A New Family of Generalized Distributions on the Unit Interval: The 𝑻 − 𝐊𝐮𝐦𝐚𝐫𝐚𝐬𝐰𝐚𝐦𝐲 Family of Distributions

The so-called Kumaraswamy distribution is a special probability distribution developed to model doubled bounded random processes for which the mode do not necessarily have to be within the bounds. In this article, a generalization of the Kumaraswamy distribution called the T-Kumaraswamy family is defined using the T-R {Y} family of distributions framework. The resulting T-Kumaraswamy family is obtained using the quantile functions of some standardized distributions. Some general mathematical properties of the new family are studied. Five new generalized Kumaraswamy distributions are proposed using the T-Kumaraswamy method. Real data sets are further used to test the applicability of the new family.


Introduction
developed a double-bounded probability distribution to model random processes which are limited to interval of finite length for which the mode doesn't necessarily have to be within the Interval. A special case of the interval being (0,1) has been studied extensively and called the Kumaraswamy distribution. The Kumaraswamy distribution which closely mimics the beta distribution has been thought of as a good alternative to the beta distribution due to the circumstance that it has both closed form cumulative distribution function (cdf), and probability density function (pdf), a characteristic which the beta distribution do not possess (For details on some important properties of the Kumaraswamy distribution, see Jones, 2009;Mitnik, 2013). Many probability distributions have been generated in the literature using the Kumaraswamy distribution as the generator (see. Cordeiro (Elgarhy et al. 2018). For random processes that assume values on the interval (0,1), there is a great need to develop flexible and highly adaptive distributions to model such processes. Areas of application of distributions defined on (0,1) include but not limited to serving as conjugate prior to some of the classical discrete distribution in Bayesian inference and modeling of the random behavior of percentages and proportions.
The − family of distributions was developed by Alzaatreh et al. (2013a). They utilized a random variable defined on the interval [ , ], −∞ ≤ < ≤ ∞ with cdf and pdf ( ) and ( ) respectively, and another random variable with pdf and cdf ( ) and ( ) respectively. Using a transformation ( ( )) of the cdf of , they defined a new class of distributions by the cdf of the form where (. ) satisfies the conditions i.
is differentiable and monotonically non-decreasing, iii. = ( ( ( ))), (2) where ( ) is the quantile function of the random variable . Observe that in (2), is used as a random variable having cdf ( ) and at the same time having cdf ( ) which may be confusing. This made Alzaatreh et al. (2014) to re-define the − { } as − { }and proposed several generalizations of the normal distribution using the − { } framework.
In section 2 the − Kumaraswamy family of distributions is defined. General mathematical properties of the proposed family are presented in section 3. Some members of the new family are specified in section 4 alongside their properties. In section 5 some applications to real data sets is carried out and the paper closes in section 6 with summary and conclusions. (3) The corresponding pdf associated with (3) is

The
where ′ ( ) =   The hazard function of the random variable can be written as where ℎ (. ) and ℎ (. ) are the hazard functions of the random variable and respectively.

The − Kumaraswamy {exponential} distribution
The corresponding pdf is The corresponding pdf is

The −Kumaraswamy {log-logistic} distribution
If follows the standard log-logistic distribution with quantile function The corresponding pdf is

General Mathematical
follows the − Kumaraswamy {log-logistic} distribution in (12). Proof. The proof follows from Remark 1(i). The results obtained in Lemma 1 enable one to establish a relationship between the random variable following the −Kumaraswamy distribution and the random variable .
Consequently, random samples from the −Kumaraswamy distribution can be simulated by first simulating random samples from the distribution of the random variable and applying the transformation accordingly.
Proof. The proof readily follows from Remark 1(ii). Theorem 1. The mode(s) of the −Kumaraswamy family of distributions is/are the solution(s) of the equation for . Proof. The proof follows from setting the derivative of the pdf given in (4) to zero. }.

Remark 2.
The mode obtained using the result in Theorem 1 may not be unique. It is possible for there to exist more than one value satisfying (14). Shannon (1948) defined the entropy of a random variable as {−log( ( ))}, where ( ) is the pdf of the random variable. The entropy of the random variable measures variation of uncertainty (Rényi, 1961).
where is the Shannon entropy of the distribution of the random variable . Proof. From Remark 1(i), it follows that = (1 − (1 − ) ) and hence the pdf in where is the mean of the random variable . Observe that the results in Corollary 2 (i-iv) follow from the fact that ( ) = e − , e (1 + e ) −2 , e e −e and (1 + T) −2 for the exponential, logistic, extreme value and log-logistic distribution respectively. ( 2 ), where ( ) =  (16) is obtained. The results of (17) - (19) can be obtained by applying the same technique. (16) and (19) hold if the support of the random variable T is on the positive real line, while (17) and (18) hold if T is on the entire real line. These fully validate the choice of ( ( )) as opined by Alzaatreh et al. (2013a), for a given distribution T.

Remark 3. The results in
The dispersion and the spread in a population from the center are often measured by the deviation from the mean, and the deviation from the median. Denote the mean deviation from the mean ( ) by ( ) and the mean deviation from the median ( ) by ( ).

