A note on an extreme left skewed unit distribution: Theory, modelling and data fitting


 In probability and statistics, unit distributions are used to model proportions, rates, and percentages, among other things. This paper is about a new one-parameter unit distribution, whose probability density function is defined by an original ratio of power and logarithmic functions. This function has a wide range of J shapes, some of which are more angular than others. In this sense, the proposed distribution can be thought of as an “extremely left skewed alternative” to the traditional power distribution. We discuss its main characteristics, including other features of the probability density function, some stochastic order results, the closed-form expression of the cumulative distribution function involving special integral functions, the quantile and hazard rate functions, simple expressions for the ordinary moments, skewness, kurtosis, moments generating function, incomplete moments, logarithmic moments and logarithmically weighted moments. Subsequently, a simple example of an application is given by the use of simulated data, with fair comparison to the power model supported by numerical and graphical illustrations. A new modelling strategy beyond the unit domain is also proposed and developed, with an application to a survival times data set.


Introduction
In recent years, there has been a significant increase in the development of unit distributions. They work on the modelling of a variety of phenomena involving unit data, such as proportions, probabilities, and percentages, among other things. There are also compositional data to consider (see Aitchison (1982)). The analysis of unit data, in particular, necessitates the development of parametric, semi-parametric, and regression models. Among the most useful unit distributions, there are the famous power (Po) distribution, unit gamma (UG) distribution established in Consul and Jain (1971), log-Lindley (LL) distribution proposed in Gómez-Déniz et al. (2014), unit Weibull (UW) distribution developed in  and later refined in Mazucheli et al. (2020), unit Gompertz (UG) distribution developed in Mazucheli et al. (2019), unit Birnbaum-Saunders (UBS) distribution studied in , log-xgamma (LXG) distribution established in Altun and Hamedani (2018), unit inverse Gaussian (UIG) distribution introduced in Ghitany et al. (2019), unit generalized half-normal (UGHN) distribution proposed in Korkmaz (2020), unit Johnson SU (UJSU) distribution established in Gündüz and Korkmaz (2020), log-weighted exponential (LWE) distribution developed in Altun (2020), unit Rayleigh (UR) distribution studied in Bantan et al. (2020), unit modified Burr-III (UMBIII) distribution examined in Haq et al. (2020), arcsecant hyperbolic normal (ASHN) distribution proposed in , unit Burr-XII (UBXII) distribution developed in , transmuted unit Rayleigh (TUR) distribution studied in  and unit half-normal (UHN) distribution created in Bakouch et al. (2021).
The majority of these unit distributions are the result of complex mathematical transformations of wellknown flexible distributions with larger domains (gamma, Lindley, Weibull, normal, half-normal, among others). Depending on the modelling goals, they have different complexity structures. In this article, beyond the transformation scheme, we propose a new and simple unit distribution combining the following interesting properties: (i) it depends on a single positive parameter, (ii) its probability density function (pdf) is defined as an original ratio of power and logarithmic functions, (iii) its pdf is increasing and can be highly asymmetric on the left, with different types of angular and J forms, which is a relatively uncommon property for a oneparameter unit distribution, (iv) it enjoys strong results in stochastic orders, also involving the Po distribution, (v) its cumulative distribution function (cdf) and hazard rate function (hrf) have a closed-form depending on well-referenced integral functions, (vi) simple expressions exist for diverse moments related quantities, such as ordinary moments, moment generating function and incomplete moments, (vii) simple expressions are found for the logarithmic and logarithmically weighted moments, which is a rare property in a unit distribution, (viii) the behavior of the moments skewness and kurtosis of the distribution are quite manageable and (ix) it has a high degree of applicability and can serve as generator for the creation of new statistical models. Theoretical results, graphics, and numerical works are used to describe these statements in detail. From a theoretical standpoint, the new and Po distributions are contrasted and discussed, revealing some relevant relationships. The new distribution can be referred to as an "extreme left skewed alternative" to the Po distribution because of immediate anaytical similarities and a large panel of J shapes in its pdf. Following that, a part is devoted to the inference of the proposed model. Extremely left-skewed data are used to perform a fitting analysis, which yields much better results than the Po model. Furthermore, based on the proposed unit distribution, new general and flexible models are developed and applied to a real data set. Finally, we would like to point out that the article contains a number of inequalities that are used as intermediates in the proofs but may be of interest on their own.
The reminder of the article is as follows. Section 2 defines the new unit distribution through its pdf and cdf, with important properties related to these functions. Section 3 is devoted to various moments related quantities. An example of statistical application is provided in Section 4. A new general modelling strategy is proposed in Section 5, beyond the unit domain. The article concludes in Section 6.

