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Research article

A note on Kaliman's weak Jacobian Conjecture

  • We improve Kaliman's weak Jacobian Conjecture by the Hurwitz formula and resolution of singular curves. Furthermore, we give a more general form of Kaliman's weak Jacobian Conjecture.

    Citation: Yan Tian, Chaochao Sun. A note on Kaliman's weak Jacobian Conjecture[J]. AIMS Mathematics, 2024, 9(11): 30406-30412. doi: 10.3934/math.20241467

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  • We improve Kaliman's weak Jacobian Conjecture by the Hurwitz formula and resolution of singular curves. Furthermore, we give a more general form of Kaliman's weak Jacobian Conjecture.



    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 asymmetric-bimodal 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 BXII distribution assists insurers in setting premiums by offering insights into the potential distribution of claim amounts. In summary, the BXII distribution is a versatile tool in the insurance industry, influencing underwriting, reserving, solvency analysis, and risk transfer decisions. Its capacity to model various claim severities and capture tail risk makes it a valuable asset for assessing and managing risks associated with insurance claims. The following cumulative distribution function (CDF) is related to the BXII model:

    Fα1,α2(z)|(z0)=1(zα2+1)α1, (1.1)

    where both α2>0 and α1>0 controls the shape of the model. From Eq (1.1), if α2=1 (α1=1) we obtain the standard one-parameter Lomax (Lx) (the standard one-parameter log-logistic (LL) model). Details and many mathematical properties, applications, and more useful BXII extensions see [1,2,3,4,5,6,7,8,9]. Recently, many authors considered the extension of the BXII model such as [7] (beta BXII (B BXII)), [8] (Kumaraswamy BXII (KMBXII)) and [9] (Marshall-Olkin extended BXII (MOEBXII)), [10] (Marshall-Olkin Weibull-Burr XII), [11] (Compound class of unit Burr XII), [12] (Unit-Power Burr X), [13] (inverse exponentiated Lomax power series), [14] (Chen Burr-Hatke exponential), for more details [16,17,18,19,20]. After inverting the CDF of the type-X Burr-G (BX-G) family of [15], we substitute the CDF of the BXII model in (1.1), then the CDF of the IBXBXII model can be expressed as

    Fζ,α1,α2(z)=1(1exp{[(zα2+1)α11]2})ζ. (1.2)

    Staying in (1.2) and if ζ=1, we get the inverted Rayleigh BXII (IRBXII) model. For α2=1, we get the inverted BX Lomax (IBXLx) model. If α1=1, we get the inverted BX log-logistic (BXLL) model. For ζ=α2=1, we get the inverted Rayleigh Lomax (IRLx) model. If ζ=α1=1, we get the inverted Rayleigh log-logistic (IRLL) model. The PDF of the IBXBXII is given by

    fζ,α1,α2(z)=2ζα1α2zα11[1(zα2+1)α1]3Aζ,α1,α2(z)(zα2+1)2α11exp{[(zα2+1)α11]2}, (1.3)

    where Aζ,α1,α2(z)=(1exp{[(zα2+1)α11]2})ζ1. The hazard rate function (HRF) of the IBXBXII model can be derived directly using (1.2) and (1.3) via the following formula fζ,α1,α2(z)1Fζ,α1,α2(z). The updated distribution was applied in three ways. Firstly, it effectively evaluated entropy through four models, demonstrated via numerical comparisons. Secondly, its superior quality in various fields was highlighted by comparing it to competing distributions, particularly in applied modeling. Real-world applications were demonstrated with four datasets. Thirdly, the distribution analyzed actuarial data, assessing risks, and determining maximum insurance claims and revenue losses, showcasing its suitability for modeling and actuarial risk assessment, especially in right-skewed data. The IBXBXII model outperformed the MOOP estimator for positively skewed data but was surpassed for negatively skewed data. Regression analysis indicated a pronounced right skew, suggesting the distribution's preference for right-skewed insurance data. Computer simulations confirmed its effectiveness in mathematical modeling and actuarial risk assessments, showcasing its wide applicability against common distributions.

    ● The paper highlights the new distribution's versatility with four applications in medicine, engineering, dependability, and economics, showcasing its adaptability across diverse industries. Comparisons with established models in applied modeling reveal instances where the new distribution outperforms existing ones, providing valuable insights for practitioners. This aids in the selection of the most suitable distribution for specific needs, contributing to informed decision-making in various contexts.

    ● The study will compare the BXBXII distribution with various well-known BXII extensions, including Marshall-Olkin BXII (MARBXII), Topp-Leone BXII (TOLBXII), 5-paramters beta BXII (FBBXII), Beta BXII, beta exponentiated BXII (BEXBXII), 5-paramters Kumaraswamy BXII (FKMBXII), Zografos-Balakrishnan BXII (ZOBBXII), and Kumaraswamy modified BXII (KMBXII), in modeling veterinary medicine data (specifically, the survival times of guinea pigs). The evaluation will be based on criteria such as Akaike information, Bayesian information, Hannan-Quinn information, and Consistent Akaike information.

    ● The study focuses on modeling engineering data, specifically breaking stress data, through a comparison of the BXBXII distribution with various well-known BXII extensions. These extensions include the MARBXII distribution, TOLBXII distribution, ZOBBXII distribution, Beta BXII distribution, FKMBXII distribution. The assessment will employ criteria such as Akaike information, Bayesian information, Hannan-Quinn information, and Consistent Akaike information.

    ● In modeling econometrics data (the revenue data data), the BXBXII distribution will be compared with many well-known BXII extensions such as the standard BXII, MARBXII, TOLBXII, ZOBBXII, FBBXII, Beta BXII, BEXBXII, FKMBXII, and KMBXII distributions under the the Akaike information criteria (INFC), Bayesian INFC, Hannan-Quinn INFC, Consistent Akaike INFC.

    ● In modeling medicine data (the leukemia data), the BXBXII distribution will be compared with many well-known BXII extensions such as the standard BXII, MARBXII, TOLBXII, ZOBBXII, FBBXII, Beta BXII, BEXBXII, FKMBXII and KMBXII distributions under the Akaike INFC, Bayesian INFC, Hannan-Quinn INFC, Consistent Akaike INFC.

    ● The inclusion of two case studies, one on Value-at-Risk (VaR) modeling and the other on Mean of Order-P analysis underscores the practical relevance of the research. These case studies allow for the evaluation of the new distribution's performance in real-life scenarios, which can be invaluable for decision-makers and analysts.

    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.

    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.

    Figure 1.  Plots of PDF and HRF for the IBXBXII distribution.

    The quantile function (QF) of the new model is obtained by inverting (1.2) as Q(u;ζ,α1,α2)=F1(u;ζ,α1,α2), u(0,1). Then, the QF of the IBXBXII distribution is provided after some reductions by

    Q(u;ζ,α1,α2)=[(1+{log[1(1u)1ζ]}1α2)1α11]1α2. (2.1)

    Based (2.1), Bowley's skewness (γ1) and Moor's kurtosis (γ2) are given by

    γ1=Q(78;ζ,α1,α2)Q(58;ζ,α1,α2)+Q(38;ζ,α1,α2)Q(18;ζ,α1,α2)Q(68;ζ,α1,α2)Q(28;ζ,α1,α2), (2.2)

    and

    γ2=Q(68;ζ,α1,α2)2Q(48;ζ,α1,α2)+Q(28;ζ,α1,α2)Q(68;ζ,α1,α2)Q(28;ζ,α1,α2), (2.3)

    respectively. The plots of the γ1 and γ2 for the IBXBXII distribution is given in Figure 2 at ζ = 0.5.

