Abstract
In this paper, by including a trigonometric function, we propose a family of heavy-tailed distribution called the arcsine Kumaraswamy generalized-X family of distributions. Based on the proposed approach, a four-parameter extension of the Lomax distribution called the arcsine Kumaraswamy generalized Lomax (ASKUG-LOMAX) distribution is discussed in detail. Maximum likelihood, bootstrap, and Bayesian estimation are used to estimate the model parameters. A simulation study is used to evaluate ASKUG-LOMAX performance. The flexibility and usefulness of the proposed ASKUG-LOMAX distribution to predict unique symmetric and asymmetric patterns is demonstrated by analyzing real data. The results show that the ASKUG-LOMAX model is a good candidate for analyzing claims based on heavy-tailed data.
1. Introduction
In the applied sciences, distributions that are heavy-tailed play an important role in modeling data, especially in the fields of industry, economics, finance, banking, and risk management. However, the quality of these methods depends primarily on the assumed probability model of the phenomenon under consideration. In the applied fields, industrial datasets are generally positive [], right-skewed [], unimodal [], and with strong outliers []. Right-skewed data can be adequately modeled by skewed distributions []. Therefore, a set of unimodal positively-skewed parametric distributions can be used to model such datasets [,]. When the CDF is an underlined distribution, the heavy-tailed models are those with probabilities at the right tail that are heavier than the exponential one (see Beirlant et al. [] and Resnick []), satisfying
Dutta and Perry [] conducted an empirical analysis of loss distributions to estimate risk using various approaches. They rejected the use of the exponential, gamma, and Weibull models because of their poor results, and concluded that the best choice would be to use a model that is flexible enough in its structure. These results encouraged researchers to propose new flexible models that provide greater accuracy in data fitting. To overcome the problems associated with modeling based on classical distributions, new families of distributions (see [,,,,,,]) have been introduced.
In this context, Mudholkar and Srivastava [] have discussed the exponential family via the composition of a shape parameter to obtain a more adaptive extension of the basic model. The CDF of the random variable X over the exponential family is provided by
where is the CDF of the underline distribution, is the parameter vector, and is a shape parameter. Moreover, the Kumaraswamy generalised (KUG) family is a more adaptive family proposed by Cordeiro and de Castro [] via the composition of two shape parameters. The CDF of the random variable X over the KUG family is provided by
The arcsine exponentiated X family of distributions appears here as a special case when . It is based on Equation (3); for a contribution, see Mead and Afify []. With respect to Equation (3), the KUG expansion of the current distributions has been discussed in the literature; see Mansour et al. [] and Ahmad et al. []. The arcsine KUG-X family is denoted by (ASKUG-X). The CDF of the random variable X over ASKUG is
where and are two additional shape parameters. The main motivations for using the ASKUG-X family in practice are the following:
- (i)
- To develop the flexibility and properties of basic models.
- (ii)
- A suitable procedure for adding two extra parameters in expanded models with potent outliers, which is very useful when modeling industrial data (see Section 5).
- (iii)
- To introduce the extended version of a basic model with closed forms for the cdf and hazard rate function, the special submodels of this family can be used in the analysis of censored data sets.
- (iv)
- Compared to existing competing models, the special cases of the ASKUG-X approach are able to model data sets with high-tailed content in factorial habits.
The respective PDF, survival function (SF), and hazard rate function (HRF) of the random variable X via ASKUG are provided as follows:
and
where and Via the new CDF of ASKUG-X, many new heavy-tailed flexible models can be obtained. A number of newly contributed models based on the ASKUG-X approach are presented in Table 1.