The unit power-log distribution
The proposed distribution is presented in this section.

Probability density function
The following proposition introduces a special function with pdf-like properties.
The distribution related to the pdf fα(x) is called the unit power-log (UPL) distribution or UPL(α) when the parameter α needs to be mentioned. That is, we say that a random variable X follows the UPL distribution over a probability space formally denoted by (Ω, F, P) if, for any set A into R, we have P(X ∈ A) = ∫︀ A fα(x)dx. The interesting properties of the UPL distribution are the objects of the rest of the study.
Although it is presumed in the sequel that α > 0, the case α ∈ (−1, 0) is not excluded, and will be briefly discussed in a separate subsection (see Subsection 2.5).
First of all, some basic properties of fα(x) are presented below.
-When α → 0, fα(x) tends to the pdf of the unit uniform distribution.
-The following asymptotic behavior of fα(x) at the boundaries holds: Proof.
-When x → 0, we have and, when x → 1, by using e y ∼ 1 + y when y → 0, we get This ends the proof of Proposition 2.2.  From Figure 1, we see that fα(x) is near angular for very small α, and increasing in all circumstances, illustrating the findings in Proposition 2.2. Also, we can notice that fα(x) can be concave, such as the green curve, and have tilde shapes or "concave then convex" shapes, such as the pink curve. It is clear that the UPL distribution is mainly left skewed, reaching some extreme behavior in this regard, such as the yellow or gray curves.
Some comparisons between the UPL and Po distributions are now formulated.
However, some stochastic connections behind fα(x) and kα(x) exist and will be presented in the next subsection.

Relevant stochastic order results
We now present relevant stochastic order results satisfied by the UPL distribution through the use of fα(x). We adopt the concept of likelihood ratio order as presented in Shaked and Shanthikumar (2007) in the continuous case. That is, let X and Y be two continuous random variables with pdfs f (x) and g(x), respectively, so that f (x)/g(x) decreases (or g(x)/f (x) increases) in x over the union of the domains of X and Y. Then, in the likelihood ratio order, X is smaller than Y.
Proposition 2.3. Let X be a random variable with the UPL(α 1 ) distribution and Y be a random variable with the UPL(α 2 ) distribution, with α 2 ≥ α 1 . Then X is smaller than Y in the likelihood ratio order.

Proof.
Proving that X is smaller than Y in the likelihood ratio order is equivalent to prove that the following ratio function is decreasing with respect to x: In order to study the sign of this function, let us set θ(u) = ux u−1 /(1 − x u ) for u > 0 and x ∈ (0, 1). Then, we have Owing to the following inequality: log(1 + y) < y for y ∈ (−1, 0), applied with y = x u − 1 ∈ (−1, 0), we have As a result, ∂θ(u)/∂u < 0 and θ(u) is strictly decreasing. That is, for Therefore, based on Equation (3), we have ∂qα 1 ,α2 (x)/∂x ≤ 0, so qα 1 ,α2 (x) is decreasing. It is established that the desired likelihood ratio order exists. Figure 3, where the ratio function fα 1 (x)/fα 2 (x) is plotted with α 1 = α, α = 0.5, 5, and α 2 = α + ϵ with various positive values for ϵ such that α 2 ≥ α 1 . We now link the UPL and Po distributions by using the likelihood ratio order.

Proposition 2.4. Let X be a random variable with the UPL(α) distribution and Y be a random variable with the Po(α) distribution. Then X is smaller than Y in the likelihood ratio order.
Proof. Let us prove that the following ratio function is decreasing with respect to x: With the same arguments to Equation (4), we have α log(x)+1− x α < 0 for x ∈ (0, 1), implying that ∂qα(x)/∂x < 0, so qα(x) is decreasing. The desired likelihood ratio order is provided.
Proposition 2.4 has several consequences on other characteristics of the UPL and Po distributions that will be presented along the article.