    Figure 2.  Plots of the γ1 and γ2 of the IBXBXII distribution at ζ = 0.5.

    If |ν1ν2|<1 and ν3>0 is a real non-integer, the following power series holds

    (1ν1ν2)ν3=ς=0(1)ςΓ(1+ν3)ς!Γ(1+ν3ς)(ν1ν2)ς. (2.4)

    Applying (2.4) to Aζ,α1,α2(z) and inserting the expansion of Aζ,α1,α2(z) into (1.3), we get

    fζ,α1,α2(z)=2ζα1α2zα11[1(zα2+1)α1]3(zα2+1)2α11ς=0(1)ςΓ(ζ)ς!Γ(ζς)exp[(ς+1)[(zα2+1)α11]2]Bς,α1,α2(z). (2.5)

    Then, applying the power series to Bς,α1,α2(z) and inserting the expansion of Bς,α1,α2(z) into (2.5), the Eq (2.5) can be summarized as

    fζ,α1,α2(z)=2ζα1α2zα11(zα2+1)α11×ς,κ=0(1)ς+κ(ς+1)κΓ(ζ)ς!κ!Γ(ζς)[1(zα2+1)]2κ3[(zα2+1)]2κ1Bκ,α1,α2(z). (2.6)

    Applying (2.4) to Bκ,α1,α2(z), Eq (2.6) can be written as

    fζ,α1,α2(z)==0Δgα1,α2(z) (2.7)

    where

    Δ=ς,κ,j=02ζ(1)ς+κ+j+(ς+1)κΓ(ζ)Γ(2κ+2)Γ(j2(κ+1))ς!κ!j!!Γ(ζς)Γ(2κ+2j)Γ(j2(κ+1))(1+),

    and gα1,α2(z)=α1α2zα11(zα2+1)α11 is the PDF of the BXII model with parameters α1=α1(1+) and α2. Similarly, the CDF of the IBXBXII can also be expressed as a mixture of BXII CDFs given by

    Fζ,α1,α2(z)==0ΔGα1,α2(z)

    where Gα1,α2(z) is the CDF of the BXII model with parameters α1 and α2. Let W be a random variable having the BXII distribution with parameters α1 and α2. Then, the nth ordinary and incomplete moments of W are, respectively, given by μn|n<α1α2=α1B(α1nα2,nα2+1), and Υn(z)|n<α1α2=α1B(zα2;α1nα2,nα2+1), where B(a1,a2)=0ta11(1+t)(a1+a2)dt, and B(z;a1,a2)=z0ta11(1+t)(a1+a2)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 nth ordinary moment of Z is given by

    μn,Z=E(Zn)==0Δ0zngα1,α2(z)dz.

    Then,

    μn,Z=E(Zn)==0Δα1B(α1nα2,nα2+1)|n<α1α2. (2.8)

    The nth incomplete moment, say Υs(t), of the IBXBXII distribution is given by

    Υs(t)==0Δα1B(tα2;α1sα2,sα2+1)|s<α1α2. (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 (-26.0701, 51.49378) however SBXII(Z) of the BXII model (0.86282, 4.64757). The K{IBXBXII}(Z) of the IBXBXII model (-602.017 to 2957.860) however the KBXII(Z) of the BXII model (3.07, 73.80). 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).

    Table 1.  E(Z),V(Z), S(Z) and K(Z) for the IBXBXII model.
    ζ α1 α2 E(Z) V(Z) S(Z) K(Z)
    1 3 2 0.81543166 0.022730081 1.607457 8.11645
    20 0.62372778 0.000991441 0.032010 3.01156
    50 0.60097949 0.000596520 -0.149218 3.01841
    100 0.58720877 0.000432940 -0.252641 3.07572
    500 0.56232022 0.000237130 -0.423748 3.25830
    1000 0.55368533 0.000191146 -0.473076 3.15282
    2000 0.54594558 0.000157149 -0.523666 3.41765
    5000 0.53682011 0.000124283 -0.574841 3.51511
    10000 0.53060272 0.000105681 -0.607892 3.58415
    150 1 5 0.07414597 4.74804×105 -0.133600 2957.86
    2 0.27199933 0.000162344 -0.274436 3.08417
    3 0.41969961 0.000172811 -0.322432 3.13870
    4 0.52139129 0.000150478 -0.346657 3.16935
    5 0.59389776 0.000125188 -0.361268 3.18888
    3 1 1 0.8674569 0.110564801 2.297332 17.2822
    2 0.3617870 0.012992946 1.599782 8.576392
    5 0.1305731 0.001346009 1.308430 6.46725
    10 0.0631470 0.000291877 1.223986 16.9348
    25 0.0247642 4.29055×105 -5.10736 113.501
    35 0.0176226 2.15464×105 -26.0701 285.1312
    45 0.0136780 1.29135×105 30.28077 -392.66
    50 0.0123013 1.04287×105 51.49378 -602.02

     | Show Table
    DownLoad: CSV
    Table 2.  E(Z), V(Z), S(Z) and K(Z) for the BXII model.
    a b E(Z) V(Z) S(Z) K(Z)
    1 5 0.250000 0.1041667 4.64751 73.8001
    5 0.682424 0.0289951 0.040149 3.07002
    15 0.873845 0.0058751 -0.553252 3.71623
    35 0.942924 0.0013088 -0.755678 4.26811
    50 0.959545 0.0006708 -0.804447 4.42885
    75 0.972765 0.0003089 -0.843332 4.56514
    100 0.979473 0.0001768 -0.862816 4.60851
    0.831252 0.0049489 -0.673595 3.83723
    15 0.5 1.113879 0.0442417 2.131816 15.3416
    1 1.007348 0.0151023 0.598998 5.10838
    10 0.873846 0.0058750 -0.553252 3.71623
    25 0.780286 0.0041837 -0.742638 3.92925
    50 0.744512 0.0037574 -0.765126 3.96261
    100 0.710636 0.0034003 -0.776273 3.97973
    200 0.678424 0.0030882 -0.781829 3.98853
    500 0.638154 0.0027274 -0.785153 3.99380
    1000 0.609315 0.0024848 -0.786258 3.99556
    5000 0.547307 0.0020034 -0.787144 3.99698
    10000 0.522589 0.0018267 -0.787255 3.99832

     | Show Table
    DownLoad: CSV
    Figure 3.  3D plots for the IBXBXII distribution at ζ =2.5.

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

    The Rényi entropy (RE) (see [28]) measure is determined using the following the following formula:

    Rω=11ωlog[Jω(ζ,α1,α2)],ω>0,ω1, (3.1)

    where Jω(ζ,α1,α2)=0[fζ,α1,α2(z)]ωdz. Now we have to compute Jω(ζ,α1,α2). Then,

    Jω(α1,α2,ζ)=(2ζα1α2)ω0zwα1ω(1+zα2)2ωα1ω[1(1+zα2)α1]3ω×exp{ω[1(1+zα2)α1]2}(1exp{ω[1(1+zα2)α1]2})ωζωdz.

