A new losses (revenues) probability model with entropy analysis, applications and case studies for value-at-risk modeling and mean of order-P analysis

: This study introduces the Inverse Burr-X Burr-XII (IBXBXII) distribution as a novel approach for handling asymmetric-bimodal claims and revenues. It explores the distribution’s statistical properties and evaluates its performance in three contexts. The analysis includes assessing entropy, highlighting the distribution’s signiﬁcance in various ﬁelds


Introduction
Insurance companies use historical data for risk assessment.Asymmetric-bimodal data analysis helps insurers understand varied risk levels and identify rare but severe events (tail risk) for better preparation.Analyzing claims data distribution improves claims management, fraud detection, and resource allocation.Reinsurance decisions, crucial for risk management, are guided by asymmetricbimodal analysis.The introduced IBXBXII model efficiently addresses bimodal and asymmetric actuarial data, extending the adaptable BXII distribution commonly used in insurance.In summary, asymmetric-bimodal insurance data analysis facilitated by the IBXBXII model is vital for accurate risk assessment, pricing, compliance, and competitiveness, enabling data-driven decisions for financial stability and long-term success.These parameters can be adjusted to fit data with different shapes and tail characteristics (see [1][2][3][4]).
The remaining parts of the paper can be structured in the following manner: Section 2 presents some properties.Section 3 gives some measures of entropy with numerical analysis.The main risk indicators are given in Section 4, the MOOP methodology for risk analysis is illustrated in Section 4. Section 5 presents a simulation study for assessing the estimation method.Section 6 offers a comparative study under four applications.Two actuarial case studies are presented in Section 7. Section 8 assesses the MOOP value at risk.Some conclusions are offered in Section 9.

Exploring flexibility and the quantile function
Figure 1 gives some plots of PDF for the IBXBXII distribution.Due to Figure 1, it is seen that the new PDF of the IBXBXII distribution can an unimodal PDF with a right tail. Figure 1 gives some plots of HRF for the IBXBXII distribution.Due to Figure 1, it is seen that the new HRF of the IBXBXII distribution can be J-HRF and upside-down HRF. Figure 1 illustrates the importance of the BXBXII distribution in modeling the real-life data sets that have upside-down HRF.
(2.6) Applying (2.4) to B κ,α 1 ,α 2 (z), Eq (2.6) can be written as where and Similarly, the CDF of the IBXBXII can also be expressed as a mixture of BXII CDFs given by where G α * 1 ,α 2 (z) is the CDF of the BXII model with parameters α * 1 and α 2 .Let W be a random variable having the BXII distribution with parameters α 1 and α 2 .Then, the n th ordinary and incomplete moments of W are, respectively, given by dt are the beta and the incomplete beta functions of the second type, respectively.So, several structural properties of the IBXBXII model can be obtained from (2.5) and those properties of the BXII distribution.The n th ordinary moment of Z is given by Then, (2.8) The n th incomplete moment, say Υ s (t), of the IBXBXII distribution is given by (2.9) Table 1 gives expected value (E(Z)) , variance (V(Z)) , skewness (S (Z)) and kurtosis (K(Z)) for the IBXBXII model.Table 2 gives E(Z), V(Z), S (Z) and K(Z) for the BXII model.From Tables 1 and 2 we note that the S IBXBXII (Z) of the IBXBXII model ∈ (-  Table 1.E(Z),V(Z), S (Z) and K(Z) for the IBXBXII model.

Measures of entropy
One of the most important measures of uncertainty is entropy.Many different types of entropy may be utilized to assess risk and reliability.

Rényi entropy
The Rényi entropy (RE) (see [28]) measure is determined using the following the following formula: where By employing the binomial expansion to the last term in the above equation, we get By applying the exponential expansion to the last term in the last equation, we have Again utilizing the binomial theory, then where where κ (α 1 ;ω, k) = (2ω + k) α 1 + ω and κ (α 1 , α 2 ;w,ω) where κ (α 1 , α 2 ;w, This approach can lead to more balanced and robust portfolios see [35,36] for mare details.

