On a Special Weighted Version of the Odd Weibull-Generated Class of Distributions

: In recent advances in distribution theory, the Weibull distribution has often been used to generate new classes of univariate continuous distributions. They ﬁnd many applications in important disciplines such as medicine, biology, engineering, economics, informatics, and ﬁnance; their usefulness is synonymous with success. In this study, a new Weibull-generated-type class is presented, called the weighted odd Weibull generated class. Its deﬁnition is based on a cumulative distribution function, which combines a speciﬁc weighted odd function with the cumulative distribution function of the Weibull distribution. This weighted function was chosen to make the new class a real alternative in the ﬁrst-order stochastic sense to two of the most famous existing Weibull generated classes: the Weibull-G and Weibull-H classes. Its mathematical properties are provided, leading to the study of various probabilistic functions and measures of interest. In a consequent part of the study, the focus is on a special three-parameter survival distribution of the new class deﬁned with the standard exponential distribution as a reference. The exploratory analysis reveals a high level of adaptability of the corresponding probability density and hazard rate functions; the curves of the probability density function can be decreasing, reversed N shaped, and unimodal with heterogeneous skewness and tail weight properties, and the curves of the hazard rate function demonstrate increasing, decreasing, almost constant, and bathtub shapes. These qualities are often required for diverse data ﬁtting purposes. In light of the above, the corresponding data ﬁtting methodology has been developed; we estimate the model parameters via the likelihood function maximization method, the efﬁciency of which is proven by a detailed simulation study. Then, the new model is applied to engineering and environmental data, surpassing several generalizations or extensions of the exponential model, including some derived from established Weibull-generated classes; the Weibull-G and Weibull-H classes are considered. Standard criteria give credit to the proposed model; for the considered data, it is considered the best.


Introduction
This section presents the background, motivations, contributions, and structure of the study.

Background
General classes of univariate continuous distributions are involved in various facets of statistical modeling. They offer solutions to practitioners who wish to understand and best explain their subject of study from the data observed. In fact, most of these general classes are based on well-known distributions serving as generators. Basic works and recent developments on this topic are discussed in [1][2][3]. Among the most interesting classes in terms of heterogeneous distributions, are the Weibull-generated-type classes where G(x) denotes the cdf of an arbitrary or targeted reference continuous distribution with support on (a, b), then the cdf given as Equation (2) defines the Weibull-X class by [4]. By considering the logistic distribution as reference, it is proved in [16] that the corresponding Weibull-X distribution is flexible enough to perfectly fit the glass fiber data set of [17]. In particular, the fit of the proposed Weibull logistic model outperforms the adjustment criteria of the logistic, skewed logistic, and skewed logistic "with location" models. Further, when H(x) denotes the following "odd ratio" function: , the cdf given as Equation (2) characterizes the Weibull-G class by [5]. The foundational work of [5] shows that the distributions of the Weibull-G class are applicable in various settings. In particular, the Weibull-G models defined with the exponential, log-logistic, and Burr XII models as references are appropriate for the adjustments of the glass fiber data set of [17] and the fatigue time data set of [18]. As a valuable indicator of its usefulness, the Weibull-G class has been cited in more than 400 references. On the other hand, when H(x) denotes the following modified odd ratio function: H(x) = H 3 (x) = G(x) 1 − G(x)(1 + G(x))/2 , the cdf given as Equation (2) defines the MOW-G class by [6]. The MOW-G class is illustrated with the reference distributions: gamma, Weibull, and Lindley distributions. The related models are proved to be efficient for fitting the guinea pig data set by [19], the fatigue time data set by [18], and the failure time data set by [20]. Thanks to their flexible properties, these models are preferable to several competitors based on the same reference models. A deep relationship exists between the Weibull-X, Weibull-G, and MOW-G classes.
In particular, they are complementary in the following stochastic dominance sense: For any x ∈ R, we have Thus, this hierarchical order indicates that, for a given reference distribution and data set, the objectives of the corresponding estimated cdfs of the classes differ somewhat; one estimated cdf may be more adequate than another relative to the empirical cdf of the data.

