Asymmetric Right-Skewed Size-Biased Bilal Distribution with Mathematical Properties, Reliability Analysis, Inference and Applications

: Asymmetric distributions, as opposed to symmetric distributions, may be more resilient to extreme values or outliers. Furthermore, when data show substantial skewness, asymmetric distributions can shed light on the underlying processes or phenomena being investigated. In this direction, the size-biased Bilal distribution (SBBD) is suggested in this study as a generalization to the Bilal distribution. The length-biased and area-biased Bilal distributions are discussed in detail as two special cases. The main statistical properties of the distribution including the r th moment, coef-ﬁ cients of variation, skewness, kurtosis, moment generating function, incomplete moments, moments of residual life, harmonic mean, Fisher’s information, and the Rényi entropy as a measure of uncertainty are presented. Graphical representations of the cumulative distribution, probability density, odds, survival, hazard, reversed hazard rate, and cumulative hazard functions are presented for further explanation of the distribution behavior. In addition, the methods of moments and maximum likelihood estimates are taken into account for estimating the model parameters. A simulation study is carried out to see the e ﬃ ciency of the maximum likelihood in terms of standard errors and bias. Real data sets of precipitation and myeloid leukemia patients are considered to show the practical signi ﬁ cance of the suggested distributions as an alternative to some well-known distributions such as the Rama, Rani, Bilal, and exponential distributions. It is found that the size-biased Bilal distribution is right-skewed and has a superior ﬁ tt ing performance compared to the other distributions in this study.


Introduction
One of the most crucial objectives of a statistician is to describe real-world situations using probability distributions. In many real-world applications, including the fields of medicine, actuarial science, engineering, industry, and finance, modeling and interpreting lifetime data are crucial. This has led researchers to concentrate on creating families of probability distributions in recent decades. The size-biased distributions appear when the gathered data in actual life scenarios are assigned based on a weight function rather than randomly. Size-biased distribution refers to the distribution that results in sampling process selecting sample units with a probability that is proportional to some size measure of the unit. The concept of weighted distributions was first proposed by [1] to accurately model real data. Over the last 25 years, some fitting models for scanned data have been collected using the weighted distributions technique. Assume that the random variable X has a probability density function (pdf) ( ) f x ; then, the pdf of the weighted random variable w X is obtained using the following: E w X exists and the normalization factor, ( ) w x , makes the overall probability equal to 1. The concept of length-biased sampling was first introduced by [2], and then [3] formulated and explored this idea broadly in the context of modeling statistical data when the conventional method of employing standard distributions was proven to be ineffective. Ref. [3] introduced a special case of the weighted distribution known as size-biased distribution of order t and if the weighted function has the form ( ) , t w x x = the pdf is obtained using the following: t t x f x f x E X = Furthermore, due to the importance of the weighted distributions, many researchers studied it in various cases. Ref. [4] talked about using weighted binomial distribution to model human families and determine the size of wildlife families. The relationship between initial reliability measures and size-biased distribution was outlined by [5]. Ref. [6] provided the theory of bivariate weighted distribution. Ref. [7] proposed the length-biased weighted Maxwell distribution. Ref. [8] suggested size-biased weighted transmuted Weibull distribution. Length and area-biased Maxwell distributions were introduced by [9]. Ref. [10] proposed size-biased Ishita distribution. Weighted power Lomax distribution and its length-biased form were proposed by [11]. For modeling skewed and heavy-tailed data, [12]presented the normal weighted inverse Gaussian distribution. Ref. [13] offered weighted generalized Quasi Lindley distribution. Ref. [14] suggested generalized weighted exponential distribution. Ref. [15] introduced weighted Gamma-Pareto distribution, and [16] investigated a modified weighted Pareto distribution of Type I.
Ref. [17] offered a new continuous lifetime distribution, called Bilal distribution (BD), which is demonstrated to belong with the class of novel renewal failure rates that are better than average. The Bilal distribution's density function is always unimodal, with around 25% and 28% less skewness and kurtosis than the exponential distribution's density function, respectively. The distribution function, qth quantiles, and failure rate function, on the other hand, are in compact form, and the distinct moments are expressed explicitly in terms of the exponential function. He defined the probability density function of BD as follows: Proof: To prove the theorem, the rth moment is defined as follows:   (7), we obtain the first four moments, respectively, for the origin of X: Proof: The proof is similar to the proof of Theorem (1), considering the following: The rth incomplete moment of the SBBD is defined as follows: Proof. Recall the pdf in (5), the incomplete rth moment of the SBBD can be proved as follows: ( 1 ) 1 , Rényi entropy is a statistical mechanics and information theory concept that gauges how random or uncertain a system is. The Rényi entropy of the random variable X as defined by [18] is expressed using Proof: The Rényi entropy of the size-biased distributed random variable X can be derived as follows: ( 1) 3 2