Arimoto entropy
Due to [31], the Arimoto entropy (AE) measure is determined using the following the following formula: A For more details see [32].

Tsallis entropy
The Tsallis entropy (TE) (see [31]) measure is determined using the following the following formula: T Inserting (3.3) into (3.7),then the TE of the IBXBXII distribution is For more details see [35].

Havrda-Charvat entropy
Due to [25], the Havrda and Charvat entropy (HCE) measure is determined using the following the following formula: Inserting (3.3) into (3.9),then the HCE of the IBXBXII distribution is Havrda-Charvat entropy is a valuable tool for optimizing portfolio diversification by considering not only individual assets' risk and return but also their pairwise relationships, see [33] for more details.Tables 3 and 4 mention some numerical results of R ω , A ω , T ω , and HC ω for the IBXBXII distribution at ζ=2.0.Table 3 gives some numerical results of R ω , HC ω , T ω , and A ω for the IBXBXII distribution at ω = 0.3 and 0.6.Table 4 gives some numerical results of R ω , HC ω , T ω , and A ω for the IBXBXII distribution at ω = 1.2 and 2.0.We can note that from Tables 3 and 4 when α 2 is fixed and the values of α 1 are increases then the values of R ω , A ω , T ω and HC ω are decreases.Also, when α 1 is fixed and the values of α 2 are increases then the values of R ω , A ω , T ω , and HC ω are decreases.Also when α 1 and α 2 are fixed and ω is increase then the values of R ω , A ω , T ω , and HC ω are decreases.Table 3.Some numerical results of R ω , HC ω , T ω , and A ω for the IBXBXII distribution at ω = 0.3 and 0.6.4. Some numerical results of R ω , HC ω , T ω , and A ω for the IBXBXII distribution at ω = 1.2 and 2.

Main risk indicators
In this work, we consider the main four indicators of risks called the value-at-risk (V 1 ), tail valueat-risk (V 2 ), tail variance (V 3 ) and tail mean variance (V 4 ).These indicators are used for analyzing the actuarial data sets below.Let Z denotes a random variable of losses (or gains).The value-at-risk of Z at the 100q% level, say V 1 or π (q), is the 100q% quantile of the distribution of Z under the IBXBXII distribution can be expressed as: According to [34], the indicator V 1 satisfies all coherence requirements if the distribution of losses (or distribution of gains) is restricted to the normal distribution.In most cases, the data sets that deal with insurance have a bias either toward the left or the right, and on occasion, it is bimodal.Because of this, it is inappropriate to apply the normal distribution to the process of describing insurance claims.For this purpose, the indicator V 2 may be a useful indicator in such cases, where where B π (q) α 2 ; The indicator V 2 refers to average of all V 1 at the confidence level q, which means that the indicator V 2 provides more actuarial information about the tail of the IBXBXII distribution, see [29].Due to [29,30] it can also be expressed as , where e (V 1 ) is the function of mean excess loss evaluated at the 100q%th quantile.[24] presented an explicit expressions for the indicator V 2 indicator under the multivariate normal distribution, the indicator V 2 under the IBXBXII distribution can be obtained as , where Finally, due to [26], the V 4 indicator can be estimated from According to [23], the Mean of Order-P (MOOP) methodology serves as an alternative approach in value-at-risk (VaR) analysis, offering a distinct perspective by considering the order or rank of returns.VaR is a widely used risk management statistic to quantify potential investment or portfolio losses over a specific time horizon with a certain confidence level.

