Revista Colombiana de Estadística - Applied statistics Julio 2020, volumen 43, no. 2, pp. 285 a 313 http://dx.doi.org/10.15446/rce.v43n2.83598

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

Distribución alpha-power-Kumaraswamy: propiedades, simulación y aplicación

Mohamed Ali Ahmeda

Department of Statistics, Mathematics and Insurance, Al Madina Higher Institute of Management and Technology, Giza, Egypt.

Abstract 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. Key words: Alpha power transformation; Maximum likelihood estimation; Moments; Orders statistics; The Kumaraswamy distribution.

Resumen Agregar nuevos parámetros a las distribuciones clásicas se convierte en uno de los métodos más importantes para aumentar la flexibilidad de las distribuciones, especialmente en estudios de simulación y conjuntos de datos reales. En este documento, se utiliza la transformación de potencia alfa (TPA) y es aplicada a la distribución de Kumaraswamy (K) y a una distribución propuesta, denominada distribución de energía alfa de Kumaraswamy (AK). Se derivan algunas propiedades matemáticas, y se muestra la estimación de parámetros de la distribución AK utilizando el método de máxima verosimilitud. Un estudio de simulación y un conjunto de datos reales se utilizan para ilustrar la flexibilidad de la distribución AK en comparación con otras distribuciones. Palabras clave: Transformación de potencia alfa; Estimación de máxima verosimilitud; Momentos; Estadísticas de pedidos; La distribución de Kumaraswamy. aPh.D. E-mail: [email protected]

285 286 Mohamed Ali Ahmed 1. Introduction

Kumaraswamy (1980) 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,

θ F (x) = 1 − 1 − xβ ; 0 < x < 1; β, θ > 0, (1) and

θ−1 f(x) = βθXβ−1 1 − xβ . (2)

Jones (2009) illustrated that the K distribution has some properties like the beta distribution such as both of their densities are unimodal, uniantimodal, increasing, decreasing or constant depending on its parameters, where the CDF and PDF of the beta distribution are

F (x, a, b) = Ix(a, b); 0 < x < 1; a > 0, b > 0, and xa−1 (1 − x)b−1 f(x, a, b) = , B(a, b) where, Ix is the regularized incomplete beta function. Also, Jones (2009) highlighted several advantages of the K distribution over beta distribution: the K distribution is very simple to use especially in simulation studies due to the simple closed form of both its quantile function and cumulative distribution function, the normalized constant of the K distribution is very simple, his explicit formulae for the quantile function, distribution function and moments of order statistics and has a simple formulae for L-moments. Mahdavi & Kundu (2017) presented, for the first time, a transformation of the parent CDF by adding a new parameter to derive a proposed family of distributions, this method is called the APT method. The main object of this study is to propose and study a new distribution called APK distribution based on the APT method. The paper is organized as follows: In section 1, the introduction of the study is presented. In section 2, the proposed model is derived, its special cases are presented and its asymptotes are studied. In section 3, some properties are obtained. In section 4, the Hazard function is given. In section 5, the Rényi entropy is obtained. In section 6, the stress strength model is given. In section 7, order statistics are studied. In Section 8, the distribution parameters are estimated by the maximum likelihood estimation (MLE) method. In Section 9, a simulation study is illustrated. Finally, in Section 10, an application is used to investigate, practically, the flexibility of the proposed distribution.

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 On the Alpha Power Kumaraswamy Distribution 287 2. The New AK Distribution

Since, the CDF and PDF of the APT, Mahdavi & Kundu (2017), for a continuous random variable X, respectively, are ( αF (x)−1 ̸ α−1 ; α > 0, α = 1, FAP T (x) = (3) F (x); α = 1, and ( log α f(x) αF (x); α > 0, α ≠ 1, f (x) = α−1 (4) AP T f(x); α = 1.

The AK distribution can be derived, as follows: substituting (1) into (3) gives

β θ α1−(1−x ) − 1 F (x) = , 0 < x < 1; α, β, θ > 0; α ≠ 1, (5) αK α − 1 differentiating last equation (w.r.t. x) yields
βθ log α β θ θ−1 f (x) = α1−(1−x ) xβ−1 1 − xβ , (6) αK α − 1 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.

