A New Exponential-<i>X</i> Family: Modeling Extreme Value Data in the Finance Sector

Ahmad, Zubair; Mahmoudi, Eisa; Roozegar, Rasool; Alizadeh, Morad; Afify, Ahmed Z.

doi:https://doi.org/10.1155/2021/8759055

Mathematical Problems in Engineering

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Robust Estimation Methods in the Presence of Extreme Observations

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Research Article | Open Access

Volume 2021 | Article ID 8759055 | https://doi.org/10.1155/2021/8759055

A New Exponential-X Family: Modeling Extreme Value Data in the Finance Sector

Zubair Ahmad,¹Eisa Mahmoudi,¹Rasool Roozegar,¹Morad Alizadeh,²and Ahmed Z. Afify³

Academic Editor: Amer Al-Omari

Received22 Jul 2021

Revised04 Oct 2021

Accepted06 Oct 2021

Published21 Oct 2021

Abstract

In this paper, a family of statistical models, namely, a new exponential-X family is proposed. A subcase of the introduced family, called the new exponential-Weibull (NE-Weibull) model, is studied. The NE-Weibull model is very competent and possesses heavy-tailed properties. The maximum likelihood estimators of its parameters are derived. The consistency and efficiency of these estimators are assessed in a brief simulation study. Finally, the effectiveness of the NE-Weibull distribution is illustrated by modeling real insurance claims data. The practical analysis shows that the NE-Weibull distribution outclassed other distributions and it can be a better choice for modeling data in the finance sector.

1. Introduction

The extreme value phenomena such as financial returns and other related events can be modeled effectively by extreme value methods. The heavy-tailed (HT) distributions have been proven to be substantial in modeling HT and extreme value data. Researchers have shown a deep concern in the financial sector to study new HT distributions. Among the applicability of the statistical distributions in the applied area, the HT distributions have received much attention for modeling financial phenomena. Most of the data sets in the financial sector possesses HT behavior with long right tail (see Lane [1]; Cooray and Ananda [2]; Wang et al. [3]; Ahn et al. [4]; Jelenković and Tan [5]; Forbes and Wraith [6]; Guo [7]; Punzo et al. [8]; Bhati and Ravi [9]; Punzo [10]; Ke et al. [11]; Dos Reis et al. [12]; and Niu et al. [13]).

Recently, different copulas and new approaches of introducing HT distribution have been studied due to the importance of HT distributions in the financial sector (for example, Bladt et al. [14]; Tikhomirov [15]; Lugosi and Mendelson [16]; and Yousri et al. [17]). For more information about the usefulness of statistical distributions, one can refer to studies by Ramos et al. [18, 19]; He et al. [20]; Alfaer et al. [21]; Afify et al. [22]; and Al Mutairi et al. [23].

A statistical distribution with SF (survival function), say , is said to be a HT model, if its SF verifiesfor all . For more details, see the study by Resnick [24].

An interesting characteristic of the HT distributions is the regularly varying behavior. A statistical distribution is said to possess the regularly varying behavior, if it satisfieswhere , and it is also known as an index of regular variation.

The statistical models possessing such property are very prominent models for modeling HT phenomena in the financial sector [25]. Furthermore, actuaries are very interested in looking for new flexible HT models (see Nadarajah and Bakar [26]; and Ahmad et al. [27]).

Alzaatreh et al. [28] proposed a prominent approach called the T-X family method. Let represent the PDF (probability density function) of T, where T is a RV (random variable) belonging to and . Suppose that is a function of of a RV-X and it has the following conditions:(i)(ii) is a differentiable function as well as monotonically increasing(iii) for and for

The CDF (cumulative distribution function) of the T-X distributions iswhere fulfills the aforementioned conditions. The corresponding PDF of equation (3), say , reduces to

More details about the T-X approach can be found in the work of Ahmad et al. [29]. By implementing the T-X method, the survival distribution family [30] can be obtained via the CDF:where , representing the SF of X.

In this study, a new exponential-X (NE-X) family is proposed based on equation (3). Suppose ; then, its respective CDF is

Linking this to equation (6), the PDF isBy using and in equation (3), the CDF of the NE-X family is as follows:where is a baseline CDF with parametric space . Next, in Propositions 1 and 2, we are going to prove that the expression provided in equation (8) is a CDF.