    By employing the binomial expansion to the last term in the above equation, we get

    Jω(α1,α2,ζ)=(2ζα1α2)ωi=0(1)i(ωζωi)0zwα1ω(1+zα2)2ωα1ω×[1(1+zα2)α1]3ωexp{(i+1)ω[1(1+zα2)α1]2}dz.

    By applying the exponential expansion to the last term in the last equation, we have

    Jω(ζ,α1,α2)=(2ζα1α2)ως,j=0(1)ς+j[(ς+1)ω]jj!(ωζως)×0zwα1ω(1+zα2)2ωα1ω[1(1+zα2)α1]3ω2jdz.

    Again utilizing the binomial theory, then

    Jω(ζ,α1,α2)=ς,j,k=0ς,j,k0zwα1ω(1+zα2)(2ω+k)α1ωdz, (3.2)

    where

    ς,j,k=(1)ς+j[(ς+1)ω]jj!(2ζα1α2)ω(ωζως)(3ω+2j+k1k).

    Let v=zα2, then

    Jω(ζ,α1,α2)=ς,j,k=0ς,j,kα20vwα1α2ωα2+1α21(1+v)(2ω+k)α1ωdv,

    then

    Jω(ζ,α1,α2)=ς,j,k=0ς,j,kα2B(ϰ(α1,α2;w,ω),[ϰ(α1;ω,k)ϰ(α1,α2;w,ω)]), (3.3)

    where ϰ(α1;ω,k)=(2ω+k)α1+ω and ϰ(α1,α2;w,ω)=wα1α2+ωα21α2>0. Inserting (3.3) in (3.2), then the RE of the IBXBXII distribution is

    Rω=11ωlog[ς,j,k=0ς,j,kα2B(ϰ(α1,α2;w,ω),[ϰ(α1;ω,k)ϰ(α1,α2;w,ω)])],ω>0,ω1, (3.4)

    where ϰ(α1,α2;w,ω)=wα1α2ωα2+1α2. This approach can lead to more balanced and robust portfolios see [35,36] for mare details.

    Due to [31], the Arimoto entropy (AE) measure is determined using the following the following formula:

    Aω=ω1ω[(Jω(ζ,α1,α2))1ω1],ω>0,ω1. (3.5)

    Inserting (3.3) into (3.5), then the AE of the IBXBXII distribution is

    Aω=ω1ω[(ς,j,k=0ς,j,kα2B(ϰ(α1,α2;w,ω),[ϰ(α1;ω,k)ϰ(α1,α2;w,ω)]))1ω1],ω>0,ω1. (3.6)

    For more details see [32].

    The Tsallis entropy (TE) (see [31]) measure is determined using the following the following formula:

    Tω=1ω1[1Jω(ζ,α1,α2)],ω>0,ω1. (3.7)

    Inserting (3.3) into (3.7), then the TE of the IBXBXII distribution is

    Tω=1ω1[1ς,j,k=0ς,j,kα2B(ϰ(α1,α2;w,ω),[ϰ(α1;ω,k)ϰ(α1,α2;w,ω)])],ω>0,ω1. (3.8)

    For more details see [35].

    Due to [25], the Havrda and Charvat entropy (HCE) measure is determined using the following the following formula:

    HCω=121ω1[(Jω(ζ,α1,α2))1ω1],ω>0,ω1. (3.9)

    Inserting (3.3) into (3.9), then the HCE of the IBXBXII distribution is

    HCω=121ω1[(ς,j,k=0ς,j,kα2B(ϰ(α1,α2;w,ω),[ϰ(α1;ω,k)ϰ(α1,α2;w,ω)]))1ω1],ω>0,ω1. (3.10)

    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 Rω, Aω, Tω, and HCω for the IBXBXII distribution at ζ = 2.0. Table 3 gives some numerical results of Rω, HCω, Tω, and Aω for the IBXBXII distribution at ω = 0.3 and 0.6. Table 4 gives some numerical results of Rω, HCω, Tω, and Aω for the IBXBXII distribution at ω = 1.2 and 2.0. We can note that from Tables 3 and 4 when α2 is fixed and the values of α1 are increases then the values of Rω, Aω, Tω and HCω are decreases. Also, when α1 is fixed and the values of α2 are increases then the values of Rω, Aω, Tω, and HCω are decreases. Also when α1 and α2 are fixed and ω is increase then the values of Rω, Aω, Tω, and HCω are decreases.

    Table 3.  Some numerical results of Rω, HCω, Tω, and Aω for the IBXBXII distribution at ω = 0.3 and 0.6.
    α2 α1 ω=0.3 ω=0.6
    Rω HCω Tω Aω Rω HCω Tω Aω
    1.5 0.5 0.831 137.364 4.022 36.765 -0.122 -0.534 -0.265 -0.256
    1.0 0.220 3.612 0.607 0.967 -0.337 -1.264 -0.667 -0.606
    1.2 0.115 1.369 0.291 0.366 -0.379 -1.382 -0.737 -0.662
    1.5 0.003 0.028 0.007 0.007 -0.425 -1.500 -0.810 -0.719
    2 0.5 0.383 10.925 1.219 2.924 -0.465 -1.597 -0.871 - 0.765
    1.0 0.029 0.266 0.067 0.071 -0.468 -1.604 -0.875 -0.769
    1.2 -0.038 -0.296 -0.085 - 0.079 -0.464 -1.594 -0.869 - 0.764
    1.5 -0.110 -0.713 -0.231 - 0.191 -0.456 -1.575 -0.857 - 0.755
    2.5 0.5 0.146 1.907 0.379 0.510 -0.683 -2.034 -1.168 - 0.975
    1.0 -0.097 -0.651 -0.207 - 0.174 -0.564 -1.813 -1.013 - 0.869
    1.2 -0.143 -0.861 -0.295 - 0.230 -0.532 -1.747 -0.969 - 0.837
    1.5 -0.193 -1.033 -0.382 - 0.277 -0.493 -1.661 -0.912 - 0.796
    3 0.5 -0.014 -0.114 -0.031 -0.030 -0.840 -2.268 1.347 -1.087
    1.0 -0.192 -1.030 -0.380 - 0.276 -0.640 -1.959 -1.114 - 0.939
    1.2 -0.225 -1.124 -0.435 - 0.301 -0.591 -1.866 -1.049 - 0.894
    1.5 -0.260 -1.206 -0.490 - 0.323 -0.530 -1.743 -0.966 - 0.835
    3.5 0.5 -0.133 -0.819 -0.276 -0.219 -0.961 -2.414 1.468 -1.157
    1.0 -0.268 -1.222 -0.501 - 0.327 -0.705 -2.069 -1.194 - 0.991
    1.2 -0.293 -1.269 -0.537 - 0.340 -0.642 -1.962 -1.116 - 0.940
    1.5 -0.318 -1.311 -0.573 - 0.351 -0.567 -1.819 -1.017 - 0.872