Motivations and Contributions
In this paper, we contribute to the subject by proposing a new motivated Weibullgenerated-type class, called the weighted odd Weibull-generated class (WOW-G class for short), which has deep connections with the Weibull-X, Weibull-G, and modified odd Weibull-G classes. It respects the definition of the cdf given as Equation (2) with the following weighted odd ratio function: Thus, H 4 (x) is a weighted version of H 2 (x); H 4 (x) = w(x)H 2 (x), precisely. The WOW-G class is specified by the following cdf: F H 4 (x) = 0 for x ≤ a, and F H 4 (x) = 1 for x ≥ b. Here, we note specific motivations for considering the particular function H 4 (x): (i) As a basic remark, in relation to H 1 (x), H 2 (x) and H 3 (x), the analytical complexity of H 4 (x) is quite acceptable, with the weight function w(x) remaining simple. (ii) Let us now underline the mathematical connections behind the functions H 1 (x), H 2 (x), H 3 (x), and H 4 (x). By virtue of the result in [21], the following logarithmic holds: On the other side, for any y ∈ [0, 1], we have 1 − y 2 /4 < 1, which is equivalent to 1 − y/2 < 2/(2 + y); so, Therefore, the following chain of inequalities holds: implying the following first stochastic dominance result: In this sense, the proposed WOW-G class can be seen as an alternative to the Weibull-X, Weibull-G, and MOW-G classes. This information is important enough to warrant an investigation of the WOW-G class. (iii) In relation to the literature on distribution theory, we notice that H 4 (x) can be expressed as with c = 1/2 and γ = 1. Therefore, the WOW-G class appears to be a subclass of the extended odd-G class championed by [22].
(iv) Last but not least, the inverse function of H 4 (x) is quite manageable; after some operations, we find where G −1 (y) denotes the quantile function (qf) of the reference distribution. This implies that the qf of the WOW-G class has an analytical expression, which will be provided later.
In this study, we derive the theoretical and practical properties of the proposed WOW-G class. We emphasize a special distribution of the class based on the standard exponential distribution as a reference. Among its features, it possesses the following simple cumulative hazard rate function: R(x) = α[sinh(φx)] β for x > 0, with α > 0, φ > 0 and β > 0, where sinh(x) denotes the standard hyperbolic sine function: sinh(x) = (e x − e −x )/2. Thus defined, it constitutes a special three-parameter survival distribution that takes advantage of the structure of the WOW-G class to achieve a high level of flexibility. In particular, its probability density function (pdf) exhibits all the asymmetric qualities; the triplet "left skewed-almost symmetrical-right skewed" is observed, as well as various monotonic decreasing shapes and reversed N shapes. Its sister function in terms of modeling, the hazard rate function (hrf), presents increasing concave and convex shapes, decreasing, or almost constant shapes. This flexibility is also observed in diverse moment measures, such as the mean, variance, moment skewness, and moment kurtosis. This new survival distribution is thus adapted for the adjustment of versatile data sets. We illustrate this claim by considering engineering and environmental data, estimating the model parameters using the maximum likelihood (ML) method. By following the standards, we use established statistical tools to prove that the fit of the proposed model surpasses those of several extended exponential models. Statistical comparisons with those derived from the Weibull-X, Weibull-G, and MOW-G classes are discussed. Numerical tables and graphics support the findings.

Structure of the Paper
Section 2 formally presents the most useful functions of the WOW-G class, as well as the mentioned special distribution. In Section 3, we describe some relevant properties of the class. Statistical considerations are given in Section 4. The application to two practical data sets is given in Section 5. Summary and further research are posed in Section 6.

The WOW-G Class
Here, we refine the presentation of the WOW-G class and show what we can call its "distributional richness".

Presentation
We recall that the WOW-G class is defined by the cdf given as Equation (2), which will be denoted in the next section as F(x) = F H 4 (x), to simplify the notations. From F(x), we derive the pdf by differentiation according to x in the almost everywhere sense; the pdf of the WOW-G class is given as where f (x) = 0 for x ∈ (a, b). We recall that g(x) refers to the pdf of the reference distribution. As an important reliability function, the survival function (sf) of the WOW-G class is given as We can also express the main hazard functions of the WOW-G class, defined by the cumulative hrf and hrf. Thus, the cumulative hrf is expressed as where R(x) = 0 for x ≤ a, and R(x) = +∞ for x ≥ b, and the hrf is obtained by differentiation of R(x) according to x in the almost everywhere sense: and r(x) = 0 for x ∈ (a, b).

