New Weighted Burr XII Distribution: Statistical Properties, Applications, and Regression

In this study, a three-parameter modifcation of the Burr XII distribution has been developed through the integration of the weighted version of the alpha power transformation family of distributions. Tis newly introduced model, termed the modifed alpha power-transformed Burr XII distribution, exhibits the unique ability to efectively model decreasing, right-skewed, or unimodal densities. Te paper systematically elucidates various statistical properties of the proposed distribution. Te estimation of parameters was obtained using maximum likelihood estimation. Te estimator has been evaluated for consistency through simulation studies. To gauge the practical applicability of the proposed distribution, two distinct datasets have been employed. Comparative analyses involving six alternative distributions unequivocally demonstrate that the modifed alpha power-transformed Burr XII distribution provides a better ft. Additionally, a noteworthy extension is introduced in the form of a location-scale regression model known as the log-modifed alpha power-transformed Burr XII model. Tis model is subsequently applied to a dataset related to stock market liquidity. Te fndings underscore the enhanced ftting capabilities of the proposed model in comparison to existing distributions, providing valuable insights for applications in fnancial modelling and analysis.


Introduction
Te development of novel or new statistical models is a key area of study in the application of distribution theory.Tese distributions' usefulness has led to much research into their theory and the development of new distributions.Te idea of generating new continuous distributions by modifying the existing distributions with one or more shape or scale parameters has gained attention in recent years.Tis parameter introduction has been shown to improve the ability of the developed distributions to ft varied real-life datasets with high degrees of skewness and kurtosis.Some of these newly developed distributions include the modifed alpha power transformed Weibull [1], general two-parameter [2], truncated inverse power Ailamujia [3], half-logistic modifed Kies exponential [4], truncated inverse power Lindley [5], Marshall-Olkin-Weibull-Burr XII [6], generalised unit half-logistic geometric [7], Chen Burr-Hatke exponential [8], modifed XLindley [9], arctan power [10], harmonic mixture Fréchet [11], sine-Weibull geometric [12], bounded odd inverse Pareto exponential [13], new extended Chen [14], power XLindley [15], extended Poisson-Fréchet [16], exponentiated Fréchet loss [17], Gompertz-Makeham [18], and logistic exponential [19] distributions.
Te authors of [20] introduced a new method by adding an additional parameter called the alpha power transformation (APT) family.Te APT family has been used to develop several modifed distributions, including the APT Fréchet [21], APT extended exponential distribution [22], APT inverse Lomax distribution [23], APT log-logistic distribution [24], APT inverse Lindley distribution [25], and APT Pareto distribution [26], among others.With the aim of improving the fexibility of the APT family of distributions, Alotaibi et al. [27] modifed the APT family of distributions and obtained a new family of distributions called the modifed alpha power transformed method (MAPT).According to Alotaibi et al. [27], the cumulative distribution function (CDF) and probability density function (PDF) of the MAPT are given as follows: and Tis study proposes a modifcation of the Burr XII distribution using the MAPT proposed byAlotaibi et al. [27].Te Burr XII distribution introduced by Burr [28] is widely used in reliability analysis, actuarial studies, medicine, and agriculture.Te PDF and CDF of the Burr XII distribution can be expressed, respectively, as follows: and We are motivated to contribute to the ongoing eforts to enhance the versatility of statistical distributions, thus providing researchers with a powerful tool to analyse and model diverse data scenarios efectively as no single distribution is omnibus.Specifcally, our motivations for developing the modifed alpha power transformed Burr XII (MAPTBXII) distribution are as follows: (i) Develop an extension of the Burr XII distribution that provides a good parametric ft to data with complex traits (ii) Propose a new Burr XII distribution with closed form CDF and tractable quantile function that facilitates easy generation of random observations for simulation experiments (iii) Formulate a location-scale regression model using the proposed distribution Te subsequent sections of the paper are organised as follows.In Section 2, we develop the MAPTBXII distribution.Section 3 is devoted to deriving various statistical properties of the MAPTBXII distribution.Te parameters of the MAPTBXII distribution are estimated through the maximum likelihood estimation method, as detailed in Section 4. Section 5 delves into the discussion of the MAPTBXII regression model.A comprehensive simulation study is presented in Section 6. Real-world applications of the MAPTBXII distribution on two datasets are presented in Section 7. Finally, the paper is concluded in Section 8.