Figure 1: The AK density functions.

2.1. The Expansions for the CDF and PDF

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

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 288 Mohamed Ali Ahmed

2.1.1. An Expansion for the CDF

Using exponential expansion for (5) gives ! X∞ i h i 1 [log α]
θ i F (x) = 1 − 1 − 1 − xβ , αK 1 − α i! i=0 then, using binomial expansion for last equation yields     X∞ i Xi 1 [log α] i
jθ F (x) = 1 − (−1)j 1 − xβ  , αK 1 − α i! j i=0 j=0 since, X∞ Xi X∞ X∞ = , i=0 j=0 j=0 i=j then,   X∞ 1
jθ F (x) = 1 − w 1 − xβ  , (7) αK 1 − α j j=0 where, ∞   X [log α]i i w = (−1)j . j i! j i=j

2.1.2. An Expansion for the PDF

Differentiating (7) (w.r.t. x) gives X∞ βθ
jθ−1 f (x) = w xβ−1 1 − xβ , αK 1 − α j j=1 shifting j leads to X∞ βθ
(j+1)θ−1 f (x) = w xβ−1 1 − xβ , (8) αK 1 − α j+1 j=0 where, ∞   X [log α]i i w = (−1)j+1 ; i ≥ j + 1. j+1 i! j + 1 i=j+1

Conditions of the Expansion for the PDF. Since,

X∞ Z1 θ
(j+1)θ−1 w β xβ−1 1 − xβ dx = 1, 1 − α j+1 j=0 0

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 On the Alpha Power Kumaraswamy Distribution 289 then, " # ∞
(j+1)θ−1 1 −θ X 1 − xβ w = 1, 1 − α j+1 (j + 1) θ j=0 0 moreover, P∞ wj+1 j=0 = 1, (1 − α)(j + 1) hence, X∞ wj+1 = (1 − α)(j + 1) . (9) j=0 One can obtain this condition in different form as follows: since, X∞ Z1 θ
(j+1)θ−1 w β xβ−1 1 − xβ dx = 1, 1 − α j+1 j=0 0 then, ∞ θ X w B (1, (j + 1) θ) = 1, 1 − α j+1 j=0 substituting (9) into last equation gives

θ (j + 1) B (1, (j + 1) θ) = 1, hence, 1 B (1, (j + 1) θ) = . (10) (j + 1) θ

2.2. The Asymptotes of the CDF and PDF

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

2.2.1. The Asymptotes of the CDF

First: as x converges to zero. Using only first and second terms of exponential expansion for the CDF gives n h i o 1
θ F (x) ∼ 1 − 1 − xβ log α , αK α − 1 using only first and second terms of binominal expansion gives 1 
 F (x) ∼ 1 − 1 − θxβ log α , αK α − 1

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 290 Mohamed Ali Ahmed then, 1 β F (x) ∼ log αθx . αK α − 1 Second: as x converges to 1. Using only first and second terms of binominal expansion leads to   1 β F (x) ∼ αθx − 1 . αK α − 1

2.2.2. The Asymptotes of the PDF

First: as x converges to zero. Since,

θ−1 β θ lim 1 − xβ = 1, lim α1−(1−x ) = 1, x→0 x→0 then, βθ log α f (x) ∼ xβ−1. αK α − 1 Second: as x converges to 1. Since,

β θ lim xβ−1 = 1, lim α1−(1−x ) = α, x→1 x→1 then, αβθ log α
θ−1 f (x) ∼ 1 − xβ , αK α − 1 using only first and second terms of binominal expansion gives

αβθ log α   f (x) ∼ 1 − (θ − 1) xβ . αK α − 1

3. Some Properties of the AK Distribution

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

3.1. The r-th Moment

Generally, the r-th moment of a continuousR random variable X, (Johnson, Kotz & Balakrishnan 1995), is given by E (Xr) = xrf (x) dx,substituting (8) into last x equation yields