     | Show Table
    DownLoad: CSV
    Table 4.  Some numerical results of Rω, HCω, Tω, and Aω for the IBXBXII distribution at ω = 1.2 and 2.
    α2 α1 ω=1.2 ω=2
    Rω HCω Tω Aω Rω HCω Tω Aω
    1.5 0.5 3.032 5.312 3.763 4.126 1.175 1.483 0.933 1.483
    1.0 1.690 3.687 2.704 2.864 0.468 0.833 0.659 0.833
    1.2 1.391 3.195 2.365 2.481 0.311 0.602 0.511 0.602
    1.5 1.044 2.551 1.909 1.981 0.130 0.278 0.259 0.278
    2 0.5 3.221 5.481 3.866 4.257 1.105 1.440 0.922 1.440
    1.0 1.508 3.394 2.503 2.636 0.327 0.627 0.529 0.627
    1.2 1.123 2.704 2.018 2.100 0.153 0.323 0.297 0.323
    1.5 0.676 1.765 1.338 1.371 -0.047 -0.112 -0.115 -0.112
    2.5 0.5 3.307 5.553 3.909 4.313 1.041 1.397 0.909 1.397
    1.0 1.373 3.164 2.343 2.457 0.220 0.447 0.397 0.447
    1.2 0.937 2.333 1.752 1.812 0.036 0.081 0.080 0.081
    1.5 0.430 1.176 0.899 0.914 -0.176 -0.449 -0.499 -0.449
    3 0.5 3.347 5.587 3.930 4.339 0.983 1.355 0.896 1.355
    1.0 1.267 2.974 2.210 2.310 0.134 0.285 0.265 0.285
    1.2 0.797 2.035 1.536 1.581 -0.057 -0.135 -0.139 -0.135
    1.5 0.251 0.709 0.546 0.551 -0.276 -0.749 -0.890 -0.749
    3.5 0.5 3.364 5.601 3.938 4.350 0.932 1.316 0.883 1.316
    1.0 1.180 2.814 2.096 2.185 0.062 0.137 0.133 0.137
    1.2 0.686 1.788 1.355 1.389 -0.133 -0.331 -0.359 -0.331
    1.5 0.112 0.324 0.251 0.252 -0.359 -1.022 -1.284 -1.022

     | Show Table
    DownLoad: CSV

    In this work, we consider the main four indicators of risks called the value-at-risk (V1), tail value-at-risk (V2), tail variance (V3) and tail mean variance (V4). 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 V1 or π(q), is the 100q% quantile of the distribution of Z under the IBXBXII distribution can be expressed as:

    V1=Pr(Z>Q(u;ζ,α1,α2))|(q=99%)=1%.

    According to [34], the indicator V1 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 V2 may be a useful indicator in such cases, where

    V2=11qς(Z;π(q),),

    where ς(Z;π(q),)= π(q)zfζ,α1,α2(z)dz. Then,

    V2=α11q=0Δ(1+)[B(α11α2,1α2+1)B(π(q)α2;α11α2,1α2+1)]|1<α1α2,

    where B(π(q)α2;α11α2,1α2+1)>0. The indicator V2 refers to average of all V1 at the confidence level q, which means that the indicator V2 provides more actuarial information about the tail of the IBXBXII distribution, see [29]. Due to [29,30] it can also be expressed as V2=V1+e(V1), where e(V1) is the function of mean excess loss evaluated at the 100q%th quantile. [24] presented an explicit expressions for the indicator V2 indicator under the multivariate normal distribution, the indicator V2 under the IBXBXII distribution can be obtained as

    V3=V[Z2;π(q)]V22,

    where

    V[Z2;π(q)]=α1=0Δ(1+)[B(α12α2,2α2+1)B(π(q)α2;α12α2,2α2+1)]|2<α1α2,

    where B(π(q)α2;α12α2,2α2+1)>0. Finally, due to [26], the V4 indicator can be estimated from

    V4=V2+πV3|0<π<1.

    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.

    Let Θ_=(ζ,α1,α2) be the parameter vector of our model. For determining the maximum likelihood estimations (MAXLE) of Θ_, we deriving the log-likelihood function (ζ,α1,α2). 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 α1, α2, and ζ given sample size {n = 25, 50, 100, 200 and 300.

    Step 2: Calculate the MLEs of α1, α2, 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 (C1), lower limit (C2), upper limit (C3) of 90% and 95% confidence interval and average length (C4).

    All the results of the Monte Carlo simulation for each case for the given parameters: (ζ,α1,α2) are reported in Tables 58. From these tabulated values, one can indicate that: As n increases, the C2 and C4 decreases.

    Table 5.  Point and interval simulation results for the IBXBXII distribution at α1 = 0.5, α2 = 1.5 and ζ = 0.5.
    n MAXLE C1 90% 95%
    C2 C3 C4 C2 C3 C4
    25 α1 0.5192 0.1137 0.3513 0.6871 0.3358 0.3192 0.7193 0.4001
    α2 2.0599 1.7948 -3.0266 7.1465 10.1731 -4.0006 8.1205 12.1211
    ζ 0.4743 0.2179 -0.6169 1.5655 2.1824 -0.8258 1.7745 2.6003
    50 α1 0.4991 0.0909 0.3707 0.6275 0.2568 0.3462 0.6521 0.3059
    α2 2.4692 1.4596 -1.7495 6.6879 8.4375 -2.5574 7.4958 10.0532
    ζ 0.3758 0.2129 -0.4399 1.1915 1.6314 -0.5961 1.3477 1.9438
    100 α1 0.4618 0.0056 0.3841 0.5994 0.2153 0.3635 0.6200 0.2565
    α2 1.7169 1.4084 -2.0831 5.5169 7.6000 -2.8108 6.2446 9.0554
    ζ 0.5191 0.0754 -0.3177 1.3560 1.6737 -0.4780 1.5162 1.9942
    200 α1 0.4877 0.0023 0.4035 0.5718 0.1683 0.3874 0.5879 0.2006
    α2 1.6821 1.2229 -1.0977 4.4620 5.5596 -1.6300 4.9943 6.6242
    ζ 0.4948 0.0590 -0.0846 1.0741 1.1587 -0.1955 1.1851 1.3806
    300 α1 0.5039 0.0012 0.4400 0.5679 0.1279 0.4277 0.5801 0.1524
    α2 1.5210 1.1090 -0.7268 4.5688 5.2956 -1.2338 5.0759 6.3097
    ζ 0.4420 0.0330 -0.0766 0.9605 1.0371 -0.1759 1.0598 1.2356