Table 1.
New submodules via the ASKUG-X family.
The rest of this paper is outlined as follows. Section 2 defines ASKUG-LOMAX. Section 3 provides the maximum likelihood estimators (MLEs). Section 4 provides the Bayesian estimators. Section 5 presents a discussion of our simulations and analyzes two examples of real data to illustrate the proposed ASKUG-LOMAX potentiality. Section 6 provides a brief conclusion.
2. ASKUG-Lomax Distribution
In this section, we introduce the ASKUG-Lomax distribution and examine the attitude of its PDF and HRF. First, we assess the CDF of the Lomax distribution, . Then, a random variable X is said to follow the ASKUG-Lomax distribution if its CDF takes the following form:
The respective PDF, SF, and HRF of the ASKUG-Lomax distribution are provided by
and
The ASKUG-Lomax model reduces to the AS-exponentiate-Lomax distribution when , and to the AS-Lomax distribution when .
2.1. Quantile Function
Let X be the ASKUG-Lomax random variable with the PDF from Equation (5); then, the quantile function of X, i.e., Q(u), reduces to
where u has a uniform distribution on the interval (0, 1). From the expression in Equation (12), it is evident that the ASKUG-Lomax family has a closed form solution of its quantile function.
2.2. Moments
Moments are very important in statistical analysis, and play an essential role. They help to capture important features and properties of the distribution (e.g., its central tendency, dispersion, skewness, and kurtosis). The moment of the ASKUG-X family is
Substituting Equation (5) into Equation (12), we obtain
Using the binomial expansion, we have
By replacing u with in Equation (14), we obtain
where
The moment-generating function of the ASKUG-X class has the following form:
3. Maximum Likelihood Estimation
Take the observed values of , which is a random sample from the ASKUG-X model; then, the ASKUG-X log-likelihood is
The MLE can be derived by maximizing Equation (12) (see Appendix A).
The asymptotic CIs of and can be computed. The variance–covariance matrix is provided by
The respective two-sided approximate CIs for and are provided by
where and are provided by the diagonal elements of , and is the upper percentile of a standard normal distribution.
Next, to obtain the bootstrap CI boot-p for the unknown parameters ), we apply the following algorithm, Algorithm 1;
Algorithm 1 Boot-p interval algorithm: |
|
4. Bayesian Estimation
Suppose that , and are random variables that follow the prior PDFs Gamma, Gamma, Gamma, and Gamma, respectively, where and are positive constants and . The posterior DF of and the data under the Gamma priors can take the forms
and
Then, the posterior density of and the data can be extracted as
where and J is the normalizing constant.
MCMC Method
We use the Metropolis Hastings (M-H) procedure as follows:
- Set initial values and . Then, simulate a sample of size n from ASKUG-Lomax, next set .
- Simulate and using the proposal distributions , , , and .
- Obtain the probability .
- Simulate U from Uniform (0, 1).
- If , then .If , then .
- Set .
- Iterate Steps 2–6, M repetitions, and obtain and for .
Now, we use the squared error loss function provided by where is an estimate of the unknown parameter , which against the SE loss function is the posterior mean. Using the generated random samples from the above Gibbs sampling technique and with N the nburn, the Bayes estimator of , say, , can be obtained as
The second loss function is the LINEX loss function, provided by
The approximate Bayes estimate of under the LE loss function based on the Gibbs sampling technique becomes
Finally, the general entropy (GE) loss function is provided by
The approximate Bayes estimate of the parameters is provided by
MCMC HPD-credible interval algorithm:
- Sort , and in rising values.
- The lower bounds of , and in the rank .
- The lower bounds of , and in the rank .
- Iterate the previous steps M times. Obtain the average value of the lower and upper bounds of , and .
5. Simulation Study
We generate M = 1000 samples of size n = 25, 30, 40, 50, 60, 70, 80, 90, 100 from the ASKUG-LOMAX model via the initial parameter values and . Suppose that and are random variables that follow the prior PDFs Gamma, Gamma, Gamma, and Gamma, respectively. In this simulation study, we empirically obtain the bias and expected risk (ER) of the MLEs and the Bayesian methods for different parameter combinations and each sample. The point estimations of the parameters are obtained using 200 burns MCMC methods. Two LINEX loss function are used, when and when The respective biases and ERs are provided by
and
Coverage probabilities (CPs) are calculated at the 95% and 90% HPD credible intervals. The simulation results are presented in Table 2, Table 3, Table 4 and Table 5 for the parameters , and , respectively. Based on the generated data, it can be seen that

Table 2.
Point and Interval estimation of the parameter when .

Table 3.
Point and Interval estimation of the parameter when .

Table 4.
Point and Interval estimation of the parameter when .