Z1 X∞ βθ
(j+1)θ−1 E (Xr) = xr w xβ−1 1 − xβ dx, 1 − α j+1 0 j=0

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 On the Alpha Power Kumaraswamy Distribution 291 then, X∞ Z1 θ
(j+1)θ−1 E (Xr) = w βxr+β−1 1 − xβ dx, 1 − α j+1 j=0 0 hence, ∞   θ X r + β E (Xr) = w B , (j + 1) θ , 1 − α j+1 β j=0 one can see that, setting r = 0 gives

∞
θ X E X0 = w B (1, (j + 1) θ) , 1 − α j+1 j=0 substituting (9) and (10) into last equation yields
E X0 = 1.

Mean, variance, skewness, and kurtosis of the AK distribution can be calculated, numerically, for α in Table 1.

Table 1: Mean, variance, skewness and kurtosis of AK distribution for various values of α. Measure α = 0.1 α = 0.2 α = 0.5 α = 0.9 α = 1.1 α = 1.5 α = 2 α = 10 Mean 1.383 1.289 1.218 1.216 1.223 1.242 1.27 1.556 Variance 0.23 0.167 0.116 0.112 0.117 0.131 0.150 0.358 Skewness −0.871 −0.947 −1.157 −1.206 −1.178 −1.101 −1.015 −0.767 Kurtosis −0.332 −0.097 0.534 0.671 0.5822 0.345 0.095 −0.596

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.

3.2. Moment Generating Function

Basically, the moment generating function ( MGF) of a continuous random variable X is given by Z
tx tx Mx(t) = E e = e f (x) dx, x a first representation can be obtained via substituting (8) into last equation giving

Z1 X∞ βθ
(j+1)θ−1 M (t) = etx w xβ−1 1 − xβ dx, x 1 − α j+1 0 j=0

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 292 Mohamed Ali Ahmed then, X∞ Z1 βθ
(j+1)θ−1 M (t) = w etxxβ−1 1 − xβ dx, x 1 − α j+1 j=0 0 using binomial expansion for last equation yields

∞   Z1 βθ X (j + 1) θ − 1 M (t) = w (−1)k etxxβ+βk−1dx, x 1 − α j+1 k j,k=0 0 then, the following integration, Gradshteyn and Ryzhik (2000), will be considered

Z1 F (a; a + 1; t) 1 1 = etzza−1dz, (11) a 0 using (11) gives ∞   βθ X (j + 1) θ − 1 F (β + βk; β + βk + 1; t) M (t) = w (−1)k 1 1 , x 1 − α j+1 k β + βk j,k=0 moreover, the following expansion, Gradshteyn and Ryzhik (2000), will be considered ∞ X Γ(a + u) zu F (a; b; z) = , (12) 1 1 Γ(b + u) u! u=0 using (12) yields ( ) ∑∞ − ℓ βθ k (j + 1) θ 1 Γ(β + βk + ℓ) t Mx(t) = wj+1 (−1) , (1 − α)(β + βk) k Γ(β + βk + 1 + ℓ) ℓ! j,k,ℓ=0 hence, ∞   θ X (j + 1) θ − 1 tℓ M (t) = w (−1)k . x (1 − α) (1 + k) j+1 k ℓ!(β + βk + ℓ) j,k,ℓ=0 A second representation for MGF, based on exponenatial expansion, can be given as follows: Since,
tx Mx(t) = E e , using exponential expansion, in last equation, leads to ! ∞ X (tx)k M (t) = E , x k! k=0 then, ∞ X tk
M (t) = E xk . x k! k=0

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 On the Alpha Power Kumaraswamy Distribution 293

3.3. The Quantile Function and the Median

The well-known definition of the 100 u-th is

u = P (X ≤ xu) = F (xu); xu > 0, 0 < u < 1, equating (5) to u gives

( ) 1   1 β log (1 + (α − 1) u) θ x = 1 − 1 − , log α easily, replacing u with 1/2 (the second quartile) yields the median.