     | Show Table
    DownLoad: CSV
    Table 6.  Point and interval simulation results for the IBXBXII distribution at α1 = 0.9, α2 = 0.6 and ζ = 0.4.
    n MAXLE C1 90% 95%
    C2 C3 C4 C2 C3 C4
    25 α1 0.9048 0.1077 0.6901 1.1195 0.4294 0.6489 1.1606 0.5117
    α2 0.5067 1.1646 -2.5296 3.5430 6.0725 -3.1110 4.1244 7.2354
    ζ 0.4638 0.1774 0.0255 0.9022 0.8767 -0.0585 0.9861 1.0446
    50 α1 0.9051 0.0702 0.7366 1.0736 0.3370 0.7043 1.1059 0.4015
    α2 0.7728 1.0684 -1.2408 3.4664 4.7073 -1.6915 4.9171 6.6086
    ζ 0.3462 0.1479 0.0636 0.8288 0.7652 0.0095 0.8830 0.8734
    100 α1 0.9152 0.0559 0.7910 1.0395 0.2485 0.7672 1.0632 0.2961
    α2 0.7666 0.5973 -1.9020 2.4352 4.3372 -2.4130 3.9462 6.3592
    ζ 0.3758 0.1153 0.0245 0.7271 0.7026 -0.0427 0.7944 0.8371
    200 α1 0.8843 0.0444 0.8447 1.0239 0.1792 0.8180 1.0507 0.2327
    α2 0.3641 0.4164 -1.4883 2.2165 3.7048 -1.8430 2.5712 4.4142
    ζ 0.4642 0.1076 0.2013 0.7271 0.5258 0.1510 0.7775 0.6265
    300 α1 0.9025 0.0386 0.8125 0.9926 0.1801 0.7952 1.0099 0.2146
    α2 0.6634 0.6231 -0.8207 2.1476 2.9683 -1.1049 2.4318 3.5367
    ζ 0.4097 0.0875 0.1937 0.6258 0.4322 0.1523 0.6672 0.5149

     | Show Table
    DownLoad: CSV
    Table 7.  Point and interval simulation results for the IBXBXII distribution at α1 = 0.8, α2 = 0.5 and ζ = 0.5.
    n MAXLE C1 90% 95%
    C2 C3 C4 C2 C3 C4
    25 α1 0.7908 0.0998 0.6261 0.9555 0.3295 0.5945 0.9871 0.3925
    α2 0.9671 1.0789 -1.6168 3.5511 5.1679 -2.1116 4.0459 6.1575
    ζ 0.3983 0.2643 0.0141 0.7825 0.7684 -0.0595 0.8561 0.9156
    50 α1 0.8171 0.0621 0.6729 0.9613 0.2884 0.6453 0.9889 0.3437
    α2 0.7095 1.1214 -1.5445 2.9635 4.5080 -1.9761 3.3951 5.3712
    ζ 0.5095 0.2376 0.1477 0.8713 0.7237 0.0784 0.9406 0.8622
    100 α1 0.8046 0.0321 0.6921 0.9171 0.2250 0.6705 0.9386 0.2681
    α2 0.8810 0.9905 -1.2841 3.0461 4.3301 -1.6987 3.4606 5.1593
    ζ 0.4517 0.1712 0.1169 0.7864 0.6695 0.0528 0.8505 0.7978
    200 α1 0.7986 0.0689 0.6889 0.9082 0.2193 0.6679 0.9292 0.2613
    α2 0.6635 1.2782 -1.3298 2.6569 3.9867 -1.7116 3.0386 4.7502
    ζ 0.5313 0.2214 0.2273 0.8353 0.6079 0.1691 0.8935 0.7243
    300 α1 0.8106 0.0556 0.7312 0.8901 0.1590 0.7159 0.9054 0.1894
    α2 0.6300 0.8947 -0.7001 1.9601 2.6601 -0.9548 2.2148 3.1695
    ζ 0.5055 0.1442 0.2834 0.7275 0.4441 0.2409 0.7701 0.5292

     | Show Table
    DownLoad: CSV
    Table 8.  Point and interval simulation results for the IBXBXII distribution at α1 = α2 = ζ = 0.5.
    n MAXLE C1 90% 95%
    C2 C3 C4 C2 C3 C4
    25 α1 0.4927 0.0827 0.2777 0.9078 0.6301 0.1557 0.9298 0.7742
    α2 0.3132 0.7077 -1.6453 2.2718 3.9171 -2.0204 2.6468 4.6672
    ζ 0.5506 0.1466 0.1563 0.9448 0.7886 0.0808 1.0204 0.9396
    50 α1 0.4832 0.0792 0.1100 0.6563 0.5463 0.1960 0.8703 0.6743
    α2 0.3960 1.2828 -1.1539 1.9458 3.0997 -1.4506 2.2426 3.6932
    ζ 0.4989 0.1361 0.2249 0.7729 0.5480 0.1725 0.8254 0.6529
    100 α1 0.6019 0.0192 0.3433 0.8606 0.5173 0.2938 0.9101 0.6164
    α2 0.6360 0.2203 -0.0396 1.3117 1.3514 -0.1690 1.4411 1.6101
    ζ 0.5470 0.0173 0.3951 0.6990 0.3039 0.3660 0.7281 0.3621
    200 α1 0.4839 0.0172 0.3544 0.6335 0.2791 0.3277 0.6602 0.3325
    α2 0.4683 0.1218 0.1140 0.7625 0.6485 0.0519 0.8246 0.7727
    ζ 0.5418 0.0139 0.4391 0.6845 0.2454 0.4156 0.7080 0.2924
    300 α1 0.5197 0.0065 0.4109 0.6685 0.2576 0.3862 0.6931 0.3069
    α2 0.4841 0.0130 0.1768 0.7515 0.5747 0.1217 0.8065 0.6848
    ζ 0.5113 0.0052 0.4590 0.6436 0.1846 0.4413 0.6613 0.2199

     | Show Table
    DownLoad: CSV

    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 (CAI), Bayesian INFC (CBYS), Hannan-Quinn INFC (CHQ), Consistent Akaike INFC (CCA). 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).

    Figure 4.  NKDE plots.
    Figure 5.  TTT plots.
    Figure 6.  Q-Q plots.
    Figure 7.  Box plots.

    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.

    Figure 8.  EPDF plots.
    Figure 9.  Empirical and theoretical CDFs.
    Figure 10.  P-P plots.
    Figure 11.  Kaplan-Meier survival plot.

    Table 9 (the second column) gives the MLEs, SEs and CL values, respectively, for the data set I. Table 10 (the second column) shows the MLEs, SEs and CL values, respectively, for the data set II. Table 11 (the second column) presents the MLEs, SEs and CL values, respectively, for the data set III. Table 12 (the second column) provides the MLEs, SEs, and CL values, respectively, for the data set IV. Table 9 (the third column) presents the CAI, CBYS, C HQ and CCA, Kolmogorov-Smirnov test (KS) and P-value (PV), respectively, for the data set I. Table 10 (the third column) shows the CAI, CBYS, CHQ and CCA, KS and PV, respectively, for the data set II. Table 11 (the third column) gives the CAI, CBYS, CHQ and CCA, KS and PV, respectively, for the data set III. Table 12 (the third column) shows the CAI, CBYS, CHQ and CCA, KS and PV, respectively, for the data set IV. Based on the values in Tables 912, the IBXBXII model has the best fits as compared to BXII extensions in the four applications with small values of CAI, CBYS, CHQ, CCA and KS (and biggest corresponding PV) where for data set I CAI= 291.101, CBYS=298.910, CHQ=294.152, CCA=291.121, KS= 0.076266 and PV=0.6059. For data set II CAI= 205.242, CBYS=212.102, CHQ=208.888, CCA=206.967, KS=0.10192 and PV=0.4431. For data set III CAI=381.016, CBYS=387.076, CHQ=382.985, CCA=384.667, KS=0.065129 and PV=0.9638. Finally for data set IV CAI= 310.001, CBYS=315.323, CHQ=311.201, CCA=312.901, KS=0.14037 and PV=0.5339.

