MOMENTS OF ORDER STATISTICS AND K-RECORD VALUES ARISING FROM THE BETA-LOMAX DISTRIBUTION WITH APPLICATION

Ashwaq Abdulsada Kadhim Alakraa; Ali Akbar Heydari; Hossein Jabbari Khamnei; Somayeh Makouei

MOMENTS OF ORDER STATISTICS AND K-RECORD VALUES ARISING FROM THE BETA-LOMAX DISTRIBUTION WITH APPLICATION

Abstract

The concept of $k$-record values plays a significant role in extreme value theory, which is essential for modeling rare events. In this paper, we focus on $k$-record values from the Beta-Lomax distribution($BLD$), a flexible probability distribution widely used to model extreme events in various fields. We begin by introducing the $BLD$, highlighting its key properties and characteristics. We then define the notion of $k$-record values, which represent the maximum of $k$ consecutive observations from a given dataset. The $k$-record values provide valuable insights into the extreme behavior of the data and are particularly useful for estimating tail probabilities and quantiles. Next, we discuss the statistical properties of $k$-record values from the $BLD$, including their moments, distributional properties, and asymptotic behavior. We explore various methodologies for estimating the parameters of the $BLD$ using maximum likelihood estimation ($MLE$) and discuss strategies for validating the goodness-of-fit of the resulting models. Furthermore, we present applications of $k$-record values from the $BLD$ in real-world scenarios. These include assessing the risk associated with extreme events, such as natural disasters or financial market crashes, and making informed decisions regarding prevention, mitigation, or insurance coverage. Finally, we conclude by summarizing the key findings and contributions of this study. The analysis of $k$-record values from the $BLD$ provides valuable insights into extreme event modeling and enhances our understanding of rare occurrences. The results presented in this paper can help practitioners in diverse fields accurately assess and manage risks associated with extreme events, leading to more robust decision-making processes.

Keywords

References

Abdul Sathar, E., Jose, J., (2023), Extropy based on records for random variables representing residual life, Communications in Statistics-Simulation and Computation, 52(1), pp. 196–206.
Ahmadi, J., Balakrishnan, N., (2010), Prediction of order statistics and record values from two independent sequences, Statistics, 44(4), pp. 417–430.
Ahsanullah, M., Aliev, F., (2008), Some characterizations of exponential distribution by record values, Journal of Statistical Research, 42(2), pp. 41–46.
Ahsanullah, M., Holland, B., (1987), Distributional properties of record values from the geometric distribution, Statistica neerlandica, 41(2), pp. 129–137.
Alawady, M., Barakat, H., Mansour, G., Husseiny, I., (2023), Information measures and concomitants of k-record values based on sarmanov family of bivariate distributions, Bulletin of the Malaysian Mathematical Sciences Society, 46(1), p. 9.
Alsanea, M. A. S., (2024), The role of record values in statistical inference: A review article, Reliability: Theory & Applications, 19(1), pp. 406–423.
Arnold, B., Balakrishnan, N., Nagaraja, H., (1998), Records, John Wiley and Sons, New York.
Arnold, B. C., Balakrishnan, N., Nagaraja, H. N., (1992), A first course in order statistics, SIAM, 54.

Asgharzadeh, A., Bagheri, S., Ibrahim, N., Abubakar, M., (2020), Optimal confidence regions for the two-parameter exponential distribution based on records, Computational Statistics, 35, pp. 309–326.
Balakrishnan, N., Cohen, A. C., (1992), Order statistics and inference estimation methods, Journal of the Royal Statistical Society: Series A (Statistics in Society), 155(2), p. 307.
[11] Bansal, S., Gupta, N., (2022), Weighted extropies and past extropy of order statistics and k-record values, Communications in Statistics-Theory and Methods, 51(17), pp. 6091–6108.
Barlow, R., Toland, R., Freeman, T., (1984), A bayesian analysis of stress-rupture life of kevlar 49/epoxy spherical pressure vessels, Proc. Conference on Applications of Statistics, Marcel Dekker, New York.
Chandler, K., (1952), The distribution and frequency of record values, Journal of the Royal Statistical Society: Series B (Methodological), 14(2), pp. 220–228.
David, H. A., Nagaraja, H. N., (2004), Order statistics, Encyclopedia of Statistical Sciences.
Dziubdziela, W., Kopociński, B., (1976), Limiting properties of the k-th record values, Applicationes Mathematicae, 2(15), pp. 187–190.
Eugene, N., Lee, C., Famoye, F., (2002), Beta-normal distribution and its applications, Communications in Statistics-Theory and methods, 31(4), pp. 497–512.
Gradshteyn, I. S., Ryzhik, I. M., (1994), Table of integrals, series, and products, Academic press.
Jafari, A. A., Bafekri, S., (2021), Inferences on the performance index of weibull distribution based on k-record values, Journal of Computational and Applied Mathematics, 382, p. 113060.
Jaheen, Z. F., (2004), Empirical bayes inference for generalized exponential distribution based on records, Communications in Statistics-Theory and Methods, 33(8), pp. 1851–1861.
Johnson, N. L., Kotz, S., Balakrishnan, N., (1970), Continuous univariate distributions, Houghton Mifflin Boston, 1.
Jose, J., Sathar, E. A., (2019), Residual extropy of k-record values, Statistics & Probability Letters, 146, pp. 1–6.
Khan, R., Kulshrestha, A., Khan, M., (2015), Relations for moments of k-th record values from exponential-weibull lifetime distribution and a characterization, Journal of the Egyptian Mathematical Society, 23(3), pp. 558–562.
Makouei, R., Khamnei, H. J., Salehi, M., (2021), Moments of order statistics and k-record values arising from the complementary beta distribution with application, Journal of Computational and Applied Mathematics, 390, p. 113386.
Salehi, M., Ahmadi, J., Dey, S., (2016), Comparison of two sampling schemes for generating recordbreaking data from the proportional hazard rate models, Communications in Statistics-Theory and Methods, 45(13), pp. 3721–3733.
Vidović, Z., (2019), Bayesian prediction of order statistics based on k-record values from a generalized exponential distribution, Stats, 2(4), pp. 447-456.