3.4. The Mean Deviation

Generally, the mean deviation about the mean and about the median for a random variable X, respectively, can be given from Z Z

S1 (x) = |x − µ| f (x) dx and S2 (x) = |x − M| f (x) dx, x x easily, it can be given by, Ali Ahmed (2019),

S1 (x) = 2 µ F (µ) − 2 t (µ) and S2 (x) = µ − 2 t (M) ,

Rq where T (q) = x f (x) dxis the linear incomplete moment. −∞ Substituting (8) into T (.) gives

X∞ Zq θ
(j+1)θ−1 T (q) = w βx(β+1)−1 1 − xβ dx, 1 − α j+1 j=0 0 then, ∞   θ X 1 + β T (q) = w B q; , (j + 1) θ , 1 − α j+1 β j=0 where B (.; ., .) is the incomplete beta function.

3.5. The Mode

The natural logarithm of (6) is

log fαK (x) = h i βθ log α
θ
log + 1 − 1 − xβ log α + (β − 1) log x + (θ − 1) log 1 − xβ , α − 1

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 294 Mohamed Ali Ahmed differentiating the last equation (w.r.t. x) and equating it to zero yields

h i β−1 d
θ−1 β − 1 (θ − 1) βx log f (x) = θβxβ−1 1 − xβ log α + − = 0. dx αK x 1 − xβ 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 x0 is a root for the // last equation then it must be f [log (x0)] < 0.

4. The Hazard Function of the AK Distribution

Generally, the survival function of a random variable X, Meeker & Escobar (2014), can be given by S (x) = 1 − F (x), substituting (5) into last equation gives

β θ α1−(1−x ) − 1 S (x) = 1 − ; 0 < x < 1; α, β, θ > 0. (13) αK α − 1

Simply, the Hazard function, Meeker & Escobar (2014), can be given by

f(x) H (x) = , S (x) substituting (6) and (13) into last equation yields

θ−1 βθ (log α) xβ−1 1 − xβ H (x) = , (1−xβ )θ αK α − 1 some shapes of the Hazard function for the AK distribution are illustrated in figure 2.

Figure 2: The AK Hazard functions.

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 On the Alpha Power Kumaraswamy Distribution 295 5. The Rényi Entropy of the AK Distribution

The Rényi entropy of a random variable X, Meeker & Escobar (2014), is defined by   Z 1 e (ρ) = log  [f (x)]ρ dx , R 1 − ρ x substituting (8) into last equation gives     ρ ( ) ∫1 ( )  ∞ [( ) ]   ρ ρ(θ−1) ∑ θ j 1 βθ ρ(β−1) β ∗ β eR (ρ) = log x 1 − x wj 1 − x dx , αK 1 − ρ  1 − α    0 j=0 ( ) h i ρ h i P∞ P∞ P∞
j P∞
j ∗ ∗ − β θ − β θ where wj = wj+1, since, wj 1 x = mj 1 x , j=0 j=0 j=0 j=0 ∗ρ Gradshteyn & Ryzhik (2000), where m0 = w0 , Xn 1 ∗ m = (jρ − n + j) w m − ; n ≥ 1, n n w∗ j n j 0 j=1 then,   1  ρ ∞ Z  X
− − 1 βθ ρ(β−1) − β ρ(θ 1) θj eR (ρ) = log mj x 1 x dx , αK 1 − ρ  1 − α  j=0 0 moreover,    − ∞ ∫1 ( )  ρ 1 ρ ∑ ρ(θ−1)−θj+1−1 1 β θ ρ(β−1)+1−1 β eR (ρ) = log ρ mj β x 1 − x dx , αK 1 − ρ  (1 − α)  j=0 0 then,    ∞   ρ−1 ρ X − 1 β θ ρ (β 1) + 1 − − eR (ρ) = log ρ mj B , ρ (θ 1) θj + 1 . αK 1 − ρ (1 − α) β  j=0

6. Reliability: The Stress Strength Model of the AK Distribution

Generally, the stress strength model of the distribution, Meeker & Escobar (2014), can be given by Z