Proposition 1. For the defined in equation (8), and .

Proof.

Proposition 2. The CDF is right continuous and differentiable.

Proof. From the results proved in Propositions 1 and 2, we arrive at the conclusion that the function defined in equation (8) is differentiable, right continuous, and a compact CDF. Moreover, taking into account the differentiability property of for all , we have the following theorem.

Theorem 1. Let and . With the condition , if and are differentiable, then the quotient is differentiable for all and

Proof. See “Proof of Quotient Rule.”
Taking into account the right continuity of , we have the following theorem.

Theorem 2. If is a right continuous function, so is .

Proof. Assume that for all . Hence, we haveSince ,for each . Therefore, is right continuous.
As we mentioned earlier that the regularly varying tail behavior (RVTB) is a very crucial property to characterize the HT distributions, now we provide the mathematical treatment of RVTB of the NE-X family.

Theorem 3. If is a regularly varying distribution, so is .

Proof. Suppose is finite for all . Then, using equation (8), we haveSince , we havefor ; thus, is a regularly varying distribution.
Linking this to equation (8), the PDF represented by isThe RV with PDF (16) is represented by -.
The SF, , and HRF (hazard rate function), , of X arerespectively. In the next section, we discuss a subcase of the NE-X family, called the new exponential-Weibull (NE-Weibull) distribution.

2. NE-Weibull Distribution

This section deals with the NE-Weibull distribution as a subcase of the NE-X family. Suppose that with CDF, , and PDF, , where . By inserting the CDF of the Weibull model in equation (8), the CDF and PDF of the NE-Weibull distribution take the forms

Some possible PDF behaviors for the NE-Weibull distribution are shown in Figure 1. From Figure 1, it is obvious that as the values of and increase, the NE-Weibull model becomes a HT distribution.

As mentioned above, different PDF shapes of the NE-Weibull model for various values of , , and are sketched in Figure 1. When , the NE-Weibull distribution behaves similar to the exponential distribution . For , the NE-Weibull distribution captures the characteristics of . The NE-Weibull model, however, has certain benefits over the Weibull distribution. For instance, it has heavier tails than the Weibull distribution and provides closer fit to data in the finance sector.

Some possible HRF behaviors for the NE-Weibull distribution are shown in Figure 2. The NE-Weibull distribution provides increasing, unimodal, decreasing, and bathtub HRF shapes.

3. Properties

Here, we provide a concise treatment of the mathematical properties of NE-X distribution. These properties include QF (quantile function), moments, SK (skewness), and KUR (kurtosis). Furthermore, different plots for the SK and KUR are also provided.

3.1. Quantile Function

Suppose X denotes the NE-X family with CDF presented in equation (8); then, the QF of NE-X distributions, , iswhere is the solution of .

3.2. Moments

Now, we introduce the moments of the NE-X distributions which can further be used to obtain other characteristics.

The moment of the NE-X family reduces to

By putting equation (16) in (20), we obtainwhere . For , we obtain the first four moments of the NE-X family.

Using the expressions of the moments, we obtain the mathematical form of the SK and KUR measures. The SK and KUR of the NE-Weibull distribution can be calculated, respectively, via the expressions

The effects of , , and on the SK, KUR, variance, and mean of the NE-Weibull distribution are displayed in Figures 3–5.

4. Estimation

Within this section, we derive the maximum likelihood estimators (MLEs) of the NE-Weibull parameters. Consider as the values of a sample from the NE-Weibull distribution with parameters and . The log-likelihood (LL) function of the NE-Weibull distribution takes the form

The partial derivatives of the LL function are

Equating and to zero and simultaneously solving them yield the MLEs of .

5. Simulation Study

Here, we implement the Monte Carlo simulation approach to address the MLE behavior in estimating the NE-Weibull parameters. The NE-Weibull distribution can be simulated by using equation (8). Let follow the standard uniform distribution; hence, the quantile function reduces to

The simulation is done for (i) , (ii) , and (iii) .

The simulation results are obtained by utilizing the software with the algorithms and “” with . The results obtained are based on replications for samples of size , where . Statistical tools such as biases and mean square errors (MSEs) are obtained as assessing tools. These tools are calculated as follows:where .