Cumulative distribution function
The following proposition presents the cdf of the UPL distribution through special integral functions that are well-referenced in the literature.

Proposition 2.5. The cdf of the UPL distribution is obtained as
Fα(x) = 0 for x ≤ 0 and Fα(x) = 1 for x ≥ 1, where Ei(x) and Li(x) denote the well-known exponential and logarithmic integrals, respectively, defined by Now, by the successive changes of variables t = e −y and z = (1 + α)y, we get We obtain the desired result by putting the above equalities together, concluding the proof of Proposition 2.5.
The numerical values of Ei(x), Li(x) and Fα(x) involve the use of standard numerical integration procedures available in every mathematical package. In this study, the software R is used (see R Core Team (2014)), combined with the package pracma and the function expint_Ei. Since fα(x) is strictly increasing with respect to x for x ∈ (0, 1) by Proposition 2.2, Fα(x) is a convex function only. Figure 4 illustrates the possible shapes of Fα(x) for several values of α.

On the quantile and hazard rate functions
The numerical values of the quantiles of the UPL distribution can be derived from the cdf; the quantile xu is obtained by solving the following non-linear equation with respect to u: Any mathematical software can be used to determine xu for a given value of α. The pdf and cdf of the UPL distribution allow us to define the hrf as and hα(x) = 0 for x / ∈(0, 1). At the boundaries, hα(x) has the following asymptotic behavior: Due to the complexity of the denominator term, the properties of this function are difficult to identify from an analytical point of view. However, a graphical analysis suggests that it is increasing, with diverse tilde or J shapes, as illustrated in Figure 5.

On the case α ∈ (−1, 0)
It should be noted that the pdf of the UPL distribution indicated in Proposition 2.1 is still valid for α ∈ (−1, 0). Certain properties of the UPL distribution are actually inverted in this situation. In particular, the following results are provable: f α (x) is stricly decreasing for x ∈ (0, 1), and the following limit at the origin holds: Also, the extreme left skewed property is no longer available.
hα(x) has diverse U shapes, and the following limit at the origin is valid: Some plots of fα(x), Fα(x) and hα(x) with selected negative values of α are presented in Figure 6, illustrating the results above. In addition, the following stochastic order result holds: Let X be a random variable with the UPL(α) distribution and Y be a random variable with the Po(α) distribution. Then Y is smaller than X in the likelihood ratio order. The contrary was shown for the case α > 0. More facts on the case α ∈ (−1, 0) may be the subject of an independent study, which we drop here for brevity.

Diverse moments
The moments of a distribution are essential to describe various of its characteristics and statistical capacities.
Here, diverse moments of the UPL distribution are investigated theoretically and practically.