Assessing the estimation method
Let Θ =(ζ, α 1 , α 2 ) be the parameter vector of our model.For determining the maximum likelihood estimations (MAXLE) of Θ, we deriving the log-likelihood function ( ζ,α 1 ,α 2 ).The goal of this section is to investigate the MAXLE's behavior, which was addressed in the previous section.The Monte Carlo analysis is used to assess the efficacy of the recommended estimate methods.R, a statistical programming language, will be used to do the calculation.The Monte Carlo simulation is done using a variety of approved estimating approaches.Under the following assumptions, the IBXBXII distribution may be used to generate a thousand random data elements: Step 1: Generated random data of the IBXBXII distribution from Eq (2.1) with α 1 , α 2 , and ζ given sample size n = 25, 50, 100, 200 and 300.
Step 2: Calculate the MLEs of α 1 , α 2 , and ζ utilizing the true value of these parameters.
Step 3: Repeating Step 1 to Step 1 number of times 5000 and saving all estimates.
Step 4: Calculating the statistical measures of performance for point and interval estimates: Mean square errors (C 1 ), lower limit (C 2 ), upper limit (C 3 ) of 90% and 95% confidence interval and average length (C 4 ).
All the results of the Monte Carlo simulation for each case for the given parameters: (ζ, α 1 , α 2 ) are reported in Tables 5-8.From these tabulated values, one can indicate that: As n increases, the C 2 and C 4 decreases.Table 5. Point and interval simulation results for the IBXBXII distribution at α 1 =0.5, α 2 =1.5 and ζ=0.5.Now, in order to illustrate how flexible the IBXBXII model is, we will provide four applications to four distinct collections of real data.These applications will highlight how the model may be applied to a variety of situations.For the four real-life economic, reliability, and medical data sets, we compare the IBXBXII distribution, with the standard BXII, MARBXII, TOLBXII, ZOBBXII, FBBXII, Beta BXII, BEXBXII, FKMBXII and KMBXII distributions.We consider the following well-known statistic tests (information criterion (INFC)): The Akaike INFC (C AI ), Bayesian INFC (C BYS ), Hannan-Quinn INFC C HQ , Consistent Akaike INFC (C CA ).The data set I (reliability data) refers to the breaking stress data (see [27].The data set II (reliability data) presents survival times of guinea pigs see [21] ).The data set III is taxes revenue data (economic data).The data set IV is called leukemia data (medical data).Plots and box plots, Quantile-Quantile "(Q-Q) plots, the total time in test (TTT)" plots, and the "Kernel density" are some of the many helpful graphical tools that are utilized.The Kernel density for each of the four different data sets is presented in Figure 4.The TTT for each of the four data sets is represented in Figure 5, which may be found below.In Figure 6, the Q-Q is presented for each of the four data sets individually.Box plots for each of the four genuine data sets are shown in Figure 7, which may be found here.The "box plot" is used to look for and identify outliers, which are defined as extreme observations (see Figures 6 and 7).The Q-Q plot, which can be found in Figure 6, is used to evaluate the "normality" of the four different real data sets.Through the utilization of the TTT tool, the initial HRF form can be investigated (see Figure 5).Exploration of the first PDF shape can be accomplished with the help of the Kernel density tool (see Figure 4).
According to Figure 4, it can be seen that the Kernel density has the shape of a bimodal distribution with a heavy tail for data set II, data set III, and data set IV respectively.The HRF is seen to be "monotonically increasing" for data set I, data set II, and data set III, and it is shown to be a bathtub HRF (U-HRF) for data set IV based on what is seen in Figure 5.It can be seen from Figure 6 that both data set II and data set III contain some values that are on the extreme end of the scale.On the other hand, neither data Set I nor data set IV have any numbers that are particularly extreme.Additionally, it can be demonstrated that the "normality" may be present for the data sets I and III. Figure 7 provides evidence that the findings presented in Figure 2 are accurate; yet, because of Figure 2 (the panel on the top right), it is clear that there is an extreme value.Figure 8's top right panel displays the estimated probability density function (EPDF) for data sets I.The EPDF for data set II can be found in the panel located in the top left corner of Figure 8.The EPDF for data set III can be found in the panel located in the bottom right corner of Figure 8.The EPDF for data set IV can be found in the panel located in the bottom right corner of Figure 8.As a result of examining Figure 8, we have reached the conclusion that the IBXBXII model offers an appropriate fitting to the histograms of all four data sets.The empirical and theoretical CDFs for data set I are presented in the panel located in the upper right-hand corner of Figure 9.The empirical and theoretical CDFs plot for data set II can be found in the panel located in the top left corner of Figure 9.The empirical and theoretical CDFs plot for data sets III may be seen in the panel located in the bottom right corner of Figure 9.The empirical and theoretical CDFs shown for data set IV can be seen in the panel located in the bottom right corner of Figure 9.The probability-probability (P-P) ratio for data set I is displayed in the panel located in the upper right corner of Figure 10.The P-P plot for data set II may be seen in the panel located in the top left corner of Figure 10.The P-P plot for data set III may be found in the panel located in the bottom right corner of Figure 10.The P-P plot for data set IV may be found in the panel located in the bottom right corner of Figure 10.The Kaplan-Meier survival analysis for data set I may be seen in the panel on the upper right of Figure 11.The Kaplan-Meier survival plot (KMSP) for data set II is displayed in the panel located in the top left corner of Figure 11.The KMSP for data set III can be seen in the panel located in the bottom right corner of Figure 11.The KMSP for data set IV can be seen in the panel located in the bottom right corner of Figure 11.As a result of examining Figure 8, we have reached the conclusion that the IBXBXII offers an appropriate fitting to all four data sets.The KMSP for data sets I is displayed in the panel located in the top right corner of Figure 9.The KMSP for data set II may be seen in the panel located in the top left corner of Figure 9.The KMSP for data set III can be seen in the panel located in the bottom right corner of Figure 9.The KMSP for data set IV can be seen in the panel located in the bottom right corner of Figure 9.As a result of looking at Figure 8, we have come to the conclusion that the IBXBXII offers an appropriate fitting to the empirical survival functions for all four data sets.