    Table 9.  Comparing the competing models under the data set I.
    Competing Models ˆζ,^α1,^α2,ˆα,ˆβ CAI, CBYS, CCA, CHQ,KS,PV
    BXII —, 5.9415, 0.1876, — 382.943, 388.125, 383.062, 385.052, 0.1198, 0.4446
    —, (1.2792), (0.0442), —
    MARBXII —, 1.1923, 4.8343, 838.731, — 305.782, 313.681, 306.093, 308.966, 0.08819, 0.5698
    —, (0.9524), (4.8965), (229.347), —
    TOLBXII —, 1.3503, 1.0612, 13.7228, — 323.542, 331.35, 323.772, 326.708, 0.1037, 0.4598
    —, (0.3782), (0.3837), (8.4003), —
    KMBXII 48.1034, 79.5116, 0.351, 2.7340, — 303.764, 314.250, 304.182, 308.010, 0.08990, 0.5957
    (19.3481), (58.182), (0.093), (1.0763), —
    BTBXII 359.6834, 260.094, 0.1753, 1.1235, — 305.642, 316.036, 306.061, 309.853, 0.08733, 0.5543
    (57.943), (132.203), (0.0132), (0.2433), —
    BEXBXII 0.3831, 11.944, 0.9375, 33.4021, 1.7053 305.822, 318.844, 306.433, 311.091, 0.08609, 0.5509
    (0.073), (4.634), (0.264), (6.281), (0.474)
    FBBXII 0.4214, 0.8354, 6.1115, 1.6746, 3.4505 304.264, 317.314, 304.892, 309.536, 0.08436, 0.5777
    (0.02), (0.944), (2.316), (0.227), (1.960)
    FKMBXII 0.5427, 4.2237, 5.3137, 0.4116, 4.1527 305.530, 318.552, 306.144, 310.803, 0.08743, 0.5554
    (0.132), (1.884), (2.316), (0.495), (1.993)
    ZOBBXII 123.1443, 0.3632, 139.2447, —, — 302.963, 310.718, 303.251, 306.134, 0.09064, 0.5815
    (243.03), (0.3435), (318.551), —, —
    IBXBXII 1564.531, 0.2255, 0.3834, —, — 291.101, 298.910, 291.121, 294.152, 0.076266, 0.6059
    (8.1451), (0.00248), (0.00962), —, —

     | Show Table
    DownLoad: CSV
    Table 10.  Comparing the competing models under the data set II.
    Competing Models ˆζ,^α1,^α2,ˆα,ˆβ CAI, CBYS, CCA, CHQ,KS,PV
    BXII —, 3.1027, 0.4656, —, — 209.602, 214.135, 209.772, 211.4010.1443, 0.3921
    —, (0.5383), (0.0772), —, —
    MARBXII —, 2.2593, 1.5343, 6.7605, — 209.743, 216.564, 210.093, 212.4420.1236, 0.428
    —, (0.8643), (0.9074), (4.5872), —
    TOLBXII —, 2.3934, 0.4518, 1.7967, — 211.803, 218.633, 212.132, 214.5220.1399, 0.3999
    —, (0.9071), (0.2444), (0.9156), —
    KMBXII 14.103, 7.426, 0.5256, 2.2746, — 208.763, 217.862, 209.363, 212.318, 0.1443, 0.4008
    (10.803), (11.851), (0.2702), (0.993), —
    BTBXII 2.5557, 6.0585, 1.8076, 0.29465, — 210.443, 219.543, 211.033, 214.062, 0.1341, 0.4102
    (1.8588), (10.3909), (0.9551), (0.4663), —
    BEXBXII 1.8755, 2.9910, 1.7803, 1.3414, 0.5716 212.10, 223.50, 213.020, 216.603, 0.1443, 0.3988
    (0.090), (1.711), (0.722), (0.812), (0.324)
    FBBXII 0.6217, 0.5496, 3.8386, 1.3817, 1.6656 206.803, 218.202, 207.731, 211.310, 0.1332, 0.4333
    (0.543), (1.010), (2.781), (2.313), (0.435)
    FKMBXII 0.5577, 0.3084, 3.992, 2.1314, 1.4754 206.503, 217.940, 207.414, 211.002, 0.1212, 0.4302
    (0.441), (0.311), (2.083), (1.836), (0.366)
    IBXBXII 314.671, 0.17453, 0.46435, —, — 205.242, 212.102, 206.967, 208.888, 0.1019, 0.4431
    1.0877, 0.03459, 0.039101, —, —

     | Show Table
    DownLoad: CSV
    Table 11.  Comparing the competing models under the data set III.
    Competing Models ˆζ,^α1,^α2,ˆα,ˆβ CAI, CBYS, CCA, CHQ,KS,PV
    BXII —, 5.61548, 0.07243, —, — 518.426, 522.642, 518.467, 520.084, 0.15959, 0.61871
    —, (15.0466), (0.1945), —, —
    MARBXII —, 8.0172, 0.4188, 70.3579, — 387.222, 389.38, 387.626, 389.68, 0.069958, 0.8654
    —, (22.0836), (0.3132), (63.8311), —
    TOLBXII —, 91.324, 0.0127, 141.0737, — 385.944, 392.184, 386.384, 388.403, 0.069437, 0.9301
    —, (15.071), (0.0024), (70.0287), —
    KMBXII 18.137, 6.854, 10.697, 0.0867, — 385.583, 393.940, 386.312, 388.836, 0.06823, 0.9242
    (3.611), (1.034), (1.164), (0.0125), —
    BTBXII 26.7256, 9.7555, 27.3639, 0.0210, — 385.563, 394.103, 386.340, 389.104, 0.06837, 0.9239
    (9.472), (2.718), (12.353), (0.006), —
    BEXBXII 2.9245, 2.914, 3.274, 12.485, 0.372 387.044, 397.424, 388.174, 391.09, 0.069655, 0.9111
    (0.546), (0.555), (1.249), (6.888), (0.77)
    FBBXII 30.447, 0.587, 1.086, 5.168, 7.8626 386.742, 397.144, 387.827, 390.84, 0.06811, 0.9054
    (91.74), (1.01), (1.025), (8.26), (15.02)
    FKMBXII 12.8784, 1.2255, 1.6615, 1.4112, 3.7324 386.962, 397.326, 388.019, 391.062, 0.06854, 0.9054
    (3.421), (0.1315), (0.032), (0.15), (1.11)
    IBXBXII 5.37643, 1.44954, 0.14449, —, — 381.016, 387.076, 382.985, 384.667, 0.065129, 0.9638
    0.62791, 1.4225, 0.12549, —, —