Ordinary moments
The expressions of the ordinary moments of the UPL distribution are discussed in the following result.
Proposition 3.1. Let X be a random variable with the UPL distribution and r be a positive integer. Then, the rth ordinary moment of X is obtained as Proof. By the transfer formula, we have Let us now treat the integral term by following the spirit of the proof of Proposition 2.1. We consider the following integral function: with Φr(α) = 0 for α = 0. By virtue of the Leibnitz integral rule, we get Upon integrating with respect to α, we obtain Φr(α) = log(1 + α + r) + c for a certain constant c. Since Φr(0) = 0, we have c = − log(1 + r). Therefore By dividing by log(1 + α), the given formula for mr is obtained; the proof of Proposition 3.1 ends.
Alternative proof. The proof of Proposition 3.1 can be performed via series expansions. Indeed, it follows from the series expansion of the exponential function, the Lebesgue dominated convergence theorem, the following formula: where s denotes an integer and ν a real number such that ν > −1, which is a particular case of (Gradshteyn and Ryzhik, 2007, Equation 4.2726), and the series expansion of the logarithmic function, that For the final step, it is assumed that α < 1 + r, which is a technical constraint that the first proof does not have.
It is worth noting that Proposition 3.1 holds for every real number r such that r > −1.
The following result completes Proposition 3.1; it is about the monotonicity of mr with respect to α and r.
Proposition 3.2. By considering mr defined in Proposition 3.1 as a function of α and r, the following results hold: mr is a strictly increasing function with respect to α, mr is a strictly decreasing function with respect to r. Proof.
-Based on the expression of Equation (5), after some developments, we have In order to study the sign of this function, let us set ω(x) = (1 + x) log(1 + x)/x. Then we have Based on the following general logarithmic inequality: log(1 + y) < y for y > 0, we have ∂ω(x)/∂x > 0, implying that ω(x) is a strictly increasing function. In particular, for x < y, we have ω(x) < ω(y), which is equivalent to log(1 + x) log(1 + y) < x 1 + x 1 + y y .
By taking x = α/(1 + r) and y = α such that x < y, we get This inequality combined with Equation (7) gives ∂mr /∂α > 0, proving that mr is a strictly increasing function with respect to α.
-The second point is more immediate; we have implying that mr is a strictly decreasing function with respect to r. In fact, this property holds for the rth ordinary moment for any unit distribution in general.
The proof of Proposition 3.2 ends.
Based on Proposition 3.1, the mean and variance of X are obtained as respectively. The skewness and kurtosis coefficients of X are given as respectively. Table 1 presents some numerical values for these measures by taking several values of α.  Table 1 shows that the UPL distribution is mostly left skewed, with a lot of kurtosis flexibility. Proposition 3.2 is also illustrated for the considered values; it is clear that mr increases as α increases, and mr decreases as r increases. Also, from this table, we remark that σ 2 and 1 decrease as α increases, and β 4 increases as α increases, for the considered values of α.
From Proposition 2.4, the likelihood ratio order between the UPL and Po distributions implies that, for any integer r, where m o r denotes the rth ordinary moment of a random variable with the Po distribution. We refer the reader to Shaked and Shanthikumar (2007) for the details on this consequence.
The following proposition is about the moment generating function of the UPL distribution.

Proposition 3.3. Let X be a random variable with the UPL distribution. Then, the moment generating function of X is obtained as
Proof. The proof is an immediate consequence of the following formula: combined with Proposition 3.1.
Basically, the rth ordinary moment can be re-find by the following relation: mr = ∂ r M(t)/∂t r | t=0 . Also, the rth cumulant of X is given by κr = ∂ r {log M(t)}/∂t r | t=0 . In particular, we have κ 1 = m 1 , κ 2 is the second central moment of X and κ 3 is the third one.

Incomplete moments
The incomplete moment of the UPL distribution is now considered.
Proposition 3.4. Let X be a random variable with the UPL distribution, r be a positive integer and t ∈ (0, 1).

where I(A) denotes the indicator function over an event A.
Proof. By the transfer formula, we have Let us now treat the integral term as in the proof of Proposition 2.5. We consider the following incomplete integral function: with Φr(t, α) = 0 for α = 0. By virtue of the Leibnitz integral rule, we get Now, by applying the change of variables v = 1 + u + r then w = v log(t), we arrive at for a certain constant c. Since Φr(t, 0) = 0, we get c = 0. By putting the above equalities together, we have The proof of Proposition 3.4 ends.
Based on Proposition 3.4, one may re-find the cdf and ordinary moments of the UPL distribution by the following relations: Fα(x) = m 0 (x) and mr = lim t→1 mr(t), respectively. Also, we can define the normalized incomplete moment given by )︀]︀ , t ∈ (0, 1).
In full generality, the normalized incomplete moments given as ψ 0 (t) and ψ 1 (t) are involved in many applications and provide interesting distributional information. Also, they allow to define the famous Gini coefficient defined in our setting as Other important measures can be defined in a similar way, such as the Lorenz curve, Bonferroni curve and Pietra measure. In this regard, we refer to Butler and McDonald (1989) and, in a more general setting, Cordeiro et al. (2020).

Logarithmic moments
Logarithmic moments of a distribution occur naturally in various probability quantities such as entropy and concentration inequalities. Here, we provide the expressions of the logarithmic moments of the UPL distribution.

Proposition 3.5. Let X be a random variable with the UPL distribution and r be a positive integer. Then, the rth logarithmic moment of X is obtained as
Proof. By the transfer formula and Equation (6), we obtain The proof of Proposition 3.5 comes to an end.
From this result, the mean, variance, skewness and kurtosis of Y = log(X) can be expressed and studied.