R = f1 (x; λ1) F2 (x; λ2) dx, (14) x

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 296 Mohamed Ali Ahmed where λis the vector of parameter, β and θ are common parameters, substituting (5) and (6) into last equation yields

θ Z1 1−(1−xβ ) − − β θ
− − βθ log α1 1 (1 x ) β−1 − β θ 1 α2 1 R = α1 x 1 x dx, α1 − 1 α2 − 1 0 setting R = I1 − I2 , gives

Z1 − − β θ − − β θ
− βθ log α1 1 (1 x ) 1 (1 x ) β−1 − β θ 1 I1 = α1 α2 x 1 x dx, (α1 − 1) (α2 − 1) 0 and Z1 − − β θ
− βθ log α1 1 (1 x ) β−1 − β θ 1 I2 = α1 x 1 x dx. (α1 − 1) (α2 − 1) 0 Firstly,

Z1 [ ] [ ] β θ β θ
− βθ log α1 1−(1−x ) log α1+ 1−(1−x ) log α2 β−1 β θ 1 I1 = e x 1 − x dx, (α1 − 1) (α2 − 1) 0 then,

[ ] ∫1 ( ) − − β θ ( ) − log α1 1 1 x [log α1+log α2] β−1 β θ 1 I1 = e [log α1 + log α2] θβx 1 − x dx, (α1 − 1) (α2 − 1) 0 [log α1 + log α2] hence, (log α1)(α1α2 − 1) I1 = . (α1 − 1) (α2 − 1) [log α1 + log α2] Secondly,

Z1 [ ] β θ
− 1 1−(1−x ) (log α1) β−1 β θ 1 I2 = e βθ (log α1) x 1 − x dx, (α1 − 1) (α2 − 1) 0 then, 1 I2 = . α2 − 1

An Expansion for the Reliability. Substituting (7) and (8) into (14) gives   Z1 ∞ ∞ X
− X
βθ β−1 β (j+1)θ 1 1  β jθ R = wj+1 x 1 − x 1 − wj 1 − x dx, 1 − α1 1 − α2 0 j=0 j=0 setting θ R = [I1 − I2] , (1 − α1) (1 − α2)

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 On the Alpha Power Kumaraswamy Distribution 297 yields ∞ Z1 X
− ∗ β−1 − β (j+1)θ 1 I1 = wj βx 1 x dx, j=0 0 and Z1 ∞ ∞ X
− X
∗ β−1 − β (j+1)θ 1 − β jθ I2 = β wj x 1 x wj 1 x dx, 0 j=0 j=0 P∞ P∞ ∗ where wj = wj+1. j=0 j=0 Firstly, X∞ ∗ I1 = wj B (1, (j + 1) θ) . j=0

Secondly,

Z1 ∞ ∞ X
− X
∗ β−1 − β (j+1)θ 1 − β jθ I2 = β wj x 1 x wj 1 x dx, 0 j=0 j=0 then,     Z1  ∞ h i   ∞ h i 
− X
j X
j β−1 − β θ 1 ∗ − β θ − β θ I2 = β x 1 x  wj 1 x   wj 1 x  dx, 0 j=0 j=0 since, Garthwaite, Jolliffe & Byron Jones (2002),      ∞   ∞  ∞ X h
ij X h
ij X h
ij ∗ − β θ − β θ − β θ  wj 1 x   wj 1 x  = nj 1 x , j=0 j=0 j=0

Pm ∗ where, nm = wj wm−j, j=0 then, ∞ Z1 X
− β−1 β jθ+θ 1 I2 = nj βx 1 − x dx, j=0 0 hence, X∞ I2 = njB (1, jθ + θ) . j=0

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 298 Mohamed Ali Ahmed 7. Order Statistics of the AK Distribution