Some Examples
Based on the WOW-G class, diverse distributions can be defined for specific purposes. By considering reference distributions, some relevant ones are listed in Table 1.
All the distributions presented in Table 1 deserve an in-depth mathematical treatment to reveal their capacities in a modeling context. In this study, we focus on the WOWE distribution-on the one hand, because of the simplicity and originality of this sf, and on the other hand, because of the beautiful results observed in various statistical analyses, which will be presented later.

The WOWE Distribution
As a basis, the WOWE distribution is derived from the WOW-G class by choosing the exponential distribution as a reference. Here, the exponential distribution is specified by the following cdf and pdf: G(x) = 1 − e −φx for x > 0 and G(x) = 0 for x ≤ 0, and g(x) = φe −φx for x > 0, and g(x) = 0 for x ≤ 0, respectively, where φ > 0 is a scale parameter. Let us now discuss the cdf of the WOWE distribution. Noticing that it follows from Equation (2) that and F(x) = 0 for x ≤ 0. From this expression, one can remark that the WOWE distribution is also the distribution of the random Y = arsinh(X)/φ, where X denotes a random variable following the Weibull distribution with scale parameter α and shape parameter β, and arsinh(x) denotes the area hyperbolic sine function, defined as the inverse of sinh(x)that is, arsinh(x) = ln(x + √ x 2 + 1). By differentiating F(x) according to x, the pdf of the WOWE distribution follows immediately: f (x) = 0 for x ≤ 0. We now illustrate the flexibility of f (x) in terms of curvature in Figure 1; the values of the parameters are selected to show specific shapes for f (x).  . The light blue curve shows a decreasing function with small variations; the navy blue curve shows a right-skewed function; the green curve refers to an almost symmetric function; the black curve represents a left-skewed function; and finally, the red curve presents a perfect reversed N shape.
We observe two (local) maxima, one is attained at x = 0 and another is attained in the neighborhood of x = 1. This panel of curvatures is a strong argument for recommending the WOWE model for the fit of various survival data.
The corresponding sf and cumulative hrf of the WOWE distribution are given as and S(x) = 1 for x ≤ 0, and and R(x) = 0 for x ≤ 0, respectively. The simplicity and originality of these functions are features of the WOWE distribution. By differentiating R(x) according to x, the hrf of the WOWE distribution is obtained by and r(x) = 0 for x ≤ 0. The possible curves of r(x) is informative of the data fitting ability of the WOWE model (see [23]). In this regard, Figure 2 presents various curves for r(x); the values of the parameters for r(x) are chosen to depict specific forms.  . The light blue curve shows an increasing and convex function, the navy blue curve represents a clear bathtub shape, the green curve refers to a decreasing function, the black curve shows an increasing and concave function, and the red curve presents a near-constant shape. These observations support the WOWE model for data fitting purposes.
Other properties of the WOWE distribution will be presented throughout the study.

Theoretical Aspects
This section affords some theoretical aspects and properties of the WOW-G class, with applications to the WOWE distribution when possible.

First-Order Stochastic Dominance
By construction, as already developed in the introductory section, we have Hence, by using the concept of first-order stochastic (FOS) dominance as presented in [24], the Weibull-X class FOS-dominates the WOW-G class, which itself FOS-dominates the MOW-G class, which itself FOS-dominates the Weibull-G class. In addition, an intrinsic FOS dominance property of the WOW-G class is developed in the next proposition. Proposition 1. Let F(x; α, β) = F(x). Then, the following inequalities hold: There is no FOS property for the WOW-G class according to β. Proof.
• For any α > 0, β > 0 and x ∈ R, we have The sign of this function depends on the sign of the logarithmic term. The function is This ends the proof of Proposition 1.
Naturally, further FOS dominances can be established based on the parameter(s) of the reference distribution. For instance, a result of this aspect regarding the WOWE distribution is presented below.
The desired result follows.

