Study of a Modiﬁed Kumaraswamy Distribution

: In this article, a structural modiﬁcation of the Kumaraswamy distribution yields a new two-parameter distribution deﬁned on ( 0,1 ) , called the modiﬁed Kumaraswamy distribution. It has the advantages of being (i) original in its deﬁnition, mixing logarithmic, power and ratio functions, (ii) ﬂexible from the modeling viewpoint, with rare functional capabilities for a bounded distribution— in particular, N-shapes are observed for both the probability density and hazard rate functions— and (iii) a solid alternative to its parental Kumaraswamy distribution in the ﬁrst-order stochastic sense. Some statistical features, such as the moments and quantile function, are represented in closed form. The Lambert function and incomplete beta function are involved in this regard. The distributions of order statistics are also explored. Then, emphasis is put on the practice of the modiﬁed Kumaraswamy model in the context of data ﬁtting. The well-known maximum likelihood approach is used to estimate the parameters, and a simulation study is conducted to examine the performance of this approach. In order to demonstrate the applicability of the suggested model, two real data sets are considered. As a notable result, for the considered data sets, statistical benchmarks indicate that the new modeling strategy outperforms the Kumaraswamy model. The transmuted Kumaraswamy, beta, unit Rayleigh, Topp–Leone and power models are also outperformed.


Introduction
The perfect modeling of random processes that one regularly observes in applied sciences remains in the order of utopia. Scientists and practitioners have already discussed a wide range of models for solving these challenges. When the processes take values in the interval (0, 1), a primary statistical analysis can be performed with the standard beta and Kumaraswamy (Kw) models. Before going further, a retrospective on these models is necessary. First, they are derived from the beta and Kw distributions, respectively. The beta distribution is a continuous distribution with support on the interval (0, 1) and parameterized by two positive shape parameters, α and β. It is mathematically defined by the following cumulative distribution function (cdf): 1 0 t α−1 (1 − t) β−1 dt, F beta (x; α, β) = 0 for x ≤ 0, and F beta (x; α, β) = 1 for x ≥ 1. Naturally, the values of α and β influence the shape of the distribution. The corresponding probability density function (pdf) can be U-shaped, bell-shaped, monotonic (increasing or decreasing), or even straight lines. On the other hand, the corresponding hazard rate function (hrf) can be increasing with convex shapes, or U-shaped (see [1,2]). In a wide number of areas, the beta distribution is used to model the behavior of characteristics representing percentages and proportions. See, for example, the developments in [3].
In [4], the Kw distribution was introduced to complement the beta distribution. Thus, it is also a continuous distribution with support in (0, 1) and parameterized by two positive shape parameters, α and β. The Kw distribution is conceptually based on uniform order statistics, and it has extremely simple functions that do not rely on special functions. The cdf of the Kw distribution is given as follows: where F Kw (x; α, β) = 0 for x ≤ 0, and F Kw (x; α, β) = 1 for x ≥ 1. The form of the functions related to this distribution is naturally influenced by the values of α and β. Its application domain is identical to that of the beta distribution. As a brief comparison, the Kw distribution is similar to the beta distribution, but it is much easier to work with thanks to the tractability of its functions. We refer the reader to [5] and the references therein for more information on this claim. As a fact, the beta and Kw models rarely achieve the goal of optimal performance in all cases. For these reasons, new models on the unit intervals have been proposed, derived from new distributions with support equal to (0, 1). These distributions are more or less sophisticated in the analytical sense. Contemporary works on this topic can be found in [6][7][8][9][10][11][12][13][14][15][16][17][18][19].
In this article, we propose a new two-parameter unit distribution, called the modified Kumaraswamy (MKw) distribution. The considered cdf is such that, for x ∈ (0, 1), the following holds: where T(x) denotes a certain analytical transformation that will be presented later. At this introductory step, some features of T(x) are listed below.
Feature I. First of all, the transformation T(x) has an original form involving polynomial and logarithmic functions without additional parameters. Thanks to it, some functional capabilities of the Kw distribution are extended or modified, and can be more adapted in some practical situations. In particular, the pdf of the MKw distribution can be monotonic in a nearly angular way, "strongly" left-or right-skewed and of a leptokurtic nature, or N-shaped. To our knowledge, these properties are not observed for the pdf of the Kw distribution. On the other hand, the corresponding hrf presents a wide panel of non-monotonic shapes, including N-shapes, that the hrf of the Kw distribution does not possess. All these curvature differences give credit for a more in-depth investigation of the MKw distribution from the modeling viewpoint.
Feature III. The transformation T(x) satisfies the following inequality: T(x) ≥ x for any x ∈ (0, 1). Thanks to it, the Kw and MKw distributions are involved in a simple FOS dominance result: F MKw (x; α, β) ≤ F Kw (x; α, β) for any x ∈ R. This means that the MKw distribution generates a two-parameter statistical model, which does not intersect with the Kw model in the cdf sense.
In fact, several mathematical interpretations of the MKw distribution are possible. For instance, the MKw distribution may be viewed as a special parametric weighted version of the Kw distribution. Additionally, it is deeply connected with the ratio powerlogarithmic (RPL) distribution created by [20]. Thanks to this connection, some properties of the RPL distribution may be used to determine those of the MKw distribution, among other things. These diverse interpretations are detailed in the first part of the paper, along with some theoretical properties. These properties include detailed quantile and moment analyses, illustrated numerically and graphically for some selected values of α and β. The order statistics are also discussed. The second part is devoted to the modeling characteristics of the MKw distribution. The MKw model is considered, assuming that the parameters α and β are unknown, and may be estimated via available data. We examine the estimation of these parameters by the maximum likelihood method. We show the efficiency of the obtained estimates by the use of simulated data. Then, two different data sets are analyzed, and comparisons are made between the fits of the NKw, Kw, and beta models and four other referenced models of the literature. We show that the proposed MKw model outperforms the concurrence through the use of standard statistical benchmarks. The findings are illustrated by numerous graphics.
The organization of the paper is composed of the following section. Section 2 introduces the MKw distribution with full details, along with a functional analysis of the pdf and hrf. Section 3 is devoted to quantile analysis and moment analysis. The statistical inference on the parameters is considered in Section 4. Section 5 is devoted to applications. We end the paper with a concluding part in Section 6.