The density function f (xu : v) of the u-th order statistics for u = 1, 2, . . . , v from iid random variables X1,X2,...,Xv following the AK distribution, Arnold, Balakrishnan & Nagaraja (1992), is given by

f (x ) − − f (x ) = u F (x )u 1 {1 − F (x )}v u , u : v B (u, v − u + 1) u u using binomial expansion in last equation gives   vX−u f (x ) v − u − f (x ) = u (−1)k F (x )u+k 1 , (15) u : v B (u, v − u + 1) k u k=0 substituting (7) and (8) into (15) yields

∑∞ ( ) − β−1 β (j+1)θ 1 βθ wj+1 xu 1 − xu j=0 f (xu : v) = (1 − α) B (u, v − u + 1)   ( ) u+k−1 v−u ( ) − ∞ ( ) ∑ − u+k 1 ∑ jθ k v u 1  β  (−1) 1 − wj 1 − x , 1 − α u k=0 k j=0 using binomial expansion in last equation leads to

∞ ∑ − (j+1)θ−1   βθ w xβ 1 1−xβ   j+1 u ( u) vP−u − u+k j=0 − k v u 1 f (xu : v) = B(u,v−u+1) ( 1) 1−α k=0 k   ( )ℓ u+Pk−1 − P∞
× − ℓ u + k 1 − β jθ ( 1) wj 1 xu , ℓ=0 ℓ j=0 since, Garthwaite et al. (2002),   ℓ X∞
 X∞
− β jθ − β jθ  wj 1 xu  = mj 1 xu , j=0 j=0

Pn ℓ 1 where m0 = w , mn = (jℓ − n + j) wjmn−j; n ≥ 1, then, 0 n w0 j=1

f (x ) = u : v       β−1 β θ−1 vP−u u+k u+Pk−1 βθx (1−x ) k v − u ℓ u + k − 1 u u (−1) 1 (−1) B(u,v−u+1) k 1−α ℓ (k=0 )( ℓ=0 ) P∞
P∞
× ∗ − β jθ − β jθ wj 1 xu mj 1 xu , j=0 j=0

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 On the Alpha Power Kumaraswamy Distribution 299 since, Garthwaite et al. (2002),     X∞
 X∞
 X∞
∗ − β jθ − β jθ − β jθ  wj 1 xu   mj 1 xu  = qj 1 xu , j=0 j=0 j=0

P∞ P∞ Ps ∗ ∗ where wj = wj+1, qs = wj ms−j, hence, j=0 j=0 j=0 X∞ βθ
jθ+θ−1 f (x ) = p xβ−1 1 − xβ , (16) u : v B (u, v − u + 1) j u u j=0       vP−u − u+k u+Pk−1 − − k v u 1 − ℓ u + k 1 where pj = qj ( 1) 1−α ( 1) . k=0 k ℓ=0 ℓ 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 Xr = R u : v r xu f (xu) dxu, substituting (16) into last equation gives x   ∞ Z X
− r θ r+β−1 − β jθ+θ 1 E X = pj βxu 1 xu dxu , u : v B (u, v − u + 1) j=0 x then,   X∞   r θ r + β E X = pj B , jθ + θ . u : v B (u, v − u + 1) β j=0

8. Estimation of the AK Distribution Parameters

Let X1, X2,...,Xn 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

∑n θ n n h i n n 1−xβ Y Y θ−1 ( β θ) (log α) ( i ) β−1 β L = αi=1 x 1 − x , (α − 1)n i i i=1 i=1 the log likelihood function is given by

ℓ = ! Xn   Xn Xn h i ( β θ)n (log α)n θ + 1 − xβ log α+(β − 1) log x +(θ − 1) log 1 − xβ , (α − 1)n i i i i=1 i=1 i=1 the score function for the parameters α, β and θ are given by

n   ∂ℓ n log(α)n−1βnθn n log(α)nβnθn 1 X θ = − + 1 − xβ , (17) ∂α α (α − 1)n − n+1 α i (α 1) i=1

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 300 Mohamed Ali Ahmed   − Pn θ−1 ∂ℓ n θn log(α)nβn 1 − − β β ∂β = (α−1)n θ (log α) 1 xi xi (log xi) i=1 n n (18) P P xβ (log x ) + log x + (θ − 1) i i , i xβ −1 i=1 i=1 i and