The summary measures (SMs) of the three simulated sets of data are provided in Table 1, whereas the histograms, box plots, kernel density estimator, and estimated CDF of the NE-Weibull model are provided in Figures 6–8. The simulation results are visually displayed in Figures 9–11.

6. Data Modeling in the Finance Sector

Here, we illustrate the importance of the NE-Weibull distribution in modeling insurance claims data from the financial sector. We also calculated the risk measures such as VaR (value at risk) and TVaR (tail value at risk) for this data.

6.1. Insurance Claims Data

The insurance claims data represents the initial claims of the unemployment insurances per month from 1971 to 2018 [20]. The data set can also be retrieved from https://data.worlddatany-govns8z-xewg.

The summary statistics for the insurance claims data are reported in Table 2. Corresponding to this data, the histogram, box plot, and total time test (TTT) plots are provided in Figure 12. Figure 12 shows that the insurance claims data is unimodal, right-skewed, and HT.

The NE-Weibull distribution is applied to fit the financial data, and it is compared with the Weibull, W-Loss (Weibull-Loss), NHT-Weibull (new heavy-tailed Weibull), Ku-Weibull (Kumaraswamy Weibull), and B-Weibull (beta Weibull) distributions. The CDFs of these models are as follows:(i)Weibull model:(ii)W-Loss model:(iii)NHT-Weibull model:(iv)Ku-Weibull model:(v)B-Weibull model:

To decide about the best fitting of the applied distributions, certain criteria are taken into account. These criteria are as follows:(i)The AIC is(ii)The BIC is(iii)The HQIC is(iv)The CAIC is

Here, denotes the LL function, is a parametric space, represents the model parameters, and is the number of selected samples. In addition to the criteria measures, three goodness-of-fit tests with its corresponding values are also considered. These tests are given by the following:(i)The Anderson–Darling (AD) test statistic is where is the observation in samples, calculated after sorting the data in the ascending order.(ii)The Cramer–von Mises (CM) test statistic is(iii)The Kolmogorov–Smirnov (KS) test statistic is where represents the supremum of the set of distances and indicates the empirical CDF.

The LL function is optimized, and the goodness-of-fit measures are calculated using the algorithm method “BFGS” via optim() R-function [31]. The MLEs of the fitted models with standard errors are listed in Table 3. Tables 4 and 5 show the goodness-of-fit values for the considered models.

The fitted PDF and CDF of the NE-Weibull model are provided in Figure 13, showing that the right-skewed HT data can be adequately fitted by the NE-Weibull distribution. The Kaplan–Meier survival (KMS) and PP (probability-probability) plots of the NE-Weibull distribution are sketched in Figure 14.

6.2. VaR and TVaR for Insurance Claims Data

Now, we calculate the VaR and TVaR measures for the Weibull and proposed NE-Weibull distributions based on the MLEs and the data provided in Section 6.1. The results of the VaR and TVaR of the NE-Weibull and Weibull distributions are presented in Table 6. We also display these results graphically in Figure 15.

A model having higher values for the TVaR and VaR is considered a HT distribution. The numerical results in Table 6 and their graph in Figure 15 show that the NE-Weibull is a HT model.

7. Final Comments

During the last couple of years, the study and modeling of extreme value data has gained increased interest in numerous areas, particularly in the hydrology and finance sector. Recent studies have shown the importance and potential of the HT distributions in various areas for analyzing real data sets, especially in the finance sector. In this article, a new interesting HT extension of the Weibull distribution, called the NE-Weibull distribution, is proposed using the NE-X approach. The MLEs of the NE-Weibull parameters and are obtained. To show the applicability of the NE-Weibull model, a real-life insurance claims data from the financial sector is considered. Based on certain tests, it is showed that the NE-Weibull model offers an adequate fit to the insurance claims data than the Weibull, W-Loss, NHT-Weibull, Ku-Weibull, and B-Weibull. On the other hand, the numerical results for the VaR and TVaR measures showed that the NE-Weibull model is a HT distribution as compared to the classical Weibull distribution.

Data Availability

To replicate the findings of this paper, the data is available from the corresponding author upon request.

Disclosure

This article was drafted from the PhD work of the first author (Zubair Ahmad).

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this work.