The Modified Kw Distribution
This section focuses on the MKw distribution and its basic functional features.

Primary Definition
The following mathematical result is at the basis of the MKw distribution. Proposition 1. Let α > 0 and β > 0, and G(x; α, β) be the function defined on R by G(x; α, β) = 0 for x ≤ 0, and G(x; α, β) = 1 for x ≥ 1. Thus defined, G(x; α, β) has the required properties of a continuous cdf.
Proof. First, it is clear that G(x; α, β) is continuous on R/{0, 1}. Let us examine its behavior at the particular points x = 0 and x = 1. At the neighborhood of x = 0, by virtue of standard equivalence functions, we have the following: and, at the neighborhood of x = 1, we have Therefore, G(x; α, β) is continuous on {0, 1}, and on R. By construction, we have lim x→−∞ G(x; α, β) = 0 and lim x→+∞ G(x; α, β) = 1. Let us now prove that G(x; α, β) is increasing. By applying standard rules of differentiation, for x ∈ (0, 1), we obtain the following: The famous logarithmic inequality ln(1 + y) ≥ y/(1 + y) for y > −1, with y = (1 − x β ) α − 1 > −1 yields the following: This inequality can be rearranged as Therefore, the numerator in Equation (1) is negative. Since the denominator is also negative, we have G (x; α, β) ≥ 0, implying that G(x; α, β) is increasing. This concludes the proof of Proposition 1.
To the best of our knowledge, the cdf presented in Proposition 1 is new in the literature. It defines a modified version of the Kw distribution, described as the MKw distribution, discussed in Section 1. Some mathematical interpretations of this distribution are important to understand why the term "modified" is employed here. This is developed in the next section.

Mathematical Interpretations
Thus, the cdf of the MKw distribution is defined by the following: where F MKw (x; α, β) = 0 for x ≤ 0, and F MKw (x; α, β) = 1 for x ≥ 1, with α > 0 and β > 0. Some mathematical interpretations of this cdf are given below.