     | Show Table
    DownLoad: CSV
    Table 12.  Comparing the competing models under the data set IV.
    Competing Models ˆζ,^α1,^α2,ˆα,ˆβ CAI, CBYS, CCA, CHQ,KS,PV
    BXII —, 58.723, 0.0064, —, — 328.201, 331.139, 328.601, 329.191, 0.1300, 0.4876
    —, (42.383), (0.0044), —, —
    MARBXII —, 11.8321, 0.0777, 12.2541, — 315.543, 320.011, 316.373, 317.044, 0.1366, 0.5043
    —, (4.3686), (0.015), (7.776), —
    TOLBXII —, 0.2814, 1.8834, 50.2166, — 316.261, 320.732, 317.049, 317.763, 0.1366, 0.5121
    —, (0.2884), (2.4024), (176.54), —
    KMBXII 9.2014, 36.4254, 0.2411, 0.9421, — 317.363, 323.303, 318.719, 319.343, 0.1355, 5066
    (10.052), (35.652), (0.1645), (1.0432), —
    BTBXII 96.104, 52.121, 0.104, 1.227, — 316.462, 322.405, 317.829, 318.422, 0.1372, 0.5133
    (41.201), (33.490), (0.023), (0.326), —
    BEXBXII 0.08214, 5.0324, 1.5317, 31.2523, 0.3122 317.58, 325.064, 319.804, 320.095, 0.1350, 0.5054
    (0.071), (3.852), (0.0124), (12.943), (0.032)
    FBBXII 15.1943, 32.0477, 0.2313, 0.5814, 21.851 317.826, 325.342, 320.089, 320.363, 0.1341, 0.5021
    (11.601), (9.8721), (0.093), (0.073), (35.2)
    FKMBXII 14.7322, 15.2815, 0.2934, 0.8392, 0.0344 317.736, 325.212, 319.918, 320.262, 0.1374, 0.5000
    (12.366), (18.871), (0.222), (0.854), (0.082)
    DBXII 251.2325, 0.08153, 0.42013, —, — 310.001, 315.323, 311.201, 312.901, 0.14037, 0.5339
    2.3291, 0.0363, 0.050710, —, —

     | Show Table
    DownLoad: CSV

    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/data sets/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.

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

    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-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.

    Figure 14.  The Cullen and Frey plot, nonparametric kernel density estimation plot, TTT and box plots under the revenues data set.
    Figure 15.  The Q-Q plot, Scatter plot, the ACF plot and the partial ACF plot under the revenue data set.

    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. 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 insurance-revenue 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.

    Table 13.  Numerical description for the two actuarial data sets.
    Summery & Data Claims Revenue
    Mean 2702.572 32360452
    SD 1569.656 11641499
    Skewness 0.2986363 0.2668196
    Kurtosis 2.144497 2.264141
    Max and Min 6283.001, 340.0001 58756474, 14021480
    Median 2298.999 32090875
    Length 28 64
    Quantile (0.33%, 0.66%) 1694.9008, 3667.6184 27628461, 36090317
    Quantile (0.25%, 0.75%) 1299.4998, 3949.2490 22426547, 39929985

     | Show Table
    DownLoad: CSV

    Table 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 V1|P=2 are estimated under q=75%, 80%, 85%, 90%, 95%, 99% and 99.5%. Under actuarial claims data it is seen that:

    Table 14.  Risk analysis under the actuarial claims data set.
    q V1 V2 V3 V4 e(V1) MOOP V1|P=2
    75% 4.129296 13.30721 3.820954 44.46739 3.222634 9.3921611
    80% 6.657118 21.37147 3.773506 70.61630 3.210319 15.767623
    85% 9.342134 29.87662 3.737976 97.88235 3.198051 23.449813
    90% 15.06109 47.79871 3.691046 154.8284 3.173656 29.980094
    95% 34.07426 105.6841 3.663571 340.0352 3.101582 43.710911
    99% 226.8363 585.4409 5.879942 2883.492 2.580896 456.88933
    99.5% 513.1953 1052.658 16.45130 13087.82 2.051183 888.98398

     | Show Table
    DownLoad: CSV

    (1) The V1 indicator increases as q increases, it started with 4.129296|q=75% ended with 513.1953|q=99.5%, where

    V1|q=75%<V1|q=80%<...<V1|q=99.5%.

    (2) The V2 indicator increases as q increases, it started with 13.30721|q=75% ended with 1052.658|q=99.5%, where

    V2|q=75%<V2|q=80%<...<V2|q=99.5%.

    (3) The V3 indicator decreases as q increases, it started with 3.820954|q=75% ended with 3.663571|q=95%, then the V3 indicator increases as q increases, it started with 5.879942|q=75% ended with 13087.82|q=99.5%, where

    V3|q=75%>V3|q=80%>...>V3|q=95%<V3|q=99%<V3|q=99.5%.

    (4) The V4 indicator increases as q increases, where

    V4|q=75%<V4|q=80%<...<V4|q=99.5%.

    (5) The e(V1) indicator decreases as q increases, where

    e(V1)|q=75%>e(V1)|q=80%>...>e(V1)|q=99.5%.

    (6) The MOOP V1|P=2>V1 q|q=75%, 80%, 85%, 90%, 95%, 99% and 99.5%.

    Table 15 provides the risk analysis under the actuarial revenues data set, where q=75%, 80%, 85%, 90%, 95%, 99% and 99.5%. Moreover, the MOOP V1|P=2 are estimated under q=75%, 80%, 85%, 90%, 95%, 99% and 99.5%. Under actuarial revenues data it is seen that:

    Table 15.  Risk analysis under the actuarial revenues data set.
    q V1 V2 V3 V4 e(V1) MOOP V1|P=2
    75% 72.74761 248.409 213.2133 30981.341 3.414669 102.761889
    80% 136.3578 323.7453 712.8563 119648.02 2.374235 152.710913
    85% 411.5226 393.8717 23278.87 3354443.5 0.9571083 604.810914
    90% 423.1799 393.6161 26062.26 3710938.9 0.930139 664.909612
    95% 435.3047 393.2293 29252.05 4113263.1 0.9033427 670.109439
    99% 461.0562 392.0397 37140.02 5083795.7 0.8503079 716.221348
    99.5% 1134.023 304.2253 2786784 192215514 0.2682709 1587.87301

     | Show Table
    DownLoad: CSV

    (1) The V1 indicator increases as q increases, it started with 72.74761|q=75% ended with 1134.023|q=99.5%, where

    V1|q=75%<V1|q=80%<...<V1|q=99.5%.

    (2) The V2 indicator increases as q increases, it started with 248.409|q=75% ended with 304.2253|q=99.5%, where

    V2|q=75%<V2|q=80%<...<V2|q=99.5%.

    (3) The V3 indicator decreases as q increases, it started with 213.2133|q=75% ended with 2786784|q=99.5%, where

    V3|q=75%<V3|q=80%<...<V3|q=99.5%.

    (4) The V4 indicator increases as q increases, where

    V4|q=75%<V4|q=80%<...<V4|q=99.5%.

    (5) The e(V1) indicator decreases as q increases, where

    e(V1)|q=75%>e(V1)|q=80%>...>e(V1)|q=99.5%.

    (6) The MOOP V1|P=2>V1 q|q=75%, 80%, 85%, 90%, 95%, 99% and 99.5%.

    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 V1 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, α1 and α2 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 α1 and α2:

    Sα1,α2(z)|(z0)=(1+zα2)α1

    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 1621. Tables 1618 correspond to the negatively skewed distributions, where we selected the value of the parameter as follows: (ζ=150, α1=2, α2=5; ζ=50, α1=3, α2=2; ζ=200, α1=3, α2=2). In those cases, it is noted that the S(Z)=0.1336004, 0.1492178, 0.5236663 respectively. While Tables 1921 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: (ζ=1,α1=3,α2=2;ζ=20,α1=3,α2=2;ζ=3,α1=1,α2=5). 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.