Logarithmically weighted moments
As another interesting property, the logarithmically weighted moments of the UPL distribution have simple analytical expressions. They are presented below.
Proposition 3.6. Let X be a random variable with the UPL distribution and r be a positive integer. Then, the rth logarithmically weighted moment of X is obtained as m † r = E(X r log(X)) = − α log(1 + α) 1 (r + α + 1)(r + 1) .

Proof. By the transfer formula and classical integration techniques, we get
The proof of Proposition 3.6 ends.
To our knowledge, the simple expression of the logarithmically weighted moments remains a rare property for a unit distribution. Another remark is that, from Proposition 3.6, for any positive integer r, we have In particular, by substituting r = 0, we have the following simple relation: Therefore, the parameter α is fully determined by the logarithmically weighted moments. This property can be useful for the estimation of α in a statistical setting via a moment estimation method-like.

Application
In this section, we show how the UPL distribution can be used in a statistical context.

Estimation
The inference of the UPL model can be made by using the maximum likelihood method. The book of Casella and Berger (1990) contains a detailed description of the method as well as its statistical benefits. Let x 1 , . . . , xn be n independent values from a random variable with the UPL(α) distribution, implying that x i ∈ (0, 1) for i = 1, . . . , n, among others. It is supposed that α is unknown and must be estimated via x 1 , . . . , xn. Then, we estimate α by the maximum likelihood estimate (MLE)α defined aŝ where ℓ(α) denotes the log-likelihood function with respect to α. Mathematically, we can write The estimateα also satisfies the following non-linear equation: Then, by applying the well-known theory on the maximum likelihood method, for n large enough, the distribution of the random estimator behindα can be approximated by the normal distribution with mean α and variance V = (︀ −∂ 2 ℓ(α)/∂α 2 )︀ −1 , that is The knowledge of this limiting distribution is useful to construct important statistical objects, such as asymptotic confidence intervals and likelihood tests. On the other hand, based onα, the estimations of the unknown pdf fα(x) and cdf Fα(x) are given byf (x) = fα(x) andF(x) = Fα(x), respectively. These two estimated functions will be investigated below with an example of data analysis.

Example of data analysis
In order to highlight the interest of the UPL model, we deal with an arbitrary data set containing values of extreme left skewed nature. Thus, we generate 50 values of the random variable X = 1/(1 + Y), where Y follows the Pareto distribution specifies by the following pdf: and p η,θ (x) = 0 for x < η, with η = 0.05 and θ = 0.5. The obtained data set is presented in Table 2. With these data, based on the methodology described in Subsection 4.1 for the UPL model and the use of the function nlminb of the software R, the MLE of α is obtained asα = 9.965334. Also, with this model, -the estimated pdf underlying the data is given aŝ -based on Proposition 2.5, the estimated cdf underlying the data is given aŝ )︀ x ∈ (0, 1).
In addition, fromα and the log-likelihood function defined in Equation (8), we may derive the following well-established criteria: Akaike information criterion (AIC), corrected Akaike information criterion (AICc) and Bayesian information criterion (BIC) defined as AIC = −2ℓ(α) + 2k, AICc = AIC + 2k(k + 1) n − k − 1 , BIC = −2ℓ(α) + k log(n), respectively, where k denotes the numbers of parameters and n denotes the number of data. Here, we have k = 1, n = 50 and, after calculus, ℓ(α) = 17.01861. Hence AIC = −32.03721, AICc = −31.95388 and BIC = −30.12519. As model comparison, we consider the Po model with parameter α, with the same estimation strategy. We thus obtainα = 0.8852462, and, according to the Po model, -the estimated pdf underlying the data is given aŝ -the estimated cdf underlying the data is given aŝ We also have ℓ(α) = 8.224661, k = 1 and n = 50, and we get AIC = −14.44932, AICc = −14.36599 and BIC = −12.53730. Since it has the smallest AIC, AICc and BIC, the UPL model is considered as the best. Figure 7 plots the curves off (x) in red andk(x) in blue over the histogram of the data, and Figure 8 plots the curves ofF(x) in red andK(x) in blue over the empirical cdf of the data.  Figures 7 and 8, it is evident that the best fit of the data is given by the UPL model; the Po model has missed the concentration of the data in the neighborhood of 1. This illustrates that the UPL model is more suitable for the fit of extreme left skewed data in comparison to the former Po model. This justifies the importance of the UPL model in this regard.