Interpretation 1
As sketched in Section 1, for x ∈ (0, 1), we can write the following: with T(x) = (x − 1)/ ln(x). In this sense, the transformation T(x) modifies the mathematical structure of the cdf of the Kw distribution for different modeling perspectives.
The following facts can be deduced from the well-known logarithmic inequality ln(1 + y) ≥ y/(1 + y) for y > −1 when applied to y = x − 1: This implies that T(x) is an increasing function with respect to x. Following this is the stochastic FOS property: for any 0 < α 1 ≤ α 2 and 0 < β for any x ∈ R. In other words, there is a direct FOS dominance of the MKw distribution over the Kw distribution.
At the neighborhood of x = 0, we have T(x) ∼ −1/ ln(x) → 0. This convergence is slow, compared to the polynomial convergence to 0. This asymptotic result plays an important role to understand the possible difference in the right skewness between the MKw and Kw distributions; without more investigation, from the statistical viewpoint, it is natural to think that the MKw model is more able to capture the slow right-decay of a phenomena that the Kw model. This will be confirmed later with a skewness analysis through quantile and moment methods, as well as practical data.
Interpretation 2 As a twin viewpoint, for x ∈ (0, 1), we can write the following: . Thus, the weight function w(x; α, β) modulate the main polynomial term of the cdf of the Kw distribution. In this sense, the MKw distribution is a weighted version of the Kw distribution.

Interpretation 3
There is a deep relationship between the MKw and RPL distributions. First, we recall that the RPL distribution is defined by the following logarithmicpower cdf: Then, for a random variable Y following the RPL distribution, the random variable X = (1 − Y) 1/β follows the MKw distribution. Indeed, for x ∈ (0, 1), we have the following: By identification, the stated result is obtained. Thanks to the representation of X, some properties of the RPL distribution can be used to determine those of the MKw distribution.
Finally, because any random variable X defined on (0, 1) can be shifted to a bounded support of the form (a, b) with a < b, using the transformation (b − a)X + a, the MKw distribution can easily be extended to any bounded domain.

Functional Analysis
This part is devoted to a functional analysis of the main functions of the MKw distribution. To begin, we recall that the cdf of the MKw distribution is given by Equation (3). As any cdf, it is an increasing function. Some asymptotic results are now presented. In the neighborhood of x = 0, by virtue of standard equivalence functions at orders of 1 or 2, we have the following: Thus, F MKw (x; α, β) converges to 0 with a polynomial decay. In the neighborhood of x = 1, we have the following: This implies that F MKw (x; α, β) converges to 1 with a slow rate of convergence, due to the logarithmic term.
Let us now focus our attention on the pdf and hrf of the MKw distribution. First, the pdf is given by the following: and f MKw (x; α, β) = 0 for x ∈ (0, 1). Because of its analytical complexity, this function is difficult to investigate using ordinary mathematical techniques (derivatives, . . . ). We thus propose to study its asymptotic behaviors, then provide a graphical analysis to reveal its possible shapes. At the neighborhood of x = 0, with similar arguments as those used for the cdf, we have the following: From this result, it comes that f MKw (x; α, β) explodes to +∞ for β < 1, is equal to the constant α/2 for β = 1, and tends to 0 for β > 1. The values of β are thus of particular importance for the limit points. At the neighborhood of x = 1, we have the following: Thus, f MKw (x; α, β) explodes to +∞ for all the values of α > 0 and β > 0, more or less slowly. We complete this first approach by a graphical analysis. Figure 1  From Figures 1, we remark that f MKw (x; α, β) can be decreasing, unimodal with all the possible skewed directions, and also can be U-and N-shapes, which remains a rare property for a distribution supported on (0, 1). In particular, the almost angular monotonicity, "strongly" left-or right-skewed and N shape properties are not immediate for the Kw distribution. To this first graphical analysis, we propose to show the versatility of f MKw (x; α, β) via three-dimensional (3D) plots in Figures 2 and 3, with moving α for Figure 2, and moving β for Figure 3. The hrf of the MKw distribution is given by the following: and h MKw (x; α, β) = 0 for x ∈ (0, 1). This function is difficult to explore using traditional mathematical tools, due to its analytical complexity. We proceed in the same manner as for f MKw (x; α, β). In the neighborhood of x = 0, we have the following: As a result, h MKw (x; α, β) explodes to +∞ for β < 1 (and thus cannot be increasing), is equal to the constant α/2 for β = 1, and tends to 0 for β > 1. The values of β are thus of particular importance for the limit points. In the neighborhood of x = 1, we obtain the following: Hence, h MKw (x; α, β) exploses to +∞ for all the values of α > 0 and β > 0. A graphical analysis now completes these results, with a focus on the possible shapes of h MKw (x; α, β). The maximum number of distinct shapes of h MKw (x; α, β) can be seen in Figure 4.  From Figure 4, it is clear that h MKw (x; α, β) can be increasing with various convex properties, and can be non-monotonic with N-or U-shapes. Again, these remain as rare features for a distribution supported on (0, 1). For instance, the hrf of the Kw distribution does not possess an N shape.
To illustrate the adaptability of h MKw (x; α, β), we propose to use 3D plots in Figures 5 and 6, with moving α for Figure 5 and moving β for Figure 6.