Quantile Function
In full generality, the qfs are involved in many statistical applications and simulation techniques. The following result determines the qf of the WOW-G class. Proposition 3. The qf of the WOW-G class is obtained by inverting F(x). That is, where G −1 (u) denotes the proper qf of the reference distribution.
Proof. In order to determine Q(u), let us investigate the equation F(x) = u. Then, the following chain of equivalences holds: The desired result follows immediately from Equation (3).
From Proposition 3, it follows that the median of the WOW-G class is given by Further, by considering a random variable U with a uniform distribution over (0, 1), Q(U) becomes a random variable with the cdf of the WOW-G class. This distributional property is at the basis of the inverse transform sampling technique, which allows the generation of numerical values from Q(U). In addition, the established qf enables the definitions of various asymmetry and plateness measures, such as the Galton skewness and Moors kurtosis. These aspects are described in detail in [25].
As a concrete example of application, the qf of the WOWE distribution follows from The median of the WOWE distribution is given by The other quartiles, as well as Galton skewness and Moors kurtosis, can be expressed in a similar manner.

Asymptotic and Form Analysis
Let us now examine the asymptotic behavior and form analysis of f (x) and r(x), as already sketched graphically in Figures 1 and 2. Such asymptotic analysis is useful to highlight the roles of the parameters on the possible explosion or not of these functions. In the case of G(x) → 0, the following equivalences hold: We see that α and β have a strong impact on the asymptotic behavior of the functions, especially in the case of G(x) → 1. These asymptotic results can be completed by an analytical study. First, the local maximum(a) of f (x) are of modeling interest, representing possible "peak(s) of values" behind an observed phenomenon. Further, a local maximum corresponds to a mode of the WOW-G class. Such a local maxima-e.g., x m -satisfies the following equation: In addition, as a local maximum, Clearly, the definition and number of modes depend on g(x) and the parameters α and β.
Similarly, a critical point for r(x) satisfies dr(x)/dx = 0, or equivalently, d ln[ The sign of this function at a solution point can indicate its nature. Nevertheless, from the general equations above, the expressions of the solutions are difficult to exhibit. As a more concrete study, let us now investigate the asymptotics, modes, and shapes of the pdf and hrf of the WOWE distribution. In the case of x → 0, we have Both of f (x) and r(x) explode to +∞ when β < 1, are equal to αβφ β when β = 1, and tend to 0 when β > 1. Further, in the case of x → +∞, it follows that Thus, for all the values of the parameters, f (x) tends to 0 whereas r(x) explodes to +∞. Now, the derivative of the logarithmic transformation of the corresponding pdf is obtained as We already know that x = 0 is a local maximum for f (x) when β < 1. In this case or for β ≥ 1, another possible mode-e.g., If β > 1, this function is positive, implying that ln[r(x)] and r(x) are increasing. If β < 1, a critical point is given as the solution of the following equation: x = atanh( 1 − β)/φ. These mathematical facts confirm some observations made in Figures 1 and 2.

Mixture Representation
Following the approach of [5], mixture representations of the main functions of the WOW-G class are proved in the next proposition.

Proposition 4.
The cdf and pdf of the WOW-G class can be expanded in the following manner: Similarly, assuming that the interchange of the derivative and sum signs is valid in the mathematical sense, we have Proof. Based on the standard exponential series expansion, we obtain Moreover, by applying the generalized binomial formula twice, we obtain By expressing G(x) as G(x) = 1 − K(x) and using the standard binomial formula, it results as follows: which corresponds to the announced result. When differentiating with respect to x, the expansion of f (x) follows.
One can remark that K(x) is literally the sf of the reference distribution. Thus, Proposition 4 shows that the cdf and pdf of the WOW-G class can be expressed in terms of crucial functions of the exponentiated reference distribution. This relationship can be used for further mathematical developments.
For the WOWE distribution, based on this proposition, the corresponding cdf and pdf become expressed in terms of inverse exponential functions as follows: Under some integrability conditions, one infers that, for a function h(x) defined on (0, +∞), we have h(x)e −tx dx denotes the Laplace transform of h(x). This expansion can serve for computational purposes for various probabilistic measures.

