ZERO-INFLATED POISSON TRANSMUTED WEIGHTED EXPONENTIAL DISTRIBUTION: PROPERTIES AND APPLICATIONS
DOI:
https://doi.org/10.51200/bsj.v44i2.4702Keywords:
Mixed Poisson distribution, mixing distribution, rank transmutation map, weighted exponential distribution, excess zero countsAbstract
Count observations with high frequencies of zero counts abound in diverse fields. In
actuary science, for example, insurance claims are often underreported, leading to a higher frequency
of zero counts. This invariably reduces the mean and leads to over-dispersion. Different techniques to
model such occurrences exist. This study uses the cubic rank transmutation map to compound the
weighted exponential distribution and obtain a new count distribution in the mixed Poisson paradigm.
The zero-inflated form of the proposition is obtained along with some mathematical properties. Simulated skewness, kurtosis, and dispersion index for the new distributions reveal they are suitable for model dispersed observation with positive skewness. Five datasets with high frequencies of zero counts are used to assess the performance of the new distribution along with some popular count distributions by using the maximum likelihood for parameter estimation. Results show that the natural form of the new proposition performs creditably better than its zero-inflated forms, even when there is a higher proportion of zero counts. The classical negative binomial distribution is also observed to outperform its zero-inflated form in most cases. In contrast, the zero-inflated Poisson distribution better fits the datasets than the classical Poisson distribution.
References
Adetunji, A. A., & Sabri, S. R. M. 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 Babylon
University, 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(6), 1–15.
Asamoah, K. 2016. On the Credibility of Insurance Claim Frequency: Generalized Count Models and
Parametric Estimators. Insurance: Mathematics and Economics, 70, 339–353.
Aslam, M., Hussain, Z., & Asghar, Z. 2018. Cubic Transmuted-G family of distributions and its
properties. Stochastic and Quality Control, De Gruyte, 33(2), 103–112.
Conceição, K. S., Louzada, F., Andrade, M. G., & Helou, E. S. 2017. Zero-Modified Power Series
Distribution and its Hurdle Distribution Version. Journal of Statistical Computation and
Simulation, 87(9), 1842–1862.
Das, K. K., Ahmed, I., & Bhattacharjee, S. 2018. A New Three-Parameter Poisson-Lindley Distribution
for Modelling Over-dispersed Count Data. International Journal of Applied Engineering Research,
(23), 16468–16477.
De Jong, P., & Heller, G. Z. 2008. Generalized Linear Models for Insurance Data. Cambridge University
Press.
Greenwood, M., & Yule, G. U. 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.
Gupta, R. D., & Kundu, D. 2009. A New Class of Weighted Exponential Distributions. Statistics, 43(6),
–634.
Karlis, D., & Xekalaki, E. 2005. Mixed Poisson distributions. International Statistical Review, 73(1),
–58.
Lambert, D. 1992. Zero-Inflated Poisson Regression, with an Application to Defects in Manufacturing.
Technometrics, 34, 1–14.
Meytrianti, A., Nurrohmah, S., & Novita, M. 2019. An Alternative Distribution for Modelling
Overdispersion Count Data: Poisson Shanker Distribution. ICSA - International Conference on
Statistics and Analytics 2019, 1, 108–120.
Nash, J. C., Varadhan, R., & Grothendieck, G. 2019. optimr Package: A Replacement and Extension of
the “optim” Function.
Omari, C. O., Nyambura, S. G., & Mwangi, J. M. W. 2018. Modeling the Frequency and Severity of
Auto Insurance Claims Using Statistical Distributions. Journal of Mathematical Finance, 8(1),
–160.
Ong, S. H., Low, Y. C., & Toh, K. K. 2021. Recent Developments in Mixed Poisson Distributions. ASM
Science Journal, 14(3), 1–10.
R-Core Team. 2020. R: A language and Environment for Statistical Computing. R Foundation for
Statistical Computing, Vienna, Austria. https://www.r-project.org/
Rahman, M. M., Al-Zahrani, B., Shahbaz, S. H., & Shahbaz, M. Q. 2019. Cubic Transmuted Uniform
Distribution: An Alternative to Beta and Kumaraswamy Distributions. European Journal of Pure
and Applied Mathematics, 12(3), 1106–1121.
Sellers, K. F., & Raim, A. 2016. A flexible zero-inflated model to address data dispersion. Computational
Statistics and Data Analysis, 99, 68–80.
Shahmandi, M., Wilson, P., & Thelwall. 2020. A New Algorithm for Zero-Modified Models Applied to
Citation Counts. Scientometrics, 125(2), 993–1010.
Shanker, R., & Hagos, F. 2015. On Poisson-Lindley distribution and its Applications to Biological
Sciences. Biometrics and Biostatistics International Journal, 2(5), 1–5.
Shaw, W. T., & Buckley, I. R. C. 2007. The alchemy of probability distributions: beyond Gram-Charlier
expansions, and a skew-kurtotic-normal distribution from a rank transmutation map. Research
Report.
Tajuddin, M. R. R., & Ismail, N. 2022. Frequentist and Bayesian Zero-Inflated Regression Models on
Insurance Claim Frequency: A comparison study using Malaysia’s Motor Insurance Data.
Malaysian Journal of Science, 41(2), 16–29.
Wagh, Y. S., & Kamalja, K. K. 2017. Comparison of methods of estimation for parameters of generalized
Poisson distribution through simulation study. Communications in Statistics: Simulation and
Computation, 46(5), 4098–4112.
Zamani, H., Ismail, N., & Faroughi, P. 2014. Poisson-Weighted Exponential Univariate Version and
Regression Model with Applications. Journal of Mathematics and Statistics, 10(2), 148–154
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