Mathematical Analysis
The MKw distribution is now mathematically treated, with emphasis on the quantile and moment features.

Quantile Analysis
The fact that the cdf of the KMw distribution is invertible is one of its most notable features. The quantile function (qf) can be expressed in closed form in terms of the principal branch of the Lambert function, as described in the next proposition. We may refer to [21] for further details on the Lambert function. Proposition 2. The qf of the MKw distribution is defined by the following: where W 0 (x) denotes the principal branch of the Lambert function.
. Therefore, we can determine it by solving the following nonlinear equation: F MKw (y; α, β) = u with respect to y. We proceed as follows: We obtain the expected result.
The expression of the qf is essential to determine the main quartle of the MKw distribution, such as the median defined by the following: Similarly, the first and third quartiles of the MKw distribution are given by . These special quantile values can serve as the main tool to describe the skewness and kurtosis of the distribution. For instance, we can measure the skewness via the quantile coefficient of Bowley defined by the following: On the other hand, we can measure the kurtosis by the quantile coefficient of Moors defined by the following: The sign of QS indicates the skewness of the MKw distribution: if QS > 0, the distribution is right-skewed; if QS < 0, the distribution is left-skewed; and if QS approaches 0, the distribution is nearly symmetric. Concerning the coefficient of Moors, we can compare it to the value of 1.233, which corresponds to the quantile coefficient of Moors associated with the standard normal distribution. As a result, if QK < 1.233, the distribution is platykurtic; if QK > 1.233, the distribution is leptokurtic; and if QK approaches 1.233, the distribution is mesokurtic.
The 3D plots for QS and QK for varying α and β are displayed in Figure 7. From Figure 7, we see that the behavior of the quantile coefficients of skewness and kurtosis is consistent with what was determined in Figures 1-3: the MKw distribution exhibits all sorts of skewness since QS might be positive, almost equal to 0, or positive. Furthermore, the fact that QK can be less than, nearly equal to, or larger than 1.2333 demonstrates that all kurtosis levels are attained. In addition, the expression of the qf is particularly useful for simulations. Indeed, pseudo-random data from the MKw distribution may be easily computer-generated using the inverse transform approach. Since the Lambert function is accessible in computer systems such as Maple, MATLAB and R, this approach is quite feasible. Further details on the above quantile analysis can be found in [22].

Moment Analysis
This section is devoted to the moment properties of the MKw distribution, beginning with the raw moments. Hereafter, we designate by X a random variable with the MKw distribution.

Raw Moments
For any integer r, the rth raw moment of X can be expressed in an integral form as follows: This integral can be implemented in any mathematical software, and can be calculated numerically for given values of α and β. Some mathematical developments of this integral are proposed in the next proposition. Proposition 3. Two representations of the rth moment of X are given below.