Table 5.
Point and Interval estimation of the parameter when .
- The MLEs seem to behave as expected, i.e., the MSE values and the estimated biases decrease as n increases. Moreover, the mean values of the estimates tend to the true values as n increases, showing the consistency property of the MLEs.
- It is well known that the Bayesian estimation method provides better results in practice than the classical one, especially when the sample size is relatively small, which is exactly what the results show. The standard deviation of the MLE is greater than the Bayesian estimate for .
- The interval width of the MLE for a given confidence level is greater than the Bayesian estimate in most cases.
- A General Entropy Loss Function is a suitable alternative to the Modified LINEX loss function. The approximate Bayes estimate of the parameters based on the general entropy loss function provides better results in most cases.
Application of the ASKUG-Lomax Model
We evaluate the usefulness of the ASKUG-Lomax model by analyzing two examples of lifetime data.
Example 1.
Industry lifetime data taken from []. The observations represent the reliability time of a coating machine: 1.00, 1.00, 5.00, 5.50, 12.50, 16.75, 17.75, 20.75, 22.50, 22.75, 25.00, 25.00, 27.25, 30.25, 43.75, 45.00, 48.00, 48.25, 97.50, 99.75, 136.75, 143.50, 207.75, 215.00, 225.50, 235.00, 283.50, 567.00, 970.50.
Example 2.
Business data provided by Nigm and Hamdy [] and Wong []: 1.01, 1.05, 1.08, 1.14, 1.28, 1.30, 1.33, 1.43, 1.59, 1.62.
After analyzing the data, the estimate results are
Example 3.
The average per capita carbon dioxide emissions (in metric tons) in the Arab world, provided by []: 0.609268, 0.662618, 0.727117, 0.853116, 0.972381, 1.13867, 1.252, 1.31609, 1.45773, 1.76705, 1.79794, 1.99733, 2.11931, 2.19399, 2.2808, 2.40051, 2.5815, 2.64469, 2.71073, 2.75889, 2.80602, 2.86087, 2.87078, 2.91291, 2.9245, 2.95929, 2.97, 3.04479, 3.08222, 3.0918, 3.1313, 3.16119, 3.16314, 3.1633, 3.16669, 3.18374, 3.19605, 3.21329, 3.2388, 3.27108, 3.2779, 3.28251, 3.34723, 3.36173, 3.47424, 3.7043, 3.80165, 3.88314, 4.09354, 4.19327, 4.30874, 4.322, 4.43872, 4.49519, 4.51219, 4.52835, 4.57031, 4.60019, 4.61796.
The data are presented in Table 6, Table 7 and Table 8 for the parameters , and , respectively. Table 9, Table 10 and Table 11 compare the ASKUG-Lomax distribution via several recognition criterion: the Akaike information criterion (AIC), Bayesian information criterion (BIC), Hannan–Quinn information criterion (HQIC), and consistent Akaike information Criterion (CAIC). The goodness-of-fit results of the ASKUG-Lomax model are compared with several other models in Table 1, including Arcsine exponentiated Lomax (ASEXG-Lomax), Arcsine Lomax (AS-Lomax), Exponentiated Weibull (EX-Weibull), and Weibull distribution.

Table 6.
Point and Interval estimation of industry data.

Table 7.
Point and Interval estimation of business data.

Table 8.
Point and Interval estimation of carbon dioxide emissions data.

Table 9.
Relative quality of ASKUG-Lomax on the industry lifetime data vs. competing models.

Table 10.
Relative quality of ASKUG-Lomax on the business data vs. competing models.