Two actuarial case studies
Risk analysis plays a crucial role in managing bimodal insurance claims data, offering assistance to insurance companies and risk managers in navigating the unique challenges presented by this distribution.Bimodal distributions in such data often stem from distinct groups of claims, such as low-frequency high-severity and high-frequency low-severity claims.The Value at Risk (VaR) indicator holds particular significance in this scenario, as it quantifies potential losses within a specified confidence interval, providing insights into adverse market movements or events.The presence of fat tails in bimodal insurance claims data suggests a heightened likelihood of extreme events, and VaR analysis aids insurers in quantifying and managing associated risks by offering valuable information on the size and frequency of extreme losses.In this Section, we consider new actuarial data sets, the first one is a new bimodal insurance claims data set and the second referred as the insurance revenue data (see https://data.world/datasets/insurance).These insurance claims data have been described graphically in Figures 12 and 13. Figure 12 gives the Cullen and Frey plot, nonparametric kernel density estimation plot, TTT, and box plots under the bimodal insurance-claims data set.Figure 13 gives the Q-Q plot plot, Scatter plot, the autocorrelation function (ACF) plot, and the partial autocorrelation function (partial ACF) plot under the the bimodal insurance-claims data set.Due to the Cullen and Frey plot, it is seen that the insurance-claims data set does not follow any of the mentioned distributions.Figure 12 (the top right panel) gives the nonparametric kernel density estimation plot under the bimodal insurance-claims data set.According to the nonparametric kernel density estimation plot it is seen that the insurance-claims data set is a bimodal data.Figure 12 (the bottom left panel) gives the TTT plot under the bimodal insurance-claims data set.Due to Figure 12 (the bottom left panel), the TTT plot under the bimodal insurance-claims data set indicates that the bimodal insurance-claims data set has an increasing HRF. Figure 12 (the bottom right panel) gives the box plot under the bimodal insurance-claims data set which indicates that the bimodal insurance-claims data set has no extreme observations.Figure 13 (the top left panel) gives the Q-Q plot under the bimodal insurance-claims data set which indicates that the bimodal insurance-claims data set has no extreme observations.Figure 13 (the top right panel) gives the scatter plot under the bimodal insurance-claims data set.Figure 13 (the bottom left panel) gives the ACF plot under the bimodal insurance-claims data set.The partial autocorrelation function (partial ACF) plot is a graphical representation used in time series analysis to understand the relationship between a time series and its lagged values while controlling for intermediate lags.Figure 13 (the bottom right panel) gives the partial ACF plot under the bimodal insurance-claims data set.The insurance insurance-revenue data have been described graphically in Figures 14 and 15. Figure 14 gives the Cullen and Frey plot, nonparametric kernel density estimation plot, TTT, and box plots under the insurance-revenue data set.Figure 15 gives the Q-Q plot plot, Scatter plot, the ACF plot, and the partial ACF plot under the the insurance-revenue data set.Due to the Cullen and Frey plot, it is seen that the insurance-revenue data set does not follow any of the mentioned distributions.Figure 14 (the top right panel) gives the nonparametric kernel density estimation plot under the insurance-revenue data set.According to the nonparametric kernel density estimation plot it is seen that the insurance revenue is semi-symmetric data.Figure 14 (the bottom left panel) gives the TTT plot under the insurance-revenue data set.Due to Figure 14 (the bottom left panel), the TTT plot under