Some Members of the − Family of Distributions
, , , > 0, 0 < < 1. When = 1 and −1 = ∈ , the pdf in (34) reduces to the distribution of the minimum order statistics, (1) , from a Kumaraswamy random sample of size .
The graphs of the various shape of the WKUM distribution are provided in Figure 1.  The following are some of the properties of WKUM distribution using the general properties discussed in section 3.  (2) Mode: Using Corollary 1, the mode of the WKUM distribution is the solution of the equation (3) Shannon entropy: Using the result in Corollary 2 and given that = Γ(1 + 1/ ) and = (1 − 1/ ) + log( / ) + 1 (see Song, 2001), the Shannon entropy of the WKUM distribution can be expressed as where is the Euler-Mascheroni constant and Γ(. ) is the complete gamma function.

The log-logistic-Kumaraswamy {exponential} (LLKUM) distribution
A random variable is said to follow the log-logistic distribution with parameter if it has the cdf ( ) = 1 − (1 + ) −1 , > 0, > 0. Using (6) and (7), the cdf and pdf of the LLKUM distribution are given respectively by < < 1, , , > 0. The graph of the pdf of the LLKUM distribution is given in Figure 2.

The exponential Kumaraswamy {log-logistic} (EKUM) Distribution
A random variable is said to follow the exponential distribution with parameter if it has the cdf ( ) = 1 − − , > 0, > 0. Using (12) and (13), the cdf and pdf of the EKUM distribution are given respectively by 0 < < 1, , , > 0. The graph of the pdf of the EKUM distribution is given in Figure 3.

The normal -Kumaraswamy {logistic} (NKUM) Distribution
A random variable is said to follow the standard normal distribution if it has the cdf ( ) = Φ( ), −∞ < < ∞ and Φ(. ) is defined in terms of the error function. Using (8) and (9) the cdf and pdf of the NKUM distribution are given respectively by ϕ(. ) = Φ ′ (. ), 0 < < 1, , > 0. The graph of the pdf of the NKUM distribution is given in Figure 4.

Applications
In this section applications of some members of the generalized Kumaraswamy distributions will be carried out. Using the maximum likelihood estimation technique which involves the maximization of the log-likelihood function = ∑ log( ( )) =1 , for a random independent sample 1 , 2 … , where (. ) is the pdf of a distribution, we shall fit the proposed members of the − Kumaraswamy family alongside the beta and Kumaraswamy distributions to two real data sets and assess the performance of all the distributions. A random variable is said to follow the beta distribution with parameters > 0 and > 0, if it has the pdf ( ) = is the complete beta function. The first data set represents the first 58 observations of the failure times of Kevlar 49/epoxy strands when the pressure is at 90% stress level, obtained from Andrews and Herzberg (1985). The data set is contained in Table 1. The WKUM, LLKUM, EKUM, NKUM, LKUM, beta and Kumaraswamy (Kumar) distributions are used to fit the data set. The results which include the parameter estimates, the log-likelihood values, and the values of the Kolmogorov-Smirnov (K-S) statistic as well as its p-value for all the distributions are contained in Table 2. Figure 6 displays the histogram and fitted densities to the data set.
The second data set represents the percentage of poor children living below and equal R$140 in 1991 in 5496 Brazilian Municipal Districts. The data were extracted from the Atlas of Brazil Human Development database available at http://www.pnud.org.br. The NKUM, beta and Kumaraswamy distributions are used to fit the data set. The results of the fit which include the parameter estimates, the log-likelihood values and the values of the 232 A New Family of Generalized Distributions on the Unit Interval: The − Kumaraswamy Family of Distributions Kolmogorov-Smirnov (K-S) statistic as well as its p-value for all the distributions are contained in Table 3. Figure 7 displays the histogram and fitted densities to the data set.  From Table 2, it can be observed that all the generalized Kumaraswamy distributions as well as the beta and Kumaraswamy distributions provided adequate fit for the data by virtue of the reported p-value of the K -S statistic values with the WKUM distribution providing the best fit by possessing the highest p-value.  (Standard error of estimates in parenthesis) Results in Table 3 clearly indicate the superiority of the NKUM distribution over the beta and Kumaraswamy distributions in fitting the data set since it reported the highest pvalue value. This application clearly suggests that the 2-parameter NKUM distribution can be more flexible than the 2-parameter beta and Kumaraswamy distributions.

Summary and Conclusion
A new family of generalized univariate distributions on the unit interval called the − Kumaraswamy distributions, which generalizes the Kumaraswamy distribution has been introduced in this paper. General expression for the quantile function, mode, moments, entropy and mean deviations of the generalized family have been given. Five members of the new family have been defined and applied to real data sets to demonstrate their applicability. Results obtained indicate that the members of the new family can be used as good alternatives to the beta and Kumaraswamy distributions. In particular, the normal-Kumaraswamy {logistics} distribution proved to be more flexible than the beta and Kumaraswamy distributions. We hope that the proposed family of distribution will attract wider applications in the analysis of proportion and percentage data.