Moments
We now work with a random variable X whose distribution belongs to the WOW-G class. Let us denote by E the standard expectation. Then, if the r th order moment about the origin of X converges in the mathematical sense, it is given by After some changes of variable based on the involved qfs, it can be re-expressed as or, based on Proposition 3, In all cases, these integrals are not simply developable; if the reference distribution is known, along with the parameters α and β, numerical treatments of them are possible via mathematical software such as Matlab, Mathematica, R, Python, etc. Alternatively, one can use the expansion of f (x) in Proposition 4 for deriving an expansion of m r , as in [5].
In the setting of the WOWE distribution, the r th -order moment about the origin of X converges in the mathematical sense and the above integrals can be applied. As a remark, by introducing a random variable W following the Weibull distribution with scale parameter α and shape parameter β, we have m r = E{[arsinh(W)] r }/φ r . From these moments about the origin, one can derive the moments about the mean, variance, moment skewness, and moment kurtosis of X. They are defined by, respectively, The mean is a central measure, the variance is a dispersion measure, the moment skewness is an asymmetry measure, and the moment kurtosis is a peakedness/flatness measure.

Inferential Aspect
In this section, we briefly outline the ML method, which allows for efficient estimation of the model parameters. A simulation study illustrates this efficiency.

Parameter Estimation
Suppose that we deal with n observations x 1 , . . . , x n from a characteristic that can be modeled by a random variable with a distribution belonging to the WOW-G class. Then, by denoting (α, β, ζ) the vector of the unknown model parameters, including those of the reference model denoted by ζ, a ML estimate (MLE) of (α, β, ζ) is given as where L(α, β, ζ) denotes the likelihood function defined by Here, we assume that the MLE exists and is unique. A component of (α,β,ζ) is called the MLE of the corresponding component of (α, β, ζ). Under some configurations, it is easier to find the MLE of (α, β, ζ) through the maximization of the log-likelihood function given as If the log-likelihood function is differentiable, the desired MLEs are solutions of the following system of equations: and, by denoting ζ = (ζ 1 , . . . , ζ p ), for j = 1, . . . , p, Due to the analytical complexity of these equations, in most cases, closed form expressions for the MLEs are not possible. We can solve them numerically by using iterative procedures such as the Newton-Raphson-type algorithms. The standard errors (SEs) of the MLEs can be determined by the diagonal elements of J −1 , where J denotes the (p + 2) × (p + 2) matrix defined by , whose elements can be evaluated numerically. The setting above can be applied directly to the WOWE distribution, as illustrated in the next section. The background on the general theory and practice of the ML method can be found in [26].

Simulation
Here, based on the WOWE model, a simulation study is performed in order to investigate the behavior of the ML estimation. We start by generating N = 10,000 samples from the model, each of sample size n ∈ {35, 50, 100, 200, 300, 400, 500}, for some selected parameter values. The assessment considers the average value of the ML estimates (AE) and the root-mean-square error (RMSE) defined by respectively, where ψ = α, β, or φ. We employ the package named AdequacyModel and developed in [27] of the R software (see [28]). The results are given in Table 3. Table 3. Actual values, AEs, and RMSEs of the simulated data from the WOWE distribution for the following parameter configurations with order (φ, α, β): (1.  It can be seen from Table 3 that the performance of the ML estimation shows consistency and is quite satisfactory; the AEs converge to the corresponding actual values and the RMSEs decrease as the sample size increases.

Data Analysis
In this section, we attempt to apply the WOWE model to two real data sets, comparing its fit behavior with those of famous extended exponential models of the literature.
We discriminate these models by considering the following standard statistical measures: Conventionally, the best model is the one with the smallest value for AIC, BIC, W, CAIC, HQIC, A, KS and the largest value for the p-value related to the KS test. It is worth mentioning that the AIC, BIC, CAIC, and HQIC depend on the minus maximal estimated log-likelihood that we denote as −ˆ . Further details on these measures, including their mathematical definitions, can be found in [32]. As for the simulation study, the results are obtained by employing the AdequacyModel package of the R software.

