On the Alpha Power Kumaraswamy Distribution: Properties, Simulation and Application

Adding new parameters to classical distributions becomes one of the most important methods for increasing distributions flexibility, especially, in simulation studies and real data sets. In this paper, alpha power transformation (APT) is used and applied to the Kumaraswamy (K) distribution and a proposed distribution, so called the alpha power Kumaraswamy (AK) distribution, is presented. Some important mathematical properties are derived, parameters estimation of the AK distribution using maximum likelihood method is considered. A simulation study and a real data set are used to illustrate the flexibility of the AK distribution compared with other distributions.


Introduction
proposed a probability distribution for double bounded random processes of hydrological applications, with the following cumulative distribution function (CDF) and probability density function (PDF), respectively, and On the Alpha Power Kumaraswamy Distribution 287

The New AK Distribution
Since, the CDF and PDF of the APT, Mahdavi & Kundu (2017), for a continuous random variable X, respectively, are and The AK distribution can be derived, as follows: substituting (1) into (3) gives differentiating last equation (w.r.t. x) yields when α = 1, the AK distribution reduces to K distribution, Kumaraswamy (1980), setting θ = 1 gives the alpha power function ( AP) distribution, and setting α = 1, θ = 1 gives the power function (P) distribution, Meniconi & Barry (1996). Some shapes of the density function for the AK distribution are illustrated in figure 1.

The Expansions for the CDF and PDF
In this section, expansions for the CDF and PDF of the AK distribution will be obtained

An Expansion for the CDF
Using exponential expansion for (5) gives then, using binomial expansion for last equation yields where,

An Expansion for the PDF
shifting j leads to where, Conditions of the Expansion for the PDF. Since, One can obtain this condition in different form as follows: since,

The Asymptotes of the CDF and PDF
In this section, the asymptotes of the CDF and PDF of the AK distribution will be obtained.

The Asymptotes of the CDF
First: as x converges to zero. Using only first and second terms of exponential expansion for the CDF gives using only first and second terms of binominal expansion gives Second: as x converges to 1. Using only first and second terms of binominal expansion leads to

The Asymptotes of the PDF
First: as x converges to zero. Since, Second: as x converges to 1. Since, using only first and second terms of binominal expansion gives

Some Properties of the AK Distribution
In this section some properties of the AK distribution will be considered as follows:

The r-th Moment
Generally, the r-th moment of a continuous random variable X, (Johnson, Kotz & Balakrishnan 1995), is given by one can see that, setting r = 0 gives substituting (9) and (10) into last equation yields Mean, variance, skewness, and kurtosis of the AK distribution can be calculated, numerically, for α in Table 1. From the last table, when 0 < α < 1: as α increases, mean, variance and skewness decrease but kurtosis increases. On the other hand, when α > 1: as α increases, mean, variance and skewness increase but kurtosis decreases.

Moment Generating Function
Basically, the moment generating function ( MGF) of a continuous random variable X is given by a first representation can be obtained via substituting (8) into last equation giving then, the following integration, Gradshteyn and Ryzhik (2000), will be considered using (11) gives moreover, the following expansion, Gradshteyn and Ryzhik (2000), will be considered using (12) yields A second representation for MGF, based on exponenatial expansion, can be given as follows: using exponential expansion, in last equation, leads to

The Quantile Function and the Median
The well-known definition of the 100 u-th is equating (5) to u gives easily, replacing u with 1/2 (the second quartile) yields the median.

The Mean Deviation
Generally, the mean deviation about the mean and about the median for a random variable X, respectively, can be given from easily, it can be given by, Ali Ahmed (2019), ., .) is the incomplete beta function.

The Mode
The natural logarithm of (6) is differentiating the last equation (w.r.t. x) and equating it to zero yields The last equation is a nonlinear equation and it does not have an analytic solution with respect to x, therefore it have to be solved numerically, if x 0 is a root for the last equation then it must be f // [log (x 0 )] < 0.

The Hazard Function of the AK Distribution
Generally, the survival function of a random variable X, Meeker & Escobar (2014), can be given by

into last equation yields
some shapes of the Hazard function for the AK distribution are illustrated in figure 2.

The Rényi Entropy of the AK Distribution
The Rényi entropy of a random variable X, Meeker & Escobar (2014), is defined by

Reliability: The Stress Strength Model of the AK Distribution
Generally, the stress strength model of the distribution, Meeker & Escobar (2014), can be given by where λis the vector of parameter, β and θ are common parameters, substituting (5) and (6) into last equation yields Firstly, hence, Secondly,

Order Statistics of the AK Distribution
The density function f (x u : v ) of the u-th order statistics for u = 1, 2, . . . , v from iid random variables X 1 , X 2 , . . . , X v following the AK distribution, Arnold, Balakrishnan & Nagaraja (1992), is given by using binomial expansion in last equation gives substituting (7) and (8) into (15) yields using binomial expansion in last equation leads to Garthwaite et al. (2002),  Garthwaite et al. (2002), where The r-th Moment of Order Statistics. One, easily, finds that the r-th moment of order statistics of the AK distribution can be got by E X r