Special Cases of the SBBD
In this section, by setting (5), we obtain the pdfs of the suggested length-biased Bilal distribution (LBBD) and area-biased Bilal distribution (ABBD), respectively: For various distribution parameter values, the pdf plots of the LBBD and ABBD are shown in Figure 2. The pdfs of the recommended distributions are skewed to the right and become more flat as the values of increase. Furthermore, depending on the parameter values, the pdfs of the ABBD and LBBD can display a variety of behaviors. The CDFs of the LBBD and ABBD, respectively, are expressed using the following: Some of the CDFs plots of the LBBD and ABBD when 1, 2, 3, 4, 5, 6, 7 θ = are presented in Figure 3.

Moments and Related Measure
The moments and related measures, coefficients of variation (Cv), skewness (Sk), and kurtosis (Ku), the incomplete moments and moment of the residual lifetime, moment generating function, reliability functions of the proposed distributions are all described in this section in closed expressions. Also, the parameters estimation, Fisher's information and entropies, as well as the order statistics are presented.
The rth moments of the LBBD and ABBD by setting in Equation (7), respectively, are obtained using the following: An alternative formulation of a real-valued random variable's probability distribution is the moment-generating function (mgf). The mgf of the LBBD and ABBD, respectively, are expressed using the following: 19]) and Benforroni [20] curves are used to measure inequality using the incomplete moments of a probability distribution. The rth incomplete moments of the LBBD and ABBD from Theorem (2) by setting 1, 2 t = , respectively, are defined as follows: The residual life is the time after a component lives up to time ( 0) t ≥ until the time of failure, which is determined by the conditional random variable the rth moment of the residual lifetime (MRL) of the LBBD and ABBD, respectively, are distinct: The harmonic means of the LBBD and ABBD are defined as follows: The rth quartiles of the LBBD and ABBD, respectively, are the solutions for r Q in the following equations: Q are respectively called the first, second and third quartiles.
The coefficient of skewness is used to determine the skewness of the distribution and is expressed as ( )  Table 1 presents the values of the mean and standard deviation (SD) of the LBBD and ABBD for different selections of θ . According to Table 1, as the value of θ rises, the mean and standard deviation values of the LBBD and ABBD rise as well.

Reliability Functions
The survival function is frequently used to predict the probability that an event will not occur after a specific amount of time or threshold in domains like survival analysis and reliability engineering, among others. The survival function is defined as , and for the LBBD and ABBD, respectively, it is characterized as

θ =
The plots reveal that the survival functions curves decreases and are skewed to the right. The distances between each two curves depend on the parameters' values. In survival analysis and reliability engineering, the hazard function is a basic idea. It is utilized to simulate the instantaneous probability of an event (such as death, failure, or the occurrence of a specific event) occurring at a given time, given that it has not yet happened at that time. It is the ratio between the pdf and survival function of a random variable X, ;

Parameters Estimation
The method of moments (MOMs) and maximum likelihood technique are used in this section to estimate the LBBD and ABBD parameters. Let 1 2 , ,..., n X X X be a random sample of size n selected from a pdf ( ) f x and CDF ( ) To find the maximum likelihood estimators of the distribution parameter, let 1 2 , ,..., n x x x be the sample observations of 1 2 , ,..., n X X X from the LBBD and ABBD.
The log-likelihood functions are, respectively, obtained using the following: The partial derivatives of these equations with respect to the parameter θ are obtained using the following: The exact solution of these nonlinear equations is not easy, so one may maximize them by using some optimization approaches as the Newton-Raphson method. Table 2 provides some values of the MLE ˆA