Modified Alpha Power Transformed Burr XII Distribution
If a random variable X follows the MAPTBXII, then the PDF can be obtained by substituting equation (3) into equation ( 2), while the CDF can be obtained by substituting equation (4) into equation (1).Te CDF and PDF of the MAPTBXII can then be expressed, respectively, as follows: where α > 0, ξ > 0, c > 0, and α ≠ 1.
Te hazard function of the MAPTBXII is obtained by fnding the ratio of the PDF and complement of the CDF.Te hazard function is given by the following equation: Te various shapes (decreasing, right-skewed, leftskewed, or unimodal) of the densities of the MAPTBXII distribution are shown in Figure 1.
Te various shapes (decreasing or upside down bathtub) of the hazard function are displayed in Figure 2.
Te asymptotic nature of the CDF, PDF, and hazard function is as follows: Te quantile function of MAPTBXII distribution is given as follows: where p ∈ (0, 1) and Q (p) � x p is the quantile function.

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International Journal of Mathematics and Mathematical Sciences

Statistical Properties
3.1.Moments.Moments are a crucial component of statistical theory, and they allow for the examination of many essential characteristics of any distribution.Mathematically, the r th moments of X are given by We substitute equation (7) into equation (9), in order to obtain Letting v � x ξ , which implies x � v 1/ξ and dx � (1/ξ)v 1/ξ− 1 dv, we obtain the following expression: Utilising the identity used in [29], then gives It follows that the r th moment of the MAPTBXII distribution can be expressed as follows: where B(•, •) is given as the beta function and r � 1, 2, . ... Metrics such as variance (σ 2 ), coefcient of variation (CVAR), skewness (CSK), and kurtosis (CKUR) can be derived by utilizing moment-based computations.μ, σ 2 , CVAR, CSK, and CKUR, respectively, are obtained using the following expression: Table 1 displays the σ 2 , CVAR, CSK, and CKUR of the MAPTBXII distribution, calculated using noncentral moments for specifc parameter values.Te results suggest that the MAPTBXII distribution exhibits a varying degree of skewness, ranging from highly skewed to moderately skewed.It is noteworthy that the skewness of the distribution may difer depending on the chosen parameter values.Some combinations lead to a positively skewed distribution, while others result in a negatively skewed distribution.Tis variability underscores the distribution's  International Journal of Mathematics and Mathematical Sciences fexibility in capturing diferent skewness patterns based on the specifed parameters.Te MAPTBXII distribution is leptokurtic (CKUR > 3) and thus has heavier tails and a sharper peak.

Incomplete Moments.
Te incomplete moments can be used to calculate the Lorenz curve, the Bonferroni curve, the mean deviation, and the median deviation, among others.Mathematically, the incomplete moment is given as follows: We substitute equation ( 7) into equation ( 16), in order to obtain Letting v � x ξ , which implies x � v 1/ξ and dx � (1/ξ)v 1/ξ− 1 dv, we obtain the following expression: Utilising the identity used in [29], helps to obtain It follows that the r th incomplete moment of the MAPTBXII distribution is given by the following expression: where B(•: •, •) is given as the incomplete beta function and r � 1, 2, 3, . ...

Moment Generating
When we substitute equation ( 14) into equation ( 22), we obtain the MGF of the MAPTBXII distribution.Hence, the MGF of the MAPTBXII distribution is given by the following expression:

Estimation of Parameters
Te maximum likelihood estimation (MLE) method is used as an estimator.Te log-likelihood function is therefore given as By maximising equation (24), we obtain the parameter estimates.

Modified Alpha Power Transformed Burr XII Regression Model
Suppose X follows the MAPTBXII distribution and Y � log(βX).Te PDF of Y can be derived by substituting c � 1/σ and β � exp(μ).
4 International Journal of Mathematics and Mathematical Sciences where y ∈ R, σ > 0 is the scale parameter, c > 0 is the shape parameter, α > 0, and μ ∈ R is the location parameter.Equation ( 25) is referred to as the log-MAPT Burr XII Figure 3 shows various shapes of the densities of the LMAPTBXII distribution.
Te survival function of the LMAPTBXII distribution can be expressed as follows: Te proposed location-scale regression model is defned with the dependent variable y j and predictor variables z j ′ � (1, z j1 , . . ., z jp ), where 1 is the intercept and expressed as follows: where j � 1, 2, 3, . . ., n, β � (β 1 , β 2 , β 3 , . . .β p ) ′ are the regression parameters, and W j denotes the random error.Te log-likelihood function of the LMAPTBXII model is given by the following expression: Te parameter estimates of the model are obtained by maximising the log-likelihood function.An assessment of the Cox-Snell residuals to ascertain if they behave as a standard exponential distribution would help determine the adequacy of the model.We then diagnose the model using the goodness-of-ft measures (Cramér-von Mises, Anderson-Darling, and Kolmogorov-Smirnov) of the Cox-Snell residuals.

