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Simulation Study on Modified Weibull Distribution for Modelling of Investment Return

Hamza Abubakar and Shamsul Rijal Muhammad Sabri

Pertanika Journal of Tropical Agricultural Science, Volume 29, Issue 4, October 2021


Keywords: Extended Weibull distribution, investment growth rate, maximum likelihood, simulated annealing

Published on: 29 October 2021

The Weibull distribution is one of the most popular statistical models extensively applied to lifetime data analysis such as survival data, reliability data, wind speed, and recently in financial data, due to itsts flexibility to adaptably imitate different families of statistical distributions. This study proposed a modified version of the two-parameter Weibull distribution by incorporating additional parameters in the internal rate of return and insurance claims data. The objective is to examine the behaviour of investment return on the assumption of the proposed model. The proposed and the existing Weibull distribution parameters have been estimated via a simulated annealing algorithm. Experimental simulations have been conducted mimicking the internal rate of return (IRR) data for both short time (small sample) and long-term investment periods (large samples). The performance of the proposed model has been compared with the existing two-parameter Weibull distribution model in terms of their R-square (R2), mean absolute error (MAE), root mean squared error (RMSE), Akaike’s information criterion (AIC), and the Kolmogorov-Smirnov test (KS). The numerical simulation revealed that the proposed model outperformed the existing two-parameter Weibull distribution model in terms of accuracy, robustness, and sensitivity. Therefore, it can be concluded that the proposed model is entirely suitable for the long-term investment period. The study will be extended using the internal rate of return real data set. Furthermore, a comparison of the various Weibull distribution parameter estimators such as metaheuristics or evolutionary algorithms based on the proposed model will be carried out.

  • Abbasi, B., Jahromi, A. H. E., Arkat, J., & Hosseinkouchack, M. (2006). Estimating the parameters of Weibull distribution using a simulated annealing algorithm. Applied Mathematics and Computation, 183(1), 85-93.

  • Abbasi, B., Niaki, S. T. A., Khalife, M. A., & Faize, Y. (2011). A hybrid variable neighborhood search and simulated annealing algorithm to estimate the three parameters of the Weibull distribution. Expert Systems with Applications, 38(1), 700-708.

  • Abubakar, H., & Danrimi, M. L. (2021). Hopfield type of artificial neural network via election algorithm as heuristic search method for random boolean ksatisfiability. International Journal of Computing and Digital System, 10(2), 660-673.

  • Abubakar, H., Rijal, S., Sabri, S. R. M., Masanawa, S. A., & Yusuf, S. (2020a). Modified election algorithm in hopfield neural network for optimal random k satisfiability representation. International Journal for Simulation and Multidisciplinary Design Optimization, 16(11), 1–13.

  • Abubakar, H., M, S. A., Yusuf, S., & Abdurrahman, Y. (2020b). Discrete artificial dragonflies algorithm in agent based modelling for exact boolean k satisfiability problem. Journal of Advances in Mathematics and Computer Science, 35(4), 115-134.

  • Abubakari, A. G., Kandza-Tadi, C. C., & Moyo, E. (2021). Modified Beta Inverse Flexible Weibull Extension Distribution. Annals of Data Science, 1-29.

  • Almazah, M. M. A., Erbayram, T., Akdoğan, Y., AL Sobhi, M. M., & Afify, A. Z. (2021). A new extended geometric distribution: Properties, regression model, and actuarial applications. Mathematics, 9(12), 1336.

  • Almetwally, E. M. (2021). Extended odd weibull inverse rayleigh distribution with application on carbon fibres. Mathematical Sciences Letters, 10(1), 5-14.

  • Alrashidi, M., Rahman, S., & Pipattanasomporn, M. (2020). Metaheuristic optimization algorithms to estimate statistical distribution parameters for characterizing wind speeds. Renewable Energy, 149, 664-681.

  • Alzaatreh, A., Lee, C., & Famoye, F. (2013). A new method for generating families of continuous distributions. Metron, 71(1), 63-79.

  • Alzaeemi, S. A., & Sathasivam, S. (2020). Artificial immune system in doing 2-satisfiability based reverse analysis method via a radial basis function neural network. Processes, 8(10), Article 1295., H., Behboodian, J., & Towhidib, M. (2013). The beta weibull-geometric distribution. Journal of Statistical Computation and Simulation, 83(1), 52-67.

  • Boonta, S., & Boonthiem, S. (2019). An approximation of minimum initial capital of investment discrete time surplus process with Weibull distribution in a reinsurance company. Journal of Applied Mathematics, 2019, Article 2191509.

  • Chauhan, S. K., & Malik, S. C. (2017). Evaluation of reliability and MTSF of a parallel system with Weibull failure laws. Journal of Reliability and Statistical Studies, 10(1), 137-148.

  • Datsiou, K. C., & Overend, M. (2018). Weibull parameter estimation and goodness-of-fit for glass strength data. Structural Safety, 73, 29-41.

