## Stochastic comparisons between the extreme claim amounts from two heterogeneous portfolios in the case of transmuted-G model.(English)Zbl 1454.91203

Summary: Let $$X_{\lambda_1},\dots,X_{\lambda_n}$$ be independent and non-negative random variables belong to the transmuted-G model and let $$Y_i=I_{p_i}X_{\lambda_i}$$, $$i=1,\dots,n$$, where $$I_{p_1},\dots,I_{p_n}$$ are independent Bernoulli random variables independent of $$X_{\lambda_i}s$$, with $$E[I_{p_i}]=p_i$$, $$i=1,\dots,n$$. In actuarial sciences, $$Y_i$$ corresponds to the claim amount in a portfolio of risks. In this article, we compare the smallest and the largest claim amounts of two sets of independent portfolios belonging to the transmuted-G model, in the sense of the usual stochastic order, hazard rate order, and dispersive order, when the variables in one set have the parameters $$\lambda_1,\dots,\lambda_n$$ and the variables in the other set have the parameters $$\lambda_1^*,\dots,\lambda_n^*$$. For illustration we apply the results to transmuted exponential and the transmuted Weibull models.

### MSC:

 91G05 Actuarial mathematics
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### References:

 [1] Al-Babtain, A. A., Transmuted exponential Pareto distribution with applications, Journal of Computational and Theoretical Nanoscience, 14, 11, 5484-90 (2017) [2] Aryal, G. R.; Tsokos, C. P., Transmuted Weibull distribution: A generalization of the Weibull probability distribution, European Journal of Pure and Applied Mathematics, 4, 2, 89-102 (2011) · Zbl 1389.62150 [3] Balakrishnan, N.; Haidari, A.; Masoumifard., K., Stochastic comparisons of series and parallel systems with generalized exponential components, IEEE Transactions on Reliability, 64, 1, 333-48 (2015) [4] Balakrishnan, N.; Zhang, Y.; Zhao., P., Ordering the largest claim amounts and ranges from two sets of heterogeneous portfolios, Scandinavian Actuarial Journal, 2018, 1, 23-41 (2018) · Zbl 1416.91153 [5] Barmalzan, G.; Najafabadi., A. T.P., On the convex transform and right-spread orders of smallest claim amounts, Insurance: Mathematics and Economics, 64, 380-84 (2015) · Zbl 1348.60022 [6] Barmalzan, G.; Najafabadi, A. T. P.; Balakrishnan., N., Stochastic comparison of aggregate claim amounts between two heterogeneous portfolios and its applications, Insurance: Mathematics and Economics, 61, 235-41 (2015) · Zbl 1314.91188 [7] Barmalzan, G.; Najafabadi, A. T. P.; Balakrishnan., N., Likelihood ratio and dispersive orders for smallest order statistics and smallest claim amounts from heterogeneous Weibull sample, Statistics and Probability Letters, 110, 1-7 (2016) · Zbl 1383.62147 [8] Barmalzan, G.; Najafabadi, A. T. P.; Balakrishnan., N., Ordering properties of the smallest and largest claim amounts in a general scale model, Scandinavian Actuarial Journal, 2017, 2, 105-24 (2017) · Zbl 1401.91096 [9] Boland, P. J., Statistical and probabilistic methods in actuarial science (2007), Boca Raton, FL: Chapman & Hall/CRC Press, Boca Raton, FL · Zbl 1124.62069 [10] Bourguignon, M.; Leão, J.; Leiva, V.; Santos-Neto, M., The transmuted Birnbaum-Saunders distribution, REVSTAT Statistical Journal, 15, 601-28 (2017) · Zbl 1380.62057 [11] Denuit, M.; Frostig., E., Heterogeneity and the need for capital in the individual model, Scandinavian Actuarial Journal, 2006, 1, 42-66 (2006) · Zbl 1142.91039 [12] Elbatal, I.; Aryal, G., Transmuted Dagum distribution with applications, Chilean Journal of Statistics, 6, 12, 31-45 (2015) · Zbl 1449.60016 [13] Frostig, E., A comparison between homogeneous and heterogeneous portfolios, Insurance: Mathematics and Economics, 29, 1, 59-71 (2001) · Zbl 1072.91026 [14] Granzotto, D. C. T.; Louzada., F., The