Fisher's Information and Entropies
This subsection introduces the Fisher's information (FI) as well as the Rényi entropy of the LBBD and ABBD. Proof. Take the log of the ABBD pdf as ( ) The first and second derivatives of ( )

log ( ; )
LBBD f x θ with respect to θ respectively are as follows:   Table 3 for selected values of the parameter θ .  Table 3 reveals that the values decrease as the distribution's parameter values rise, and they eventually approach zero for high parameter values.
According to information theory, a random variable's entropy measures the average amount of uncertainty, information, or surprise that could result from its potential outcomes. From Equation (10), the Rényi entropies of the LBBD and ABBD, respectively, are obtained using the following:

Order Statistics
In statistical theory, order statistics are essential, especially in the idea of extreme value. Consider a random sample 1 , X 2 ,...,  The jth order statistics pdf is defined as follows: For simplicity, we use [ ] Therefore, the pdf for the LBBD and ABBD jth order statistics, respectively, are obtained using For 1, j m = in Equations (16) and (17), for both LBBD and ABBD, respectively, we can obtain the pdfs of the minimum (min) and maximum (max) order statistics. Figure 9 shows three-dimensional plots of the pdf of the min and max order statistics of the ABBD and LBBD when 7 m = .

Applications to Real Data
In this section, we demonstrate the applicability of the proposed ABBD and LBBD distributions using two real data sets.
The second one consists the survival times (in months) of 20 acute myeloid leukemia patients discussed by [23] The descriptive statistics for both data sets are presented in Table 5, and it is clear that both data sets are skewed to the right. The density, histogram, box and total time on test (TTT) plots for both data sets are shown in Figure 10. The P-P plots of the fitted distributions for the both data sets are displayed in Figure 11. The estimated pdfs of the suggested and competitive distributions are shown in Figure 12. Since the proposed distributions in this study have just one parameter, we compared them to certain existing distributions that likewise have just one parameter for a fair comparison.
In order to explain how adaptable LBBD and ABBD are, we are going to evaluate them in contrast to a variety of well-recognized models, including Rani ( [24]), Exponential, Rama ( [25]) and Bilal distributions with pdfs, which are respectively expressed as follows:  Table   6 and the best model is indicated by lower AIC, BIC, CAIC, HQIC values for goodness of statistics. For the first data set, the LBBD fits the data better than other distributions and similarly, the ABBD fits the second data set due to the lowest values of these measures. Therefore, both suggested models are considered as best fitted models as compared to the competitors considered in this study. These results are supported by Figures 10-12 where it is seen that the proposed models fit both data sets well.  Figure 10. The density, TTT, box and histogram plots for first (green) and second (blue) data sets. Figure 11. The P-P plots of the fitted distributions for the first (green) and second (blue) data sets. Figure 12. The estimated pdfs of the competitive distributions for the first (green) and second (blue) data sets.

Conclusions
In this study, size-biased Bilal distribution is suggested as a modification of the base Bilal distribution. Two special cases of the SBBD are derived, LBBD and ABBD. The survival function, hazard function, reversed hazard rate function, odds function, and cumulative hazard function of the LBBD and ABBD are obtained. Further statistical features include moments, incomplete moments, moment generating function, and the rth moment. Fisher's information and entropies, order statistics and harmonic mean are presented. Also, the method of moments and the maximum likelihood estimation methods are used for estimating the unknown distributions parameters. Finally, using two real data sets, the suggested model's applicability is demonstrated. According to the findings in this study, the ABBD and LBBD models may be deemed as competitive models with superior capacity than the other models studied in this paper. As expected, the new distributions are probable to garner interest from academics in other fields. In the future, the ranked set sampling techniques can be used to estimate the distribution parameters (see [26]).

Data Availability Statement:
The data used to support the findings of this study are included within the article.

Conflicts of Interest:
The authors declare no conflict of interest.