the insurance-revenue data set indicates that the insurance-revenue data set has an increasing HRF. Figure 14 (the bottom right panel) gives the box plot under the insurancerevenue data set which indicates that the insurance-revenue data set has no extreme observations.Figure 15 (the top left panel) gives the Q-Q plot under the insurance-revenue data set which indicates that the insurance-revenues data set has no extreme observations.Figure 15 (the top right panel) gives the scatter plot under the insurance-revenue data set.The primary function of the ACF plot is to visualize the serial correlation or autocorrelation in a time series.Figure 15 (the bottom left panel) gives the ACF plot under the insurance-revenue data set.Figure 15 (the bottom right panel) gives the partial ACF plot under the insurance-revenue data set.In addition to the graphical analysis previously presented, we can describe the data numerically by providing a numerical summary of the insurance data used in actuarial risk analysis.Table 13 gives a summary of the two insurance data.14 provides the risk analysis under the actuarial claims data set, where q =75%, 80%, 85%, 90%, 95%, 99% and 99.5%.Moreover, the MOOP V 1 |P = 2 are estimated under q =75%, 80%, 85%, 90%, 95%, 99% and 99.5%.Under actuarial claims data it is seen that: (1) The V 1 indicator increases as q increases, it started with 4.129296|q = 75% ended with 513.1953 − |q = 99.5%,where (2) The V 2 indicator increases as q increases, it started with 13.30721|q = 75% ended with 1052.658−|q= 99.5%,where (3) The V 3 indicator decreases as q increases, it started with 3.820954|q = 75% ended with 3.663571−|q = 95%, then the V 3 indicator increases as q increases, it started with 5.879942|q = 75% ended with 13087.82|q= 99.5%,where (4) The V 4 indicator increases as q increases, where (5) The e (V 1 ) indicator decreases as q increases, where , 85%, 90%, 95%, 99% and 99.5%.
Table 15.Risk analysis under the actuarial revenues data set.In the realm of risk assessment, the proposed distribution's performance is evaluated by comparing its distribution function to the Mean of Order-P (MOOP).This comparison is driven by the goal of assessing the distribution's effectiveness, especially in finance-related applications such as risk estimation and extreme occurrences.
In this portion, we run a numerical simulation study to evaluate the performance of the estimators of MOOP V 1 based on the QF of the of the IBXBXII model and that of the methodology of the MOOP.Specifically, we are interested in determining whether or whether the MOOP methodology produces more accurate results.We generate samples of size n (n = 50, 150, 300 and 500) from the new IBXBXII model with parameters ζ, α 1 and α 2 such that both negatively and positively skewed PDFs are obtained for assessing the estimators and their performance.Furthermore, to assess the adaptability of the suggested distribution, we generate sample data by simulating observations from the widely recognized BXII distribution, which is characterized by the following survival function with parameters α 1 and α 2 : In this scenario, we make a conscientious choice about the shape parameter in order to accomplish both a negative and a positive skewness.Because we have made this selection, we are able to test the performance of the estimators under a variety of skewness circumstances.To determine the absolute bias (ABS) and mean squared errors (MSEs) for every estimator, we run every simulation scenario several times (where N is larger than or equal to 1000 times), and then produce multiple repeats of those runs.To be more specific, we use the maximum likelihood technique for estimating purposes when dealing with the IBXBXII distribution.In spite of this, when it comes to the MOOP value at risk estimator, which is comparable to any semi-parametric estimator in extreme value statistics, the selection of the parameter k, which reflects the number of top order statistics, becomes extremely critical.This choice has an immediate and direct influence on both the bias and the variance, particularly in respect to the tail index and, as a result, the high quantile (or value at risk).In addition, for this specific estimator, an additional crucial consideration that arises regards the selection of the parameter P.
