundation for Statistical Computing; 2020. Vienna, Austria
Rahman, M. M., B. Al Zahrani, S. H. Shahbaz, and M. Q. Shahbaz (2019). Cubic Transmuted Uniform Distribution: An Alternative to Beta and Kumaraswamy Distributions. European Journal of Pure and Applied Mathematics, 12(3); 1106–1121
Sarul, L. S. and S. Sahin (2015). An Application of Claim Frequency Data Using Zero Inflated and Hurdle Models in General Insurance. Journal of Business Economics and Finance, 4(4); 732–743
Shakhatreh, M. (2012). A Two Parameter of Weighted Exponential Distributions. Statistics & Probability Letters, 82(2); 252–261
Shaw, W. T. and I. R. Buckley (2007). The Alchemy of Probability Distributions: Beyond Gram charlier Expansions, and A Skew-kurtotic-normal Distribution From A Rank Transmutation Map. Research Report
Yousof, H. M., A. Z. Afify, M. Alizadeh, N. S. Butt, and G. Hamedani (2015). The Transmuted Exponentiated Generalized-G Family of Distributions. Pakistan Journal of Statistics and Operation Research, 11(4); 441–464
Zakerzadeh, H. and A. Dolati (2009). Generalized Lindley Distribution. Journal of Mathematical Extension, 3(2); 1–17
Zamani, H., N. Ismail, and P. Faroughi (2014). Poisson weighted Exponential Univariate Version and Regression Model With Applications. Journal of Mathematics and Statistics, 10(2); 148
Adetunji, A. A. and S. R. M. Sabri (2021). Modelling Claim Frequency in Insurance Using Count Models. Asian Journal of Probability and Statistics, 14(4); 14–20
AL Kadim, K. A. (2018). Proposed Generalized Formula for Transmuted Distribution. Journal of University of Babylon for Pure and Applied Sciences, 26(4); 66–74
Altun, E. (2019). Weighted Exponential Regression Model: an Alternative to the Gamma Regression Model. International Journal of Modeling, Simulation, and Scientific Computing, 10(06); 1950035
Alzaatreh, A., C. Lee, and F. Famoye (2013). A New Method for Generating Families of Continuous Distributions. Metron, 71(1); 63–79
Aslam, M., Z. Hussain, and Z. Asghar (2018). Cubic Transmuted-g Family of Distributions and its Properties. Stochastics and Quality Control, 33(2); 103–112
Azzalini, A. (1985). A Class of Distributions Which Includes The Normal Ones. Scandinavian Journal of Statistics; 171–178
Bhati, D., D. Sastry, and P. M. Qadri (2015). A New Generalized Poisson-lindley Distribution: Applications and Properties. Austrian Journal of Statistics, 44(4); 35–51
Bourguignon, M., R. B. Silva, and G. M. Cordeiro (2014). The Weibull-G Family of Probability Distributions. Journal of Data Science, 12(1); 53–68
Cordeiro, G. M., E. Ortega, and G. Silva (2012). The Beta Extended Weibull Family. Journal of Probability and Statistical Science, 10(10); 15–40
Cordeiro, G. M., E. M. Ortega, and D. C. da Cunha (2013). The Exponentiated Generalized Class of Distributions. Journal of Data Science, 11(1); 1–27
Denuit, M. (1997). A New Distribution of Poisson Type for the Number of Claims. ASTIN Bulletin: The Journal of the IAA, 27(2); 229–242
Granzotto, D., F. Louzada, and N. Balakrishnan (2017). Cubic Rank Transmuted Distributions: Inferential Issues and Applications. Journal of statistical Computation and Simulation, 87(14); 2760–2778
Greenwood, M. and G. U. Yule (1920). An Inquiry Into the Nature of Frequency Distributions Representative of Multiple Happenings With Particular Reference to the Occurrence of Multiple Attacks of Disease or of Repeated Accidents. Journal of the Royal Statistical Society, 83(2); 255-279
Gupta, R. D. and D. Kundu (2009). A New Class of Weighted Exponential Distributions. Statistics, 43(6); 621–634
Karlis, D. and E. Xekalaki (2005). Mixed Poisson Distributions. International Statistical Review/Revue Internationale de Statistique; 35–58
Mahdavi, A. and D. Kundu (2017). A New Method for Generating Distributions with an Application to Exponential Distribution. Communications in Statistics-Theory and Methods, 46(13); 6543–6557
Meytrianti, A., S. Nurrohmah, and M. Novita (2019). An Alternative Distribution for Modelling Overdispersion Count Data: Poisson Shanker Distribution. ICSA - International Conference on Statistics and Analytics, 1; 108–120
Ohlsson, E. and B. Johansson (2010). Non-life insurance pricing with generalized linear models, volume 174. Springer
Omari, C. O., S. G. Nyambura, and J. M. W. Mwangi (2018). Modeling the Frequency and Severity of Auto Insurance Claims Using Statistical Distributions. Journal of Mathematical Finance, 8(1); 137–160
R Core Team, R. and R. C. Team (2021). R: a language and environment for statistical computing. R Foundation for Statistical Computing; 2020. Vienna, Austria
Rahman, M. M., B. Al Zahrani, S. H. Shahbaz, and M. Q. Shahbaz (2019). Cubic Transmuted Uniform Distribution: An Alternative to Beta and Kumaraswamy Distributions. European Journal of Pure and Applied Mathematics, 12(3); 1106–1121
Sarul, L. S. and S. Sahin (2015). An Application of Claim Frequency Data Using Zero Inflated and Hurdle Models in General Insurance. Journal of Business Economics and Finance, 4(4); 732–743
Shakhatreh, M. (2012). A Two Parameter of Weighted Exponential Distributions. Statistics & Probability Letters, 82(2); 252–261
Shaw, W. T. and I. R. Buckley (2007). The Alchemy of Probability Distributions: Beyond Gram charlier Expansions, and A Skew-kurtotic-normal Distribution From A Rank Transmutation Map. Research Report
Yousof, H. M., A. Z. Afify, M. Alizadeh, N. S. Butt, and G. Hamedani (2015). The Transmuted Exponentiated Generalized-G Family of Distributions. Pakistan Journal of Statistics and Operation Research, 11(4); 441–464
Zakerzadeh, H. and A. Dolati (2009). Generalized Lindley Distribution. Journal of Mathematical Extension, 3(2); 1–17
Zamani, H., N. Ismail, and P. Faroughi (2014). Poisson weighted Exponential Univariate Version and Regression Model With Applications. Journal of Mathematics and Statistics, 10(2); 148
Authors
Adetunji, A., & Sabri, S. R. M. (2023). A New Three-Parameter Mixed Poisson Transmuted Weighted Exponential Distribution with Applications to Insurance Data. Science and Technology Indonesia, 8(2), 235–244. https://doi.org/10.26554/sti.2023.8.2.235-244
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