− ∂ℓ n βn log(α)nθn 1 = ∂θ (α − 1)n Xn  θ h  i Xn   − β − β − β + (log α) 1 xi log 1 xi + log 1 xi . (19) i=1 i=1

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   2 ˆ − ∂ ℓ(Λ ) Ii j(Λ ) = E Λ=Λˆ , ∂Λi ∂Λj

−1 ˆ where Ii j (Λ)is the variance covariance matrix of the unknown parameters, the asymptotic distributions of the AK parameters is √ −1 n (Λˆ i − Λi) ≈ N3(0,I (Λˆ 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, as q ˆ ± −1 ˆ Λi z γ I (Λi); i = 1, 2, 3, 2

γ th where z γ is the upper percentile of a standard normal distribution. 2 2 The derivatives in the observed information matrix I(α, β, θ) for the unknown parameters are included in appendix I.

9. 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 X1,X2,...,Xn of sizes n = (10, 20, 30 ,50, 100, 300) by using the AK distribution.

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 On the Alpha Power Kumaraswamy Distribution 301

Step (2): Selecting different set values of the parameters as: set(1): (α = 0.5, β = 2, θ = 3), set(2): (α = 0.9, β = 2, θ = 3), set(3): (α = 1.5, β = 2, θ = 3), set(4): (α = 0.5, β = 3, θ = 3), set(5): (α = 0.5, β = 4, θ3), set(6): (α = 0.5, β = 2, θ = 4), set(7): (α = 0.5, β = 2, θ = 5). Step (3): Solving (17) to (19) via iteration to compute, biases, MLEs, RMSE (the root of mean squared error) and the Pearson type of parameters estimators, Pearson (1895), of the AK distribution. Step (4): Repeating step1 to step3, 10000 times. 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. βˆ and θˆ can be consistent, specially, when sample size increases. Moreover, the sampling distribution of αˆ can be the Pearson type IV distribution in all times, the sampling distribution of βˆ and θˆ differ according to sample size. As θ increases, for fixed values of α and β, the biases and MSEs of αˆ and βˆ decrease.

10. 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), figure (3) illustrates probability density functions for different distributions having similar skewness and kurtosis and figure (4) illustrates probability density functions for nested distributions. 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

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 302 Mohamed Ali Ahmed 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.

Table 2: The MLE of the parameter(s) and the associated AIC and BIC values. KS Log BIC AIC MLE CAIC P-value Skewness Kurtosis Likelihood parameters Distribution

α β θ

0.5 2.40 4.67 0.058 2.403 0.127 0.357 7.313 −10.627 −6.803 −10.372 AK (0.137) (0.263) (0.654)

1.201 2.717 - 0.628 2.14 0.048 0.032 0.278 3.443 7.267 3.698

beta (1.442) (0.312)

0.623 0.208 - −1.138 1.4 0.019 0.041 −0.640 5.280 9.104 5.536 (mini)

Gumbel (0.212) (2.301)

6 4.61 0.0629 0.431 2.684 0.036 0.015 1.161 3.676 9.412 4.198

McD (5.215) (1.245) (3.5)

Table 3: The log-likelihood function, the likelihood ratio tests statistic and p-values.

Distribution Parameters Log Likelihood Degrees of p-value Likelihood ℓ Ratio Test Freedom Statistics Λ DF α β θ

3 3 - −6.728 28.076 1 1.166×10−7 AP (2.157) (0.463)

- 2.2 2.5 3.515 7.59 1 5.869×10−3 K (0.331) (0.954) - 2 - −14.869 44.358 2 2.332×10−10 P (0.283) *Note that the log likelihood of the AK distribution = 7.31.

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.

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 On the Alpha Power Kumaraswamy Distribution 303

Figure 3: Probability density functions for different distributions having similar skewness and kurtosis.

Figure 4: Probability density functions for the nested distribution by the AK distribution.

11. 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.

Acknowledgements

The author thanks anyone provided any important advices or suggested any helpful comments for this study.