Representation 1
We can express m(r) in terms of the beta function as the following:

Representation 2
We can express m(r) as the following series expansion: Proof. Let us prove the two representations in turn.

Representation 1
First, by an integration by part, we have the following: where with ψ(α, r) = 0 for α = 0. Owing to the Leibnitz integral rule and the change of variables y = x β , we have the following: Since ψ(α, r) = 0 for α = 0, upon integration with respect to α, we obtain the following: The desired result follows by substituting Equation (5) in Equation (4).

Representation 2
We recall that, in the distribution sense, X can be written as X = (1 − Y) 1/β , where Y follows the RPL distribution. Hence, for any positive integer r, the general binomial theorem yields the following infinite series representation: This expansion is valid almost everywhere; the event {Y = 1} is of probability zero. Now, let us present a known result (see [20] (Proposition 6)). We obviously have E(Y 0 ) = 1, and, for any integer k ≥ 1, the kth raw moment of Y is obtained as the following: Therefore, based on Equation (6) and the linearity of the expectation operator, we have the following: The stated result is obtained.
The proof of Proposition 3 ends.
By arranging the raw moments in a suitable manner, we can determine the standard central, dispersion, skewness, and kurtosis measures of X. These measures include the mean given by m (1) 4 Var 2 .
The interpretation of the moment skewness and kurtosis is similar to those of the quantile moment and kurtosis. The only difference is that, for the kurtosis, the benchmark value of the moment coefficient of kurtosis is 3 instead of 1.233 for the quantile coefficient of kurtosis.
In addition, we can consider the coefficient of variation obtained as CV = Var 1/2 /m(1) and the index of dispersion given as ID = Var 1/2 CV. Table 1 Table 2 does the same with β = 3 and varying α. From both Tables 1 and 2, we observe that the mean is monotonic with respect to α and β, which is not the case for the variance. Furthermore, wide variations of values are seen for the mean, whereas the variance remains relatively stable and small. The behavior of the moment coefficients of skewness and kurtosis are consistent with what is concluded in Figures 1-3, as well as with the use of quantile coefficients of skewness and kurtosis; because MK can be positive, nearly equal to 0 or positive, the MKw distribution has all types of skewness. Moreover, since MK can be lower, almost equal or greater than 3, it indicates that all the kurtosis states are reached. In addition, we see a strong versatility in the numerical values of CV and ID.
All these comments are illustrated graphically in Figure 8 via 3D plots with varying α and β. In the next part, we complete this moment analysis with the expression of the mean log moments, and the incomplete moments of X.

Other Kinds of Moments
The mean logarithm of X can be expressed in an integral form as follows: This kind of moment appears in some estimation methods, and entropy. It has the advantage of having values in R, as opposed to m(1), which has values in (0, 1). This can be of interest for the construction of regression models, among other things. The integral defining m ln may be determined numerically for given values of α and β and can be implemented in any mathematical software. In the following result, the mean logarithm of X is stated as a series expansion. Proposition 4. The mean logarithm of X is the following: Proof. For this proof, we adopt the setting used in the proof of Representation 2 of Proposition 3. By using the relationship between the MKw and RPL distributions, and the logarithmic series expansion, we obtain the following: The desired expansion is proved. Now, let y ∈ [0, 1] and X y be the random variable equal to X if X ≤ y, and 0 otherwise. Then, the rth incomplete moment of X truncated at y can be expressed as the following: The incomplete moments are involved in a lot of probabilistic measures and functions, such as various mean deviations, residual life-type functions and curves. The integral defining m(r)[y] may be determined numerically for given values of α and β and can be implemented in any scientific software. The following proposition proposes a mathematical development of this integral.

Proposition 5.
Two representations of the rth incomplete moment of X truncated at y are given below.

Representation 1
We can express m(r)[y] in terms of the incomplete beta function as the following: Proof. Let us prove the two representations in turn.

Representation 1
First, by an integration by part, we obtain the following: where with ψ(α, r)[y] = 0 for α = 0. By applying the Leibnitz integral rule and the change of variables z = x β , we obtain the following: Since ψ(α, r)[y] = 0 for α = 0, upon integration with respect to α, we obtain the following: Substituting Equation (8) in Equation (7) yields the desired result.