    Table 16.  ABS and MSE under MOOP and IBXBXII with q=0.95, ζ=150,α1=2,α2=5.
    ABS MSE
    n|q=95% IBXBXII MOOP IBXBXII MOOP
    50 0.291323 0.002853 0.330868 0.083215
    150 0.194507 0.011421 0.133234 0.010731
    300 0.014406 0.003134 0.092030 0.002030
    500 0.003816 0.000925 0.010032 0.000939
    n|q=99%
    50 0.110880 0.012022 0.690910 0.900028
    150 0.023221 0.008940 0.556034 0.602821
    300 0.005154 0.006781 0.301054 0.321487
    500 0.000440 0.004343 0.100090 0.011344
    n|q=99.5%
    50 0.010091 0.008000 0.996891 0.800321
    150 0.048500 0.007124 0.676731 0.742865
    300 0.004111 0.006612 0.225555 0.420459
    500 0.000390 0.003333 0.109092 0.010310

     | Show Table
    DownLoad: CSV
    Table 17.  ABS and MSE under MOOP and IBXBXII with q=0.95, ζ=50,α1=3,α2=2.
    ABS MSE
    n|q=95% IBXBXII MOOP IBXBXII MOOP
    50 0.278712 0.033333 0.879813 0.111981
    150 0.194652 0.011219 0.776574 0.101098
    300 0.167781 0.006999 0.534332 0.001919
    500 0.100761 0.000210 0.276732 0.001111
    n|q=99%
    50 0.918434 0.301121 0.924379 0.191466
    150 0.400680 0.215133 0.194303 0.131043
    300 0.105555 0.210149 0.033123 0.041953
    500 0.006001 0.010531 0.030540 0.012522
    n|q=99.5%
    50 0.978434 0.569832 0.990325 0.391466
    150 0.404682 0.515433 0.175433 0.271043
    300 0.160093 0.300693 0.100120 0.099009
    500 0.016090 0.010532 0.006254 0.001558

     | Show Table
    DownLoad: CSV
    Table 18.  ABS and MSE under MOOP and IBXBXII with q=0.95, ζ=200,α1=3,α2=2.
    ABS MSE
    n|q=95% IBXBXII MOOP IBXBXII MOOP
    50 0.299999 0.154981 0.176763 0.331323
    150 0.149920 0.101034 0.555573 0.204315
    300 0.197819 0.100913 0.313212 0.109103
    500 0.006432 0.014315 0.200124 0.000154
    n|q=99%
    50 0.129582 0.129811 0.156767 0.175454
    150 0.111300 0.115934 0.087800 0.010777
    300 0.003232 0.003659 0.007555 0.001716
    500 0.008755 0.009149 0.005235 0.000666
    n|q=99.5%
    50 0.129554 0.199811 0.153255 0.183333
    150 0.104687 0.125432 0.074321 0.091099
    300 0.010213 0.100655 0.001708 0.001654
    500 0.006323 0.004343 0.005255 0.000931

     | Show Table
    DownLoad: CSV
    Table 19.  ABS and MSE under MOOP and IBXBXII with q=0.95, ζ=1,α1=3,α2=2.
    ABS MSE
    n|q=95% IBXBXII MOOP IBXBXII MOOP
    50 0.201392 0.354001 0.168891 0.313668
    150 0.143245 0.301054 0.105521 0.214322
    300 0.014312 0.190014 0.313212 0.119102
    500 0.000213 0.001332 0.100124 0.000159
    n|q=99.5%
    50 0.101644 0.104024 0.113338 0.355555
    150 0.040434 0.100029 0.105445 0.100004
    300 0.046667 0.029044 0.098122 0.016121
    500 0.000335 0.003333 0.000878 0.005026
    n|q=99.5%
    50 0.101432 0.100024 0.138812 0.212121
    150 0.100787 0.031022 0.011111 0.146664
    300 0.034301 0.009044 0.010243 0.100100
    500 0.000214 0.003333 0.000077 0.000122

     | Show Table
    DownLoad: CSV
    Table 20.  ABS and MSE under MOOP and IBXBXII with q=0.95, ζ=20,α1=3,α2=2.
    ABS MSE
    n|q=95% IBXBXII MOOP IBXBXII MOOP
    50 0.043344 0.114055 0.188822 0.200362
    150 0.041050 0.101022 0.100012 0.114312
    300 0.000312 0.090014 0.113214 0.010108
    500 0.000219 0.000420 0.000199 0.000051
    n|q=99.5%
    50 0.110991 0.112636 0.105144 0.112434
    150 0.098229 0.111044 0.014171 0.051545
    300 0.006364 0.005415 0.014111 0.001177
    500 0.000646 0.000486 0.000609 0.000555
    n|q=99.5%
    50 0.100451 0.102645 0.100120 0.111885
    150 0.080822 0.031002 0.015678 0.046545
    300 0.002366 0.004048 0.013993 0.001095
    500 0.000214 0.000982 0.000575 0.000425

     | Show Table
    DownLoad: CSV
    Table 21.  ABS and MSE under MOOP and IBXBXII with q=0.95, ζ=20,α1=3,α2=2.
    ABS MSE
    n|q=95% IBXBXII MOOP IBXBXII MOOP
    50 0.209354 0.200211 0.019899 0.100999
    150 0.153993 0.141454 0.012878 0.076761
    300 0.100024 0.050519 0.003003 0.002708
    500 0.000334 0.000111 0.000043 0.000044
    n|q=99.5%
    50 0.102624 0.016951 0.100555 0.124343
    150 0.043386 0.011043 0.013224 0.019543
    300 0.003363 0.009434 0.005656 0.003111
    500 0.000666 0.000765 0.000387 0.000222
    n|q=99.5%
    50 0.090043 0.013648 0.110109 0.099882
    150 0.033383 0.008143 0.017854 0.008548
    300 0.001364 0.007488 0.004343 0.002090
    500 0.000032 0.000552 0.000331 0.000121

     | Show Table
    DownLoad: CSV

    Regarding the data that are skewed in a negative direction (see Tables 1621). Tables 1618 show that the MOOP estimate for V1 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 V1 decreases as q increases.

    (2) MSE for the MOOP estimator for indicator V1< MSE for the IBXBXII estimator for indicator V1 for all negative simulated data.

    (3) MSE for the IBXBXII estimator for indicator V1 decreases as q increases.

    (4) ABS for the MOOP estimator for indicator V1< ABS for the IBXBXII estimator for indicator V1 for all negative simulated data.

    (5) MSE for the MOOP estimator for indicator V1> MSE for the IBXBXII estimator for indicator V1 for all positive simulated data.

    (6) ABS for the MOOP estimator for indicator V1< ABS for the IBXBXII estimator for indicator V1 for all positive simulated data.

    (7) For n=500, the results of the MSE for the MOOP estimator for indicator V1 get very close to the MSE for the IBXBXII estimator for indicator V1 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 V1 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 V1 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.

    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 V1|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 V1 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 V1 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.

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

    This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-RP23009).

    The authors declare no conflict of interest.



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