Histogram of the data with fits
The UPL distribution can be used to define more general models, with different domains, as described in the next section.

A note on the UPL-G models
Following the spirit of Cordeiro et al. (2020), the UPL distribution can be used to generate a plethora of new distributions by applying the compounding scheme. This direction of work is developed below.

UPL-G family
Let us consider a parental distribution with cdf G ξ (x) and pdf g ξ (x), where ξ symbolized a vector of parameters. Then, based on Equation (1), we define the corresponding UPL-G distribution by the following pdf: That is, in an expanded form, we have Natrually, the support of the related UPL-G distribution coincides with the one of the parental distribution. To our knowledge, such power-log distributions have not been the object of study in the existing literature. Also, based on Proposition 2.5 and the compounding scheme, the corresponding cdf is obtained as Examples of UPL-G distributions are given in Table 3 by their pdfs. An example of the application of a such UPL-G distribution is presented below.

The UPLE model
We now provide an application to the UPLE distribution, or UPLE(α, θ) distribution, as introduced in Table 3, and defined with the following pdf: and f α,θ (x) = 0 for x ≤ 0, with α > 0 and θ > 0. The corresponding cdf is obtained as and F α,θ (x) = 0 for x ≤ 0. As a result of the power-logarithmic mechanism, the UPLE distribution defines a new two-parameter extension of the exponential distribution. In a typical fit scenario, the parameters α and θ are assumed to be unknown, and we need to estimate them by using data. This parametric estimation can be performed by the maximum likelihood approach as described below. Let x 1 , . . . , xn be n independent values from a random variable X with the UPLE(α, θ) distribution, representing the observed data. Then, the MLEsα andθ are defined by (α,θ) = argmax (α,θ)∈(0,+∞) 2 ℓ(α, θ), where ℓ(α, θ) denotes the log-likelihood function with respect to α and θ. Mathematically, ℓ(α, θ) is given by The theory and practice of ensuring the effectiveness of MLEs are well known, and we refer to Casella and Berger (1990) in this regard once more.
Also, after calculus, we have ℓ(α,θ) = −580.0936, and we get AIC = 1164.187, AICc = 1164.289 and BIC = 1169.779. Since it has the smallest AIC, AICc and BIC, the UPLE model is the best. Also, one can remark that it has the smallest AIC, AICc and BIC in comparison to those of the three-parameter models considered in Section 7 of Ramos et al. (2013) (see (Ramos et al., 2013, three last columns of Table 1)). Figure 9 plots the curves off (x) in red andk(x) in blue over the histogram of the data. Furthermore, the curves ofF(x) in red andK(x) in blue are plotted over the empirical ecdf of the data in Figure 10.
In view of Figures 9 and 10, the best fit is attributed to the UPLE model, showing that the UPL-G scheme can outperform some classical estimation schemes.

Conclusion
In this article, we have focused on a new unit distribution called the unit power-logarithmic (UPL) distribution, which is defined by an original power-log pdf. Several of its important characteristics, such as likelihood ratio order results, an extreme asymmetry to the left, closed-form expressions for the cdf, hrf, ordinary moments, skewness, kurtosis, moments generating function, incomplete moments, logarithmic moments and logarithmically weighted moments have been determined using analytical and graphical tools. On several important characteristics, we compared the UPL with the so-called power distributions, demonstrating some non-trivial relationships. A data-handling application is provided. Then, with an application to a well-known survival data set, a more general modelling strategy based on the UPL distribution is discussed.
With an extensive theoretical study and some statistical basics, this article lays the groundwork for the UPL distribution. Some future directions of research include: (i) an in-depth analysis of the hrf, (ii) additional stochastic order properties, such as mean residual life order, dispersive order, and right-spread order, and (ii) the construction of new regression models (linear or quantile) or bivariate extensions for multivariate analysis.