Representation 2
To begin, let us introduce a known result (see [20] (Proposition 9)). The kth incomplete moment of a random variable Y with the RPL distribution at y with y ∈ [0, 1] is given by the following: Hence, by using Equation (6), we have the following: The stated result is obtained.
The proof of Proposition 5 ends.
The remainder of the work is devoted to the creation of a new data analysis paradigm based on the MKw model.

Order Statistics
In this section, we cover the fundamentals of the order statistics of the MKw distribution. Let n be a positive integer and X 1 , · · · , X n be independent and identically distributed random variables following the MKw distribution. We consider the random variables X 1:n , . . . , X n:n defined as the ordered versions of X 1 , · · · , X n , such that P(X 1:n ≤ . . . ≤ X n:n ) = 1. Then, for any j = 1, 2, . . . , n, the pdf of X j:n is given as the following: That is, in an expanded form, we have the following: and f j:n (x; α, β) = 0 for x ∈ (0, 1). In particular, if we focus on the two extreme order statistics, the pdf of X 1:n is given by and f 1:n (x; α, β) = 0 for x ∈ (0, 1), and the pdf of X n:n is given by the following: and f n:n (x; α, β) = 0 for x ∈ (0, 1). Moment analysis is possible using these formulas. However, due to the functional complexity of the pdfs, numerical evaluations appear to necessitate the employment of computer mathematical tools.

Statistical Inference
The parameters α and β of the MKw distribution are now considered to be unknown and must be calculated using data. As a result, the MKw model is examined.

Estimation Method
The maximum likelihood (MLL) technique is well adapted for this purpose. The specifics of this estimation strategy are outlined below. Let x 1 , x 2 , . . . , x n symbolize the n values lying into (0, 1) that are predicted to be seen from a random variable X with the MKw distribution and unknown parameters α and β. Then, the log-likelihood function for α and β is formulated as the following: The MLL estimates (MLLEs) of α and β, denoted byα andβ, are defined by the following: (α,β) = argmax (α,β)∈(0,+∞) 2 L(α, β).
A two-dimensional normal distribution with the mean (α, β) and matrix of covariance could be used to identify the asymptotic distribution of the random version of (α,β). To be more specific, the matrix of covariance is given by D = I(α, β) −1 | (α,β)=(α,β) , where the following holds: The matrix D can be computed quantitatively using mathematical techniques. We can establish estimated confidence intervals (CoIs) for α and β at a particular level, such as 100(1 − q)% with q ∈ (0, 1). The associated lower bounds (CoI-LBs) and upper bounds (CoI-UBs) of such intervals remain traditional; if we keep our eyes on the unknown parameter α, these interval bounds are provided by CoI-LB =α − z q SEr α and CoI-UB =α + z q SEr α , where z q is defined by P(|Z| ≥ z q ) = q, with Z a random variable having the standard Gaussian distribution, and SEr α is a term that relates to the standard error (SEr) ofα, determined by the square root of the first diagonal element of D. For more theoretical and practical achievements of the MLLEs, see [23].

Simulation
Here, we perform simulation work to measure the behavior of the constructed MLLEs. In this situation, we employ the R program, which was created by [24]. The following steps are taken into consideration:

1.
We apply the inverse transform approach to produce 10,000 random samples with values x 1 , x 2 , . . . , x n . That is, for any i = 1, . . . , n, the value x i is computed as x i = Q MKw (u i ; α, β), where u i is a value generated from the uniform distribution on (0, 1).

5.
Tables 3-9 show the outcomes that were achieved.    From the tables above, we see that the values of MLLEs are near to the values of the parameters provided in the settings as n rises. Furthermore, when n rises, the MSEr and CoI-LEN values fall, as one would anticipate.

Application
It is critical to demonstrate how the MKw model may be used in practice. This is the aim of this section.

Method
In the coming applications, the fits of the MKw model are compared with those of some competitive models, such as the transmuted Kumaraswamy (TKw), Kumaraswamy (Kw), beta (B), unit Rayleigh (UR), Topp-Leone (Topp), and Power models. The TKw model has three parameters, whereas the other models have two parameters only. These models are defined through their cdfs in Table 10. Table 10. Competent models with the MKw model.