  • Freitas de Andrade, C., dos Santos, L. F., Macedo, M. V. S., Rocha, P. A. C., & Gomes, F. F. (2019). Four heuristic optimization algorithms applied to wind energy: Determination of Weibull curve parameters for three Brazilian sites. International Journal of Energy and Environmental Engineering, 10, 1-12.

  • Dodge, Y. (2008). Kolmogorov–Smirnov test. In The concise encyclopedia of statistics (pp. 283-287). Springer.

  • Elmahdy, E. E., & Aboutahoun, A. W. (2013). A new approach for parameter estimation of finite Weibull mixture distributions for reliability modeling. Applied Mathematical Modelling, 37(4), 1800-1810.

  • Guedes, K. S., de Andrade, C. F., Rocha, P. A., Mangueira, R. D. S., & de Moura, E. P. (2020). Performance analysis of metaheuristic optimization algorithms in estimating the parameters of several wind speed distributions. Applied Energy, 268, Article 114952.

  • Guerra, R. R., Peña-Ramírez, F. A., & Bourguignon, M. (2020). The unit extended Weibull families of distributions and its applications. Journal of Applied Statistics, 1-19.

  • Hashmi, S., Ahsan, M., Haq, U., Muhammad, R., & Ozel, G. (2019). The Weibull-Moment Exponential Distribution: Properties, Characterizations & applications. Journal of Reliability and Statistical Studies, 12(1), 1-22.

  • Hirose, H. (2002). Maximum likelihood parameter estimation in the extended Weibull distribution and its applications to breakdown voltage estimation. IEEE Transactions on Dielectrics and Electrical Insulation, 9(4), 524–536.

  • Kaba, A., & Suzer, A. E. (2021). Metaheuristic data fitting methods to estimate Weibull parameters for wind speed data: A case study of Hasan Polatkan Airport. The Aeronautical Journal, 125(1287), 916-948.

  • Kellison, S. G. (2009). The theory of interest (3rd Ed.). McGraw-Hill Education.

  • Kirkpatrick, S., Gelatt, C. D., & Vecchi, M. P. (1983). Optimization by simulated annealing. Science, 220(4598), 671-680.

  • Lee, C., Famoye, F., & Alzaatreh, A. Y. (2013). Methods for generating families of univariate continuous distributions in the recent decades. Wiley Interdisciplinary Reviews: Computational Statistics, 5(3), 219-238.

  • Liao, Q., Ahmad, Z., Mahmoudi, E., & Hamedani, G. G. (2020). A new flexible bathtub-shaped modification of the Weibull model: Properties and applications. Mathematical Problems in Engineering, 2020, Article 3206257.

  • Okafor, E. G., Ezugwu, O. E., Jemitola, P. O., Sun, Y., & Lu, Z. (2018). Weibull parameter estimation using particle swarm optimization algorithm. International Journal of Engineering and Technology (UAE), 7(3), 7-10.

  • Okasha, H. M., & Basheer, A. M. (2020). On marshall-olkin extended inverse weibull distribution: Properties and estimation using type-II censoring data. Journal of Statistics Applications & Probability Letters, 7(1), 9-21.

  • Phani, K.K. (1987). A New Modified Weibull Distribution. Communications of the American Ceramic Society, 184(August), 182-184.

  • Pobočíková, I., Sedliačková, Z., & Michalková, M. (2018). Transmuted Weibull distribution and its applications. MATEC Web of Conferences, 157, 1-11.

  • Sabri, S. R. M., & Sarsour, W. M. (2019). Modelling on stock investment valuation for long-term strategy. Journal of Investment and Management, 8(3), 60-66.

  • Sarhan, A. M., & Zaindin, M. (2009). Modified Weibull distribution. Applied Sciences, 11(January 2000), 123-136.

  • Sarsour, W. M., & Sabri, S. R. M. (2020a). Evaluating the investment in the Malaysian construction sector in the long-run using the modified internal rate of return: A Markov chain approach. The Journal of Asian Finance, Economics, and Business, 7(8), 281–287.

  • Sarsour, W. M., & Sabri, S. R. M. (2020b). Forecasting the long-run behavior of the stock price of some selected companies in the Malaysian construction sector: A Markov chain approach. International Journal of Mathematical, Engineering and Management Sciences, 5(2), 296-308.

  • Sathasivam, S., Mansor, M., Kasihmuddin, M. S. M., & Abubakar, H. (2020). Election algorithm for random k satisfiability in the Hopfield neural network. Processes, 8(5), Article 568.

  • Tang, Y., Xie, M., Lai, C. D., & Goh, T. N. (2002). Statistical analysis of a Weibull extension model, communications in statistics. Theory and Methods, 32(5), 913-928.

  • Thomas, G. M. (1995). Weibull parameter estimation using genetic algorithms and a heuristic approach to cut-set analysis (Doctoral dissertation). Ohio University, USA.

  • Wang, M., & Elbatal, I. (2015). The modified Weibull geometric distribution. Metron, 73(3), 303-315.

  • Yonar, A., & Pehlivan, N. Y. (2020). Artificial bee colony with levy flights for parameter estimation of 3-p Weibull distribution. Iranian Journal of Science and Technology, Transactions: Science, 44, 851-864.

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