The outcomes of our simulation study are presented in Tables 16-21.Tables 16-18 correspond to the negatively skewed distributions, where we selected the value of the parameter as follows: In those cases, it is noted that the S (Z) = −0.1336004,−0.1492178, −0.5236663 respectively.While Tables 19-21 pertain to the positively skewed distributions when the samples are generated from the IBXBXII distribution, where we selected the value of the parameter as follows: In those cases, it is noted that the S (Z) =1.6074570, 0.0320100, 1.308430 respectively.The results are reported for various quantile values, specifically 95%, 95% and 99.5%, across different sample sizes.
Regarding the data that are skewed in a negative direction (see Tables 16-21).Tables 16-18 show that the MOOP estimate for V 1 performs significantly better than the suggested IBXBXII model estimator.In contrast, the MOOP estimator for indicator V1 is not superior to the suggested IBXBXII model estimate when applied to data with a positively skewed distribution.The results of the regressions and the average square errors both make it abundantly evident that the IBXBXII distribution has a strong tail to the right.This is the conclusion that can be drawn from the information presented here.In other words, the new distribution can be deemed more appropriate when the insurance data is skewed to the right or has a long tail to the right.This is because both of these characteristics indicate that the right side of the distribution is more prominent.It is important to point out that the outcomes of the simulation trials revealed that the new distribution is suitable for mathematical modeling and actuarial risk analysis in general.In a general sense, we are able to emphasize the following primary results: (1) MSE for the MOOP estimator for indicator V 1 decreases as q increases.
(2) MSE for the MOOP estimator for indicator V 1 < MSE for the IBXBXII estimator for indicator V 1 for all negative simulated data.
(3) MSE for the IBXBXII estimator for indicator V 1 decreases as q increases.
(4) ABS for the MOOP estimator for indicator V 1 < ABS for the IBXBXII estimator for indicator V 1 for all negative simulated data.
(5) MSE for the MOOP estimator for indicator V 1 > MSE for the IBXBXII estimator for indicator V 1 for all positive simulated data.
(6) ABS for the MOOP estimator for indicator V 1 < ABS for the IBXBXII estimator for indicator V 1 for all positive simulated data.
(7) For n = 500, the results of the MSE for the MOOP estimator for indicator V 1 get very close to the MSE for the IBXBXII estimator for indicator V 1 for all simulated data.
Nevertheless, regardless of whether the data are skewed to the right or the left, it is possible to assert that the IBXBXII distribution meets the criteria for a competitive distribution when applied to a sizeable sample.The fact that the bias and MSE of the estimators decreases as the sample size increases is an indication of the empirical consistency of the estimators in general.This is true for each and every circumstance that is taken into consideration.In addition to this, the IBXBXII model provides an estimator of the indicator V 1 that has both less of a bias and a smaller MSE value than other estimators.Therefore, it is possible to consider the recommended estimator to be suitable for the purpose of estimating indicator V 1 in the event that the underlying distribution is either positively or negatively skewed.This is because the effect of using the suggested estimator is that the indicator can be estimated.