Fit of the Drilling Machine Data
The first data set named "drilling machine data" refers to the 50 observations with holes and sheets of a certain thickness. It is extracted from [33]. The data are presented as follows: 0.04, 0.02, 0.06, 0.12, 0.14, 0.08, 0. For these data, the MLEs of the model parameters and the related SEs are presented in Table 4.  (5) and (6) and the plug-in technique, the corresponding estimated cdf and pdf are given aŝ respectively. Other estimated functions and measures of the WOWE model can be presented in a similar manner. The values of the −ˆ , AIC, BIC, CAIC, HQIC, W, A, KS, and p-value of the KS test of all the models are given in Table 5. In Table 5, according to the considered criteria, the best results are obtained for the proposed WOWE model. The main challenger is the LxE model. Figure 3 displays four different plots illustrating the nice fit of the WOWE model, which are the probabilityprobability (P-P) plot, the estimated pdff (x) superposed to the histogram of the data, the estimated cdfF(x) superposed to the empirical cdf of the data, and the quantile-quantile (Q-Q) plot.
In each of these plots, we see that the empirical object is well-adjusted by the estimated object. In particular, on the plot of the estimated pdf, we see that the kurtosis flexibility of the WOWE model is determinant; the round shape of the histogram of the data is perfectly fitted.
We complete this data analysis by comparing some descriptive statistics with the estimated ones obtained from the WOWE model. The results are presented in Table 6.  Figure 3. Plots of four estimated objects for the drilling machine data.
As expected, from Table 6, we see that the estimated statistics are close to the empirical ones. This is one more evidence of the high performance of the WOWE model.

Fit of the Daily Precipitation Data
The second data set, named "daily precipitation", is the average of the maximum daily precipitation for 30 years at 35 stations in central and western Peninsular Malaysia. It is extracted by [34]. The data are presented as follows: For these data, the MLEs of the model parameters and the associated SEs are indicated in Table 7.  (5) and (6), respectively, combined with the plug-in estimation technique.
The values of the −ˆ , AIC, BIC, W, CAIC, HQIC, A, KS, and p-value of the considered models are given in Table 8.  Table 8 indicates that the WXE model is the best, followed by the proposed WOWE model. The WOWE model is not the best, but remains a solid alternative, as it surpasses the six other models. Figure 4 displays four different plots, allowing the visual illustration of the fit of the WOWE model. From Figure 4, the observed adjustments are quite satisfactory. In particular, on the plot of the estimated pdf, we see that the left-skewed property of the WOWE model has played an important role in capturing the form of the histogram.
We finish up our data analysis by comparing some descriptive statistics to the estimated ones based on the WOWE model. Table 9 shows the results.  Table 9 shows that the predicted statistics are very close to the empirical ones, as expected. This is just another example of the strong performance of the WOWE model.

Conclusions
We have presented a new motivated Weibull-generated-type class of univariate continuous distribution, called the weighted odd Weibull-generated class. Thanks to the use of a coherent weighted version of the odd transformation, it offers a solid alternative to the Weibull-X, Weibull-G, and MOW-G classes. We have explored its main theoretical and practical facets, with a focus on a special distribution based on the exponential distribution.
The faculties of the proposed class make this special distribution particularly flexible in the modeling sense, with pdf and hrf demonstrating heterogeneous skewness and tail weights. We exploit this advantage in a data fitting scenario. By estimating the model parameters with the maximum likelihood method, we show that our model is efficient enough to fit two data sets of importance: one with engineering data and the other with environmental data. It is proved to outclass several models also based on the exponential model.
The classical Weibull-G class has been successfully applied to several significant disciplines in recent years, including medical, biology, engineering, dependability, economics, computer science, and finance. Detailed examples can be found in [5,35,36], among others. Our new Weibull-generated class can be used in similar scenarios. It may provide a more accurate model owing to its exceptional analytical versatility. As a result, it must be considered for purposes of testing beside the former Weibull-G class.
As a future potential work, we may consider extension of the new Weibull class, including its bivariate extension based on the same approach as [14], or its discrete analogue following the spirit of [15].
Author Contributions: All authors contributed equally to this article. All authors read and agreed to the published version of the manuscript.
Funding: This work is not funded.

Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.

Data Availability Statement:
The data that support the findings of this study are available from the corresponding author, upon reasonable request.