Details

Primary Language

English

Subjects

Probability Theory, Statistics (Other)

Journal Section

Research Article

Authors

Ashwaq Abdulsada Kadhim Alakraa This is me
0000-0001-9396-0980
Iraq

Ali Akbar Heydari ^* This is me
0000-0002-1236-3189
Iran

Hossein Jabbari Khamnei ^*
0000-0003-0902-0967
Iran

Somayeh Makouei This is me
0000-0001-7490-4422
Iran

Publication Date

January 8, 2026

Submission Date

December 11, 2024

Acceptance Date

August 13, 2025

Published in Issue

Year 2026 Volume: 16 Number: 1

IZ

https://izlik.org/JA48NN97LG

Cite

RIS / Bibtex

APA

Alakraa, A. A. K., Heydari, A. A., Jabbari Khamnei, H., & Makouei, S. (2026). MOMENTS OF ORDER STATISTICS AND K-RECORD VALUES ARISING FROM THE BETA-LOMAX DISTRIBUTION WITH APPLICATION. TWMS Journal of Applied and Engineering Mathematics, 16(1), 73-93. https://izlik.org/JA48NN97LG

AMA

1.Alakraa AAK, Heydari AA, Jabbari Khamnei H, Makouei S. MOMENTS OF ORDER STATISTICS AND K-RECORD VALUES ARISING FROM THE BETA-LOMAX DISTRIBUTION WITH APPLICATION. JAEM. 2026;16(1):73-93. https://izlik.org/JA48NN97LG

Chicago

Alakraa, Ashwaq Abdulsada Kadhim, Ali Akbar Heydari, Hossein Jabbari Khamnei, and Somayeh Makouei. 2026. “MOMENTS OF ORDER STATISTICS AND K-RECORD VALUES ARISING FROM THE BETA-LOMAX DISTRIBUTION WITH APPLICATION”. TWMS Journal of Applied and Engineering Mathematics 16 (1): 73-93. https://izlik.org/JA48NN97LG.

EndNote

Alakraa AAK, Heydari AA, Jabbari Khamnei H, Makouei S (January 1, 2026) MOMENTS OF ORDER STATISTICS AND K-RECORD VALUES ARISING FROM THE BETA-LOMAX DISTRIBUTION WITH APPLICATION. TWMS Journal of Applied and Engineering Mathematics 16 1 73–93.

IEEE

[1]A. A. K. Alakraa, A. A. Heydari, H. Jabbari Khamnei, and S. Makouei, “MOMENTS OF ORDER STATISTICS AND K-RECORD VALUES ARISING FROM THE BETA-LOMAX DISTRIBUTION WITH APPLICATION”, JAEM, vol. 16, no. 1, pp. 73–93, Jan. 2026, [Online]. Available: https://izlik.org/JA48NN97LG

ISNAD

Alakraa, Ashwaq Abdulsada Kadhim - Heydari, Ali Akbar - Jabbari Khamnei, Hossein - Makouei, Somayeh. “MOMENTS OF ORDER STATISTICS AND K-RECORD VALUES ARISING FROM THE BETA-LOMAX DISTRIBUTION WITH APPLICATION”. TWMS Journal of Applied and Engineering Mathematics 16/1 (January 1, 2026): 73-93. https://izlik.org/JA48NN97LG.

JAMA

1.Alakraa AAK, Heydari AA, Jabbari Khamnei H, Makouei S. MOMENTS OF ORDER STATISTICS AND K-RECORD VALUES ARISING FROM THE BETA-LOMAX DISTRIBUTION WITH APPLICATION. JAEM. 2026;16:73–93.

MLA

Alakraa, Ashwaq Abdulsada Kadhim, et al. “MOMENTS OF ORDER STATISTICS AND K-RECORD VALUES ARISING FROM THE BETA-LOMAX DISTRIBUTION WITH APPLICATION”. TWMS Journal of Applied and Engineering Mathematics, vol. 16, no. 1, Jan. 2026, pp. 73-93, https://izlik.org/JA48NN97LG.

Vancouver

1.Ashwaq Abdulsada Kadhim Alakraa, Ali Akbar Heydari, Hossein Jabbari Khamnei, Somayeh Makouei. MOMENTS OF ORDER STATISTICS AND K-RECORD VALUES ARISING FROM THE BETA-LOMAX DISTRIBUTION WITH APPLICATION. JAEM [Internet]. 2026 Jan. 1;16(1):73-9. Available from: https://izlik.org/JA48NN97LG