Conclusions
In-depth scrutiny is conducted on the inverse Burr-X Burr-XII (IBXBXII) distribution, tailored for asymmetric-bimodal loss data.This thorough investigation delves into various parameters, encompassing skewness, kurtosis, moments, and others, aligning with the paper's objectives focused on analyzing the distinctive properties of this novel distribution.The IBXBXII distribution proves beneficial in three distinct scenarios, each serving a specific purpose.Firstly, the exploration centers on entropy investigation, evaluating four entropy models -Rényi entropy, Arimoto entropy, Tsallis entropy, and Havrda-Charvat entropy-via exhaustive analytical and numerical methods.A comparative study using the IBXBXII distribution further illustrates its utility.Secondly, the research underscores the significance of the new distribution and its applicability in mathematical, statistical, and applied modeling across diverse fields, including economics, engineering, dependability, and medicine.Through a meticulous comparison with alternative distributions commonly employed in applied modeling, the IBXBXII distribution emerges as the most suitable choice, supported by four real-world data applications demonstrating its favorable outcomes across various statistical tests.The third aspect focuses on employing the IBXBXII distribution in the examination of actuarial risks, particularly in scrutinizing probability distribution tails related to actuarial data.Case studies involving bimodal actuarial data pertinent to insurance claims and revenues are incorporated, utilizing five risk indicators to assess and calculate maximum potential losses.
Overall, the study provides a comprehensive exploration of the IBXBXII distribution, showcasing its adaptability and effectiveness in diverse analytical scenarios, including entropy analysis, field applications, and actuarial risk assessments.These indicators are used in the evaluation process.The five indicators are compared under the new model with the Mean of Order-P (MOOP V 1 |P = 2) methodology.The value at risk was numerically analyzed in each case using the five actuarial indicators and using varying levels of statistical confidence.A comprehensive simulation study is presented using samples of size n (n = 50, 150, 300 and 500|q = 95%, 99% and 99.5%) from the new IBXBXII model such that both negatively and positively skewed densities are obtained for assessing the estimators and their performance.The absolute bias and mean squared errors are used for assessing the comparison between the IBXBXII model and MOOP methodology.In relation to the data that are skewed negatively, the MOOP estimate for V 1 performs noticeably better than the proposed IBXBXII model estimator.On the other hand, when applied to data with a positively skewed distribution, the MOOP estimator for indicator V 1 is not superior than the proposed IBXBXII model estimate.The IBXBXII distribution clearly has a big tail to the right, as seen by the average square errors and the regression results.The provided information suggests that the new distribution is particularly suitable for insurance data that exhibits right skewness or has a long tail to the right.In such cases, the new distribution is considered more appropriate, as these characteristics indicate a more pronounced right side of the distribution.The results of simulation tests further support the conclusion that the new distribution is well-suited for mathematical modeling and actuarial risk analysis in general.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Figure 3 (
top left panel) gives 3D plot for the skewness of the IBXBXII distribution at ζ = 2.5. Figure 3 (top right panel) presents 3D plot for the kurtosis of the IBXBXII distribution at ζ = 2.5. Figure 3 (bottom right panel) shows 3D plot for the coefficient of variation (CV) of the IBXBXII distribution at ζ = 2.5. Figure 3 (bottom left panel) displays 3D plot for the index of dispersion (ID) of the IBXBXII distribution at ζ = 2.5.Based on Figure 3 (top left panel), we note that at ζ = 2.5 the skewness of the IBXBXII distribution can have positive and negative values.Due Figure 3 (top right pane), it is seen that at ζ = 2.5 the kurtosis of the IBXBXII distribution is always positive and less that 3.According to Figure 3 (bottom right panel), we note that the CV of the IBXBXII distribution at ζ = 2.5 can have U shape.According to Figure 3 (bottom left panel), it is concluded that the ID of the IBXBXII distribution at ζ = 2.5 ∈ (0, 1) .