Monte Carlo Simulations
In this section, we ascertain the consistency of the estimators of the MAPTBXII distribution through a simulation study.Te results were obtained using sample sizes of 50, 100, 200, 250, 300, 350, 500, and 600 with parameter values α � (6.85, 10.65, 1.65), c � (0.1, 6.10, 1.90), and k � (0.5, 10.5, 6.50), respectively.It can be observed that the average biases (AB) and root mean square error (RMSE) decrease as the sample size increases as shown in Tables 2.

Applications
In this section, we provide the applications of the MAPTBXII distribution using two uncensored datasets.
Analytical measures such as Kolmogorov-Smirnov (K-S) test, Anderson-Darling test (AD), and Cramér-von Mises test (CVM), Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (AICc), and Bayesian Information Criterion (BIC) are considered in evaluating the Te maximum likelihood estimates and other goodnessof-ft statistics for the two datasets are presented in Tables 3  and 4.
Te MLEs of the MAPTBXII model for datasets 1 and 2 are both unique and represent real maxima, as demonstrated by the profle log-likelihood plots shown in Te ftted densities are presented in Figures 6 and 7, whereas the ftted CDFs are shown in Figures 8 and 9.It can be observed from these results that the MAPTBXII provides a better ft to the two datasets than the other competing lifetime distributions.

Application of Modifed Alpha Power Transformed Burr XII Regression
Model.Te LMAPTBXII model was employed in the analysis of a dataset related to stock market liquidity.Dataset can be retrieved from https://instruction. bus.wisc.edu/jfrees/jfreesbooks/Regression20Modeling/BookWebDec2010/data.html (accessed on 8 January 2023).Te competing models are the log-harmonic mixture Burr XII (LHMBXII) distribution [31] and the log-Gumbel Burr XII (LGBXII) distribution [38].Te response variable y j is the total number of shares that were traded on an exchange during a specifc period (volume), while the covariate is the number of shares outstanding as of December 31, 1984, expressed in millions of shares (shares) (z j1 ).Te ftted model is given by the following expression: We can deduce that the efect shares had on the total number of shares that were traded on an exchange during a specifc period was positively signifcant.
To assess the appropriateness of the LMAPTBXII, LGBXII, and LHMBXII models, Cox-Snell residuals were generated.Upon examination of the probability-probability (P-P) plot illustrated in Figure 10, it is evident that the residuals of the LMAPTBXII model exhibit closer alignment to the diagonal line in comparison to those of the LGBXII and LHMBXII models.Tis observation signifes that the LMAPTBXII model provides a better ft to the data.

Conclusion
In this study, we thoroughly investigated the tractability, performance, and fexibility of a novel three-parameter modifed alpha power transformed Burr XII distribution.One notable feature of this model is the discovery of a closed-form expression for its quantile function, adding to its analytical convenience.Te parameter estimation for the proposed model was carried out through the rigorous maximum likelihood estimation method.Tis approach ensures that the model parameters are optimised to best capture the characteristics of the data under consideration.To assess the performance and fexibility of the new model, a comprehensive simulation study was conducted, and the model's applicability was demonstrated through its application to two lifetime datasets.Te robust results obtained from both the simulation study and the practical applications convincingly validate the enhanced fexibility of the proposed modifed alpha power transformed Burr XII distribution.Building upon the original model, we introduced a logarithmic transformation to create a new logmodifed alpha power transformed Burr XII model.Trough thorough development and rigorous validation, we demonstrated the viability of this new model, ofering an alternative perspective for researchers working with diverse data patterns.It is worth noting that the motivation behind introducing modifcations to existing distributions, such as the Burr XII, is rooted in the need for increased fexibility.Recognising that no single distribution can perfectly ft all datasets, these modifcations empower researchers to account for a broader range of data patterns.In this context, the modifed alpha power transformed Burr XII stands out as a valuable addition, capable of accommodating decreasing, skewed, and near-symmetric datasets.In summary, our study makes a signifcant contribution to the continuous endeavour of improving the versatility of statistical distributions.As we continue to refne and expand our toolkit in statistical modelling, the MAPTBXII distribution stands out as a valuable addition, empowering researchers to address a wide range of data challenges with increased precision and adaptability.Te modifed alpha power transformed Burr XII distribution, along with its log-transformed counterpart, represents a valuable asset for researchers seeking a fexible and robust framework for data analysis and modelling.

Figure 2 :
Figure 2: Te hazard function plots of the MAPTBXII.

Table 1 :
First fve moments varying parameter values.

Table 3 :
MLEs and goodness-of-ft statistics for data 1.

Table 4 :
MLEs and goodness-of-ft statistics for data 2.

Table 5 :
Parameter estimates and selection criteria.

Table 6 :
Diagnostics results for residuals.