Table 11.
Relative quality of ASKUG-Lomax on the carbon dioxide emissions data vs. competing models.
The results in Table 9 indicate that the ASKUG-Lomax distribution provides better fits than the alternative models, and represents an adequate model for analyzing heavy-tailed industry claims data. In addition, the results in Table 10 indicate that the ASKUG-Lomax distribution provides better fits than other competing models, and is adequate for analyzing business data. The results in Table 11 again indicate that the ASKUG-Lomax distribution provides better fits than the other competing models, and is adequate for analyzing carbon dioxide emissions data.
6. Concluding Remarks
The use of the trigonometric arcsine function introduces a new family of heavy-tailed models, the Arcsine Kumaraswamy family of generalised X-distributions. The Arcsine Kumaraswamy-generalised X-distribution is very interesting, and provides a better fit for data with strong tails. A special submodel called the Arcsine Kumaraswamy-Lomax model is defined is this paper. We calculate the parameters of the Arcsine Kumaraswamy-generalised Lomax model with maximum likelihood, bootstrap, and Bayesian estimators. A simulation study and analysis of real industry data are provided to verify the performance of the Arcsine Kumaraswamy-Lomax model. The performance of the Bayesian estimators is better than that of the corresponding ML estimators. Based on our modeling of three real datasets, the results show that the Arcsine Kumaraswamy-generalised Lomax model provides a better fit than other competing models.
Author Contributions
Data curation, W.E.; Funding acquisition, Y.T.; Investigation, W.E.; Methodology, W.E.; Resources, W.E.; Supervision, W.E.; Writing—review & editing, Y.T. All authors have read and agreed to the published version of the manuscript.
Funding
This study was funded by Researchers Supporting Project number (RSP2022R488), King Saud University, Riyadh, Saudi Arabia.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
All the datasets used in this paper are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflict of interest.
Appendix A
The ASKUG-Lomax log-likelihood function is
References
- Klugman, S.A.; Panjer, H.H.; Willmot, G.E. Loss Models: From Data to Decisions; John Wiley & Sons: Hoboken, NJ, USA, 2012; Volume 715. [Google Scholar]
- Lane, M.N. Pricing risk transfer transactions. ASTIN Bull. 2000, 30, 259–293. [Google Scholar] [CrossRef]
- Cooray, K.; Ananda, M. Modeling actuarial data with a composite lognormal-pareto model. Scand. Actuar. J. 2005, 2005, 321–334. [Google Scholar] [CrossRef]
- Ibragimov, R.; Prokhorov, A. Heavy tails and copulas: Topics in dependence modelling in economics and finance. Quant. Financ. 2019, 19, 13–14. [Google Scholar]
- Bernardi, M.; Maruotti, A.; Petrella, L. Skew mixture models for loss distributions: A bayesian approach. Insur. Math. Econ. 2012, 51, 617–623. [Google Scholar] [CrossRef]
- Adcock, C.; Eling, M.; Loperfido, N. Skewed distributions in finance and actuarial science: A review. Le Eur. J. Financ. 2015, 21, 1253–1281. [Google Scholar] [CrossRef]
- Bhati, D.; Ravi, S. On generalized log-moyal distribution: A new heavy tailed size distribution. Insur. Math. Econ. 2018, 79, 247–259. [Google Scholar] [CrossRef]
- Beirlant, J.; Matthys, G.; Dierckx, G. Heavy-tailed distributions and rating. ASTIN Bull. 2001, 31, 37–58. [Google Scholar] [CrossRef]
- Resnick, S.I. Discussion of the Danish data on large fire insurance losses. ASTIN Bull. 1997, 27, 139–151. [Google Scholar] [CrossRef]
- Dutta, K.; Perry, J. A tale of tails: An empirical analysis of loss distribution models for estimating operational risk capital. SSRN Electron. J. 2006, 1, 6–13. [Google Scholar] [CrossRef]
- Afify, A.Z.; Cordeiro, G.M.; Maed, M.E.; Alizadeh, M.; Al-Mofleh, H.; Nofal, Z.M. The generalized odd Lindley-G family: Properties and applications. An. Acad. Bras. Ciˆencias 2019, 91, 1–22. [Google Scholar] [CrossRef]
- Afify, A.Z.; Alizadeh, M. The odd Dagum family of distributions: Properties and applications. J. Appl. Probab. Stat. 2020, 15, 45–72. [Google Scholar]
- Nasir, A.; Yousof, H.M.; Jamal, F.; Korkmaz, M.Ç. The exponentiated Burr XII power series distribution: Properties and applications. Stats 2019, 2, 15–31. [Google Scholar] [CrossRef]
- Jamal, F.; Nasir, M. Some new members of the TX family of distributions. In Proceedings of the 17th International Conference on Statistical Sciences, Lahore, Pakistan, 21–23 January 2019; Volume 17. [Google Scholar]
- Al-Mofleh, H. On generating a new family of distributions using the tangent function. Pak. J. Stat. Oper. Res. 2018, 14, 471–499. [Google Scholar] [CrossRef]
- Afify, A.Z.; Cordeiro, G.M.; Butt, N.S.; Ortega, E.M.M.; Suzuki, A.K. A new lifetime model with variable shapes for the hazard rate. Braz. J. Probab. Stat. 2017, 31, 516–541. [Google Scholar] [CrossRef]
- Cordeiro, G.M.; Afify, A.Z.; Ortega, E.M.M.; Suzuki, A.K.; Mead, M.E. The odd Lomax generator of distributions: Properties, estimation and applications. J. Comput. Appl. Math. 2019, 347, 222–237. [Google Scholar] [CrossRef]
- Mudholkar, G.S.; Srivastava, D.K. Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE Trans. Reliab. 1993, 42, 299–302. [Google Scholar] [CrossRef]
- Cordeiro, G.M.; de Castro, M. A new family of generalized distributions. J. Stat. Comput. Simul. 2011, 81, 883–898. [Google Scholar] [CrossRef]
- Mead, M.E.; Afify, A.Z. On five-parameter Burr XII distribution: Properties and applications. S. Afr. Stat. J. 2017, 51, 67–80. [Google Scholar] [CrossRef]
- Mansour, M.M.; Aryal, G.; Afify, A.Z.; Ahmad, M. The Kumaraswamy exponentiated Fréchet distribution. Pak. J. Stat. 2018, 34, 177–193. [Google Scholar]
- Ahmad, Z.; Hamedani, G.G.; Butt, N.S. Recent developments in distribution theory: A brief survey and some new generalized classes of distributions. Pak. J. Stat. Oper. Res. 2019, 15, 87–110. [Google Scholar] [CrossRef]
- AL-Jamal, Z.Y. Exponentiated exponential distribution as a failure time distribution. Iraqi J. Stat. Sci. 2008, 8, 63–75. [Google Scholar] [CrossRef]
- Nigm, A.M.; Hamdy, H.I. Bayesian prediction bounds for the Pareto lifetime model. Commun. Stat. Theory Methods 1987, 16, 1761–1772. [Google Scholar] [CrossRef]
- Wong, A. Approximate studentization for Pareto distribution with application to censored data. Stat. Pap. 1998, 39, 189–201. [Google Scholar] [CrossRef]
- Available online: https://databank.worldbank.org/reports.aspx?source=world-development-indicators (accessed on 20 October 2022).
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