Figure 12 .
Figure 12.The Cullen and Frey plot, nonparametric kernel density estimation plot, TTT and box plots under the the bimodal insurance-claims data set.

Figure 13 .
Figure 13.The Q-Q plot, Scatter plot, the ACF plot and the partial ACF plot under the bimodal insurance-claims data set.

Figure 12 (
Figure 12 (the top left panel) gives the Cullen and Frey plot under the insurance-claims data set.Due to the Cullen and Frey plot, it is seen that the insurance-claims data set does not follow any of the mentioned distributions.Figure 12 (the top right panel) gives the nonparametric kernel density estimation plot under the bimodal insurance-claims data set.According to the nonparametric kernel density estimation plot it is seen that the insurance-claims data set is a bimodal data.Figure 12 (the bottom left panel) gives the TTT plot under the bimodal insurance-claims data set.Due to Figure 12 (the bottom left panel), the TTT plot under the bimodal insurance-claims data set indicates that the bimodal insurance-claims data set has an increasing HRF. Figure 12 (the bottom right panel) gives the box plot under the bimodal insurance-claims data set which indicates that the bimodal insurance-

Figure 14 .
Figure 14.The Cullen and Frey plot, nonparametric kernel density estimation plot, TTT and box plots under the revenues data set.

Figure 15 .
Figure 15.The Q-Q plot, Scatter plot, the ACF plot and the partial ACF plot under the revenue data set.

Figure 14 (
Figure 14 (the top left panel) gives the Cullen and Frey plot under the insurance-revenue data set.Due to the Cullen and Frey plot, it is seen that the insurance-revenue data set does not follow any of the mentioned distributions.Figure14(the top right panel) gives the nonparametric kernel density estimation plot under the insurance-revenue data set.According to the nonparametric kernel density estimation plot it is seen that the insurance revenue is semi-symmetric data.Figure14(the bottom left panel) gives the TTT plot under the insurance-revenue data set.Due to Figure14(the bottom left panel), the TTT plot under the insurance-revenue data set indicates that the insurance-revenue data set has an increasing HRF.Figure14(the bottom right panel) gives the box plot under the insurancerevenue data set which indicates that the insurance-revenue data set has no extreme observations.
6. Reliability, economic, and medical data sets for comparing models

Table 12 (
the third column) shows the C AI , C BYS , C HQ and C CA , KS and PV, respectively, for the data set IV. Based on the values in Tables 9-12, the IBXBXII model has the best fits as compared to BXIIextensions in the four applications with small values of C AI , C BYS , C HQ , C CA and KS (and biggest corresponding PV) where for data set I C AI =291.101,C BYS =298.910,C HQ =294.152,C CA =291.121,KS =0.076266 and PV =0.6059.For data set II C AI =205.242,C BYS =212.102,C HQ =208.888,C CA =206.967,KS =0.10192 and PV =0.4431.For data set III C AI =381.016,C BYS =387.076,C HQ =382.985,C CA =384.667,KS =0.065129 and PV =0.9638.Finally for data set IV C AI =310.001,C BYS =315.323,C HQ =311.201,C CA =312.901,KS =0.14037 and PV =0.5339.

Table 9 .
Comparing the competing models under the data set I.

Table 10 .
Comparing the competing models under the data set II. , α 2 , α, β C AI , C BYS , C CA , C HQ , KS , PV

Table 11 .
Comparing the competing models under the data set III.

Table 13 .
Numerical description for the two actuarial data sets.