Authors’ Contributions

Conceptualization and methodology were carried out by Zubair Ahmad and Eisa Mahmoudi. The original draft was prepared by Zubair Ahmad and Morad Alizadeh. The original draft was edited by Eisa Mahmoudi, Rasool Roozegar, and Ahmed Z. Afify. Formal analysis was performed by Zubair Ahmad. Software was collected by Zubair Ahmad and Morad Alizadeh. The study was supervised by Eisa Mahmoudi, Morad Alizadeh, and Rasool Roozegar. Investigation was conducted by Zubair Ahmad. The manuscript was reviewed and edited by Zubair Ahmad, Eisa Mahmoudi, Morad Alizadeh, Rasool Roozegar, and Ahmed Z. Afify.

Acknowledgments

This work was supported by the Department of Statistics, Yazd University, Yazd, Iran.

References

M. N. Lane, “Pricing risk transfer transactions,” ASTIN Bulletin, vol. 30, no. 2, pp. 259–293, 2000.
View at: Publisher Site | Google Scholar
K. Cooray and M. Ananda, “Modeling actuarial data with a composite lognormal-Pareto model,” Scandinavian Actuarial Journal, vol. 2005, no. 5, pp. 321–334, 2005.
View at: Publisher Site | Google Scholar
H. J. Wang, D. Li, and X. He, “Estimation of high conditional quantiles for heavy-tailed distributions,” Journal of the American Statistical Association, vol. 107, no. 500, pp. 1453–1464, 2012.
View at: Publisher Site | Google Scholar
S. Ahn, J. H. T. Kim, and V. Ramaswami, “A new class of models for heavy tailed distributions in finance and insurance risk,” Insurance: Mathematics and Economics, vol. 51, no. 1, pp. 43–52, 2012.
View at: Publisher Site | Google Scholar
P. R. Jelenković and J. Tan, “Characterizing heavy-tailed distributions induced by retransmissions,” Advances in Applied Probability, vol. 45, no. 1, pp. 106–138, 2013.
View at: Google Scholar
F. Forbes and D. Wraith, “A new family of multivariate heavy-tailed distributions with variable marginal amounts of tailweight: application to robust clustering,” Statistics and Computing, vol. 24, no. 6, pp. 971–984, 2014.
View at: Publisher Site | Google Scholar
Z. Guo, “Heavy-tailed distribution and risk management of equity market tail events,” Journal of Risk & Control, vol. 4, no. 1, pp. 31–41, 2017.
View at: Google Scholar
A. Punzo, L. Bagnato, and A. Maruotti, “Compound unimodal distributions for insurance losses,” Insurance: Mathematics and Economics, vol. 81, pp. 95–107, 2018.
View at: Publisher Site | Google Scholar
D. Bhati and S. Ravi, “On generalized log-Moyal distribution: a new heavy tailed size distribution,” Insurance: Mathematics and Economics, vol. 79, pp. 247–259, 2018.
View at: Publisher Site | Google Scholar
A. Punzo, “A new look at the inverse Gaussian distribution with applications to insurance and economic data,” Journal of Applied Statistics, vol. 46, no. 7, pp. 1260–1287, 2019.
View at: Publisher Site | Google Scholar
Y. Ke, S. Minsker, Z. Ren, Q. Sun, and W. X. Zhou, “User-friendly covariance estimation for heavy-tailed distributions,” Statistical Science, vol. 34, no. 3, pp. 454–471, 2019.
View at: Publisher Site | Google Scholar
E. F. Dos Reis, A. Li, and N. Masuda, “Generative models of simultaneously heavy-tailed distributions of interevent times on nodes and edges,” Physical Review E, vol. 102, no. 5, Article ID 052303, 2020.
View at: Publisher Site | Google Scholar
H. Niu, J. Wei, and Y. Chen, “Optimal randomness for stochastic configuration network (SCN) with heavy-tailed distributions,” Entropy, vol. 23, no. 1, p. 56, 2021.
View at: Google Scholar
M. Bladt, B. F. Nielsen, and G. Samorodnitsky, “Calculation of ruin probabilities for a dense class of heavy tailed distributions,” Scandinavian Actuarial Journal, vol. 2015, no. 7, pp. 573–591, 2015.
View at: Publisher Site | Google Scholar
K. Tikhomirov, “Sample covariance matrices of heavy-tailed distributions,” International Mathematics Research Notices, vol. 2018, no. 20, pp. 6254–6289, 2018.