transmuted log-logistic distribution: Modelling, inference, and an application to a polled Tabapua race time up to first calving data, Communications in Statistics-Theory and Methods, 44, 16, 3387-3402 (2015) · Zbl 1330.62367 [15] Hu, T.; Ruan., L., A note on multivariate stochastic comparisons of Bernoulli random variables, Journal of Statistical Planning and Inference, 126, 1, 281-88 (2004) · Zbl 1059.60023 [16] Iriarte, Y. A.; Astorga, J. M., Transmuted Maxwell probability distribution, Revista Integración, 32, 2, 211-21 (2014) · Zbl 1308.82034 [17] Karlin, S.; Novikoff., A., Generalized convex inequalities, Pacific Journal of Mathematics, 13, 4, 1251-79 (1963) · Zbl 0126.28102 [18] Kemaloglu, S. A.; Yilmaz., M., Transmuted two-parameter Lindley distribution, Communications in Statistics-Theory and Methods, 46, 23, 11866-79 (2017) · Zbl 1384.62044 [19] Khaledi, B. E.; Ahmadi., S. S., On stochastic comparison between aggregate claim amounts, Journal of Statistical Planning and Inference, 138, 7, 3121-29 (2008) · Zbl 1134.62071 [20] Khan, M. S.; King, R.; Hudson., I. L., Transmuted Weibull distribution: Properties and estimation, Communications in Statistics-Theory and Methods, 46, 11, 5394-5418 (2017) · Zbl 1369.90060 [21] Li, C.; Li., X., Sufficient conditions for ordering aggregate heterogeneous random claim amounts, Insurance: Mathematics and Economics, 70, 406-13 (2016) · Zbl 1370.60033 [22] Li, H.; Li, X., Stochastic orders in reliability and risk (2013), New York: Springer, New York [23] Ma, C., Convex orders for linear combinations of random variables, Journal of Statistical Planning and Inference, 84, 11-25 (2000) · Zbl 1131.60303 [24] Marshall, A. W.; Olkin, I.; Arnold, B. C., Inequalities: Theory of majorization and its applications (2011), New York: Springer, New York · Zbl 1219.26003 [25] Mirhossaini, S. M.; Dolati, A., On a new generalization of the exponential distribution, Journal of Mathematical Extension, 3, 1, 27-42 (2008) · Zbl 1263.60014 [26] Mirhossaini, S. M.; Dolati, A.; Amini., M., On a class of distributions generated by stochastic mixture of the extreme order statistics of a sample of size two, Journal of Statistical Theory and Application, 10, 455-68 (2011) [27] Müller, A.; Stoyan, D., Comparison methods for stochastic models and risks (2002), New York: John Wiley & Sons, New York · Zbl 0999.60002 [28] Nadeb, H.; Torabi, H.; Dolati., A., Stochastic comparisons of the largest claim amounts from two sets of interdependent heterogeneous portfolios, Mathematical Inequalities & Applications, 35-56 (2020) · Zbl 1460.91234 [29] Okorie, I. E.; Akpanta, A. C.; Ohakwe., J., Transmuted Erlang-truncated exponential distribution, Economic Quality Control, 31, 2, 71-84 (2016) · Zbl 1390.60063 [30] Saboor, A.; Elbatal, I.; Cordeiro., G. M., The transmuted exponentiated Weibull geometric distribution: Theory and applications, Hacettepe Journal of Mathematics and Statistics, 45, 973-87 (2016) · Zbl 1359.62056 [31] Shaked, M.; Shanthikumar., J. G., Stochastic orders (2007), New York: Springer, New York [32] Shaw, W. T., and Buckley., I. R.2009. The alchemy of probability distributions: Beyond Gram-Charlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map. arXiv preprint arXiv:0901.0434. [33] Tian, Y.; Tian, M.; Zhu., Q., Transmuted linear exponential distribution: A new generalization of the linear exponential distribution, Communications in Statistics-Simulation and Computation, 43, 10, 2661-77 (2014) · Zbl 1462.62593 [34] Zhang, Y.; Zhao., P., Comparisons on aggregate risks from two sets of heterogeneous portfolios, Insurance: Mathematics and Economics, 65, 124-135 (2015) · Zbl 1348.91194
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