Estimation of the AK Distribution Parameters
Let X 1 , X 2 , . . . , X n be the iid random variables from the AK (x; Λ) distribution, where Λ = (α, β, θ), then the likelihood function for the vector of parameter Λ = (α, β, θ), Garthwaite et al. (2002), can be written as the log likelihood function is given by the score function for the parameters α, β and θ are given by ∂ℓ ∂β = n θ n log(α) n β n−1 and ∂ℓ ∂θ = n β n log(α) n θ n−1 (α − 1) n The unknown parameters of the AK distribution are obtained, using the maximum likelihood estimators (MLEs), by solving the nonlinear equations (17) to (19) but they cannot be solved analytically, so solving the equations, numerically, will be performed by using a statistical package. An iterative technique such as a Newton-Raphson algorithm may be performed to compute the estimates values.
Let Λ be the vector of the unknown parameter (α,β,θ), then elements of the 3×3 information matrix I(α, β, θ) can be approximated by is the variance covariance matrix of the unknown parameters, the asymptotic distributions of the AK parameters is √ n (Λ i − Λ i ) ≈ N 3 (0, I −1 (Λ i )) , i = 1, 2, 3, and the approximation 100(1 − γ)% confidence intervals of the unknown parameters based on the asymptotic distribution of the AK(α, β, θ) distribution are determined, respectively, aŝ where z γ 2 is the upper γ 2 th percentile of a standard normal distribution.
The derivatives in the observed information matrix I(α, β, θ) for the unknown parameters are included in appendix I.

A Simulation Study
In this study, MLEs for parameters of the AK distribution are obtained using random numbers to study the MLEs finite sample behavior. The algorithm of obtaining parameters estimates is detailed in the following steps: Step (1): Generating a random sample X 1 , X 2 , . . . , X n of sizes n = (10, 20, 30 ,50, 100, 300) by using the AK distribution.
In this study, samples of random numbers are generated using Mathcad package v15 where the conjugate gradient iteration method is performed. All results are included in tables and indicated in appendix II.
From the results, in appendix II, one can see that, as sample size increases, biases, estimators, and RMSEs decrease.β andθ can be consistent, specially, when sample size increases. Moreover, the sampling distribution ofα can be the Pearson type IV distribution in all times, the sampling distribution ofβ andθ differ according to sample size. As θ increases, for fixed values of α and β, the biases and MSEs ofα andβ decrease.

Application
A real data set is concerned to apply the empirical model, practically, using the Mathematica package version 11. In this example, some distributions are used as: the AK distribution, the AP distribution, the K distribution, the P distribution the Gumbel (mini) distribution, the beta distribution, and the McDonald (McD) distribution, McDonald (2008). The following data represents the lifetime (Hours) of T8 fluorescent lamps for 50 devices, the data are given from the UK National Physical Laboratory, for more details, one can visit: http://www.npl.co.uk/ 0. 445, 0.493, 0.285, 0.564, 0.760, 0.381, 0.690, 0.579, 0.636, 0.238, 0.149, 0.244, 0.126, 0.796, 0.405, 0.553, 0.780, 0.431, 0.184, 0.375, 0.198, 0.890, 0.192, 0.463, 0.486, 0.521, 0.366, 0.486, 0.116, 0.511, 0.612, 0.117, 0.384, 0.326, 0.057, 0.412, 0.586, 0.517, 0.570, 0.588, 0.497, 0.246, 0.234, 0.228, 0.552, 0.893, 0.403, 0.458, 0.134, 0.338 The results of some goodness of fit measures are in table (2), the results of likelihood ratio tests are in table (3) In table 2, the MLEs of distributions parameters, parameters standard error (SEs), in parentheses, Kolmogorov-Smirnov ( KS) test statistic, AIC (Akaike Information Criterion), CAIC (the consistent Akaike Information Criterion) and BIC (Bayesian information criterion), (Merovci & Puka 2014), are computed for every distribution having similar skewness and kurtosis values. The null hypothesis that the data follow the AK distribution, only, can be accepted at significance level α = 0.05, it is clear that the AK distribution has the smallest KS, AIC, CAIC, BIC and SEs, on the other hand the AK distribution has the largest log likelihood and p-value, so that, the AK distribution can be the best fitted distribution to the data compared with other distributions having similar skewness and kurtosis.  In table 3, based on the likelihood ratio test, the null hypothesis is the data follow the nested model and the alternative is the data follow the full model, where the AP distribution, K distribution and the P distribution are nested by AK distribution, it is clear that, all null hypotheses can be rejected at significance level α = 0.05.

Conclusions
The alpha power Kumarasumay distribution is a flexible distribution having several advantages as it does not have any special function, has flexible mathematical properties and generalizes three important distributions. The AK distribution works practically, well, when it is compared with other distributions, specially, in simulation studies and real data sets. The author encourages researchers to do more researches and applications on the alpha power Kumarasumay distribution.

Appendix A.
The derivatives in the observed information matrix I(α, β, θ) for the unknown parameters

Appendix B.
Results of the simulation study for different data sets: Table A1: Set (1) Table A7: Set (7)