View at: Publisher Site | Google Scholar
G. Lugosi and S. Mendelson, “Mean estimation and regression under heavy-tailed distributions: a survey,” Foundations of Computational Mathematics, vol. 19, no. 5, pp. 1145–1190, 2019.
View at: Publisher Site | Google Scholar
D. Yousri, M. Abd Elaziz, L. Abualigah, D. Oliva, M. A. A. Al-Qaness, and A. A. Ewees, “COVID-19 X-ray images classification based on enhanced fractional-order cuckoo search optimizer using heavy-tailed distributions,” Applied Soft Computing, vol. 101, Article ID 107052, 2021.
View at: Publisher Site | Google Scholar
P. L. Ramos, D. C. Nascimento, C. Cocolo et al., “Reliability-centered maintenance: analyzing failure in harvest sugarcane machine using some generalizations of the Weibull distribution,” Modelling and Simulation in Engineering, vol. 2018, Article ID 1241856, 12 pages, 2018.
View at: Publisher Site | Google Scholar
E. Ramos, P. L. Ramos, and F. Louzada, “Posterior properties of the Weibull distribution for censored data,” Statistics & Probability Letters, vol. 166, Article ID 108873, 2020.
View at: Publisher Site | Google Scholar
W. He, Z. Ahmad, A. Z. Afify, and H. Goual, “The arcsine exponentiated-X family: validation and insurance application,” Complexity, vol. 2020, Article ID 8394815, 18 pages, 2020.
View at: Publisher Site | Google Scholar
N. M. Alfaer, A. M. Gemeay, H. M. Aljohani, and A. Z. Afify, “The extended log-logistic distribution: inference and actuarial applications,” Mathematics, vol. 9, no. 12, p. 1386, 2021.
View at: Publisher Site | Google Scholar
A. Z. Afify, G. M. Cordeiro, N. A. Ibrahim, F. Jamal, M. Elgarhy, and M. A. Nasir, “The Marshall-Olkin odd Burr III-G family: theory, estimation, and engineering applications,” IEEE Access, vol. 9, pp. 4376–4387, 2021.
View at: Publisher Site | Google Scholar
A. Al Mutairi, M. Z. Iqbal, M. Z. Arshad, B. Alnssyan, H. Al-Mofleh, and A. Z. Afify, “A new extended model with bathtub-shaped failure rate: properties, inference, simulation, and applications,” Mathematics, vol. 9, no. 17, p. 2024, 2021.
View at: Publisher Site | Google Scholar
S. I. Resnick, “Discussion of the Danish data on large fire insurance losses,” ASTIN Bulletin, vol. 27, no. 1, pp. 139–151, 1997.
View at: Publisher Site | Google Scholar
J. Beirlant, G. Matthys, and G. Dierckx, “Heavy-tailed distributions and rating,” ASTIN Bulletin, vol. 31, no. 1, pp. 37–58, 2001.
View at: Publisher Site | Google Scholar
S. Nadarajah and S. A. A. Bakar, “New composite models for the Danish fire insurance data,” Scandinavian Actuarial Journal, vol. 2014, no. 2, pp. 180–187, 2014.
View at: Publisher Site | Google Scholar
Z. Ahmad, E. Mahmoudi, M. Alizadeh, R. Roozegar, and A. Z. Afify, “The exponential TX family of distributions: properties and an application to insurance data,” Journal of Mathematics, vol. 2021, Article ID 3058170, 18 pages, 2021.
View at: Publisher Site | Google Scholar
A. Alzaatreh, C. Lee, and F. Famoye, “A new method for generating families of continuous distributions,” Metron, vol. 71, no. 1, pp. 63–79, 2013.
View at: Publisher Site | Google Scholar
Z. Ahmad, G. G. Hamedani, and N. S. Butt, “Recent developments in distribution theory: a brief survey and some new generalized classes of distributions,” Pakistan Journal of Statistics and Operation Research, vol. XV, pp. 87–110, 2019.
View at: Publisher Site | Google Scholar
F. Jamal and M. Nasir, “Some new members of the TX family of distributions,” in Proceedings of the 17th International Conference on Statistical Sciences, vol. 33, Trieste, Italy, July 2019.
View at: Google Scholar
R. C. Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, 2019, https://www.Rproject.org/ Version 3.5.2.

Copyright

Copyright © 2021 Zubair Ahmad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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