Models
Cdfs References In order to compare these models, we consider well-referenced statistical measures, namely the Cramér-Von Mises (W), Anderson-Darling (A), Kolmogorov-Smirnov (KS), and Akaike information criterion (AIC) based on the minus maximal likelihood value, denoted by −ˆ . The best model is the one with the smallest values of these statistics and criteria. We refer to [26] for more information on the usage and underlying meaning of the measures W, A, KS and AIC. In addition, we extract the p-value associated with the KS test. The best model is the one with the largest p-value.
As for the simulated part, the results are obtained using the R software, and two real data sets are considered.

Maximum Flood Level Data Set
The first data set comes from [27]  Theoretically, in view of the shape ability of the related pdf, the MKw model is able to fit these data.
Following our methodology, we first determine the MLLEs and the related SEs of the considered models in Table 11. From Table 11, if we focus on the MKw model, we see thatα = 354.0108 andβ = 5.9002. By using these estimates, we derive function estimates for the pdf and cdf of the MKw model through the substitution method; f MKw (x; α, β) and F MKw (x; α, β) may be naturally estimated by f MKw (x;α,β) and F MKw (x;α,β), respectively.
The models are compared via our statistical benchmarks in Table 12.  Table 12, it is clear that the MKw model is the best, followed by the TKw model. Indeed, the MKw model has the smallest AIC, W, A, KS, and the largest p-values among all the considered models. We illustrate this claim through a graphical approach. Figure 9 plots the estimated pdfs and cdfs over the appropriate empirical objects. Based on Figure 9, we see that the fits of the MKw model are the best; the estimated pdf has well captured the form of the histogram, and well detected the lack of observations over the interval [0.5, 0.6]. Furthermore, the estimated cdf is very close to the scale shape of the empirical cdf.
First, we determine the MLLEs and the related SEs of the considered models in Table 13. Table 13 shows thatα = 11.6229 andβ = 0.9719 are the MLLEs for the MKw model. By using these estimates, we obtain the estimated pdf and cdf of the MKw model through the substitution method. Table 14 gives the necessary values to compare the models.  Table 14 clearly shows that the MKw model is the best; it has the more desirable AIC, W, A, KS, and p-values of all the models studied. We use a graphical way to demonstrate this assertion. The estimated pdfs and cdfs are plotted over the corresponding empirical objects in Figure 10. According to Figure 10, the MKw model fits best; the predicted pdf captures the form of the histogram quite well and has taken into account the values close to 1, contrary to the other models. Furthermore, the scale shape of the empirical cdf is extremely similar to the scale shape of the estimated cdf.

Conclusions
In this paper, a new distribution modifying the functional capabilities and mathematical structure of the Kumaraswamy distribution is proposed. We have called it the modified Kumaraswamy distribution. Evidence shows that it is more efficient than the Kumaraswamy distribution for the modeling of phenomena with data presenting a histogram that is monotonic in a nearly angular way, "strongly" left-or right-skewed, and of a leptokurtic nature, or N-shaped. On the stochastic plan, the modified Kumaraswamy distribution first order stochastic dominates the Kumaraswamy distribution. It is also deeply connected with the so-called ratio power-logarithmic distribution. In the first part, we have exhibited its main quantile and moment properties, with mathematical results, numerical tables, and graphics. These elements have offered the necessary comprehension of the distribution for further theoretical and practical purposes. In the statistical plan, an inferential methodology of the related model is developed. Then, two different data sets are analyzed with the proposed model, and other models of references: the Kumaraswamy, beta, unit Rayleigh, Topp-Leone and power models. The results are quite favorable to our modeling strategy. This study is thus encouraging for the use of the modified Kumaraswamy model for other statistical applications of importance in fields in full expansion, such as medicine, finance, biology, and environmental sciences. The development of likelihood inferential methods that take into account censored data, extensions to the multivariate case, incorporation of time series, spatial, and quantile regression structures in the modeling, and the development of influence diagnostic tools are some future research directions.