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 304 Mohamed Ali Ahmed Appendix A.

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

− ∂2ℓ n log (α)n (n + 1) βnθn 2n2 log(α)n−1βnθn n log (α)n 1 βnθn = − − n ∂α2 (α − 1)n+2 α (α − 1)n+1 α2 (α − 1)   Pn θ β 1 − x − i n log (α)n 2 (n − 1) βnθn − i=1 + , α2 α2 (α − 1)n

       ∂2ℓ Xn log(x2) xβ log(x2) x2β θnn log (αn)(n − 1) βn−2 i i − h i i i − 2 = β 2 (θ 1)+ n ∂β  − β  (α − 1)  xi 1 −  i=1 xi 1   Xn 
 h iθ−1 
 h iθ−2 − 2 − β β − 2 − β − 2β θ log (α) log xi 1 xi xi log xi 1 xi (θ 1) xi , i=1

n       ∂2ℓ X 2 θ n [log (α)n](n − 1) βnθn−2 = log (α) log 1 − xβ 1 − xβ + , ∂θ2 i i (α − 1)n i=1

h i 2 n−1 n−1 n ∂2ℓ n log (α) β θ n2 [log(α)n] βn−1θn = n − ∂α∂β α (α − 1) (α − 1)n+1 h iθ−1 n β − β X [log(xi)] xi 1 xi − θ , α i=1

h i 2 n−1 n n−1 ∂2ℓ n log (α) β θ n2 [log(α)n] βnθn−1 = n − ∂α∂θ α (α − 1) (α − 1)n+1 h i h iθ n − β − β X log(1 xi ) 1 xi + θ , α i=1

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 On the Alpha Power Kumaraswamy Distribution 305 and

∂2ℓ = ∂β∂θ    θ n  β β h i    X [log(xi)] x 1 − x θ−1 i i β β β log (α) − θ [log(xi)] log(1 − x ) x 1 − x  β − i i i  i=1 xi 1

θn−1βn−1n (n − 1) [log (αn)] θn−1βn−1n [log (αn)] Xn [log(x )] xβ + + + i i . (α − 1)n (α − 1)n β − i=1 xi 1

Appendix B.

Results of the simulation study for different data sets:

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 306 Mohamed Ali Ahmed Set(1): Sample Size 300 100 50 30 20 10 ( α 0 = . 5 β , Parameters α α α α α α α α α α α β β β β β β θ 2 = 0 = 0 = 0 = 0 = 0 = 0 = 3 = 2 = 2 = 2 = 2 = 2 = 2 = 3 = 3 = 3 = 3 = 3 = θ , ...... 5 5 5 5 5 5 3) = estimators enof Mean 0.717 2.956 1.162 3.115 1.476 3.316 1.636 3.695 1.787 1.789 1.989 1.973 1.995 2.031 4.852 2.96 2.09 2.31 − − − − issTtlBa RMSE Bias Total Biases − 0.217 0.662 0.115 0.976 0.316 1.136 0.695 1.287 1.289 0.039 1.852 0.09 0.31 0.044 0.014 0.027 0.035 0.04 al A1: Table 0.221 0.664 0.983 1.179 1.465 2.277 0.823 0.513 0.199 0.814 0.311 2.107 1.103 0.401 2.647 0.479 2.923 2.193 0.693 3.168 0.919 3.011 4.075 1.48 RMSE Total 0.982 2.283 2.895 3.311 3.914 5.149 ero System Pearson Coefficients − − − − − − − − 0.175 0.241 0.265 0.014 0.217 0.203 0.384 0.221 0.261 0.12 2.552 0.345 0.085 0.017 0.219 0.425 0.273 4.79 Pearson Type IV IV IV IV IV IV IV IV IV IV I I I I I I I I

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 On the Alpha Power Kumaraswamy Distribution 307 I I I I I IV IV IV IV IV IV VI IV IV IV IV IV IV Type Pearson 0.007 0.496 0.209 0.253 0.292 0.19 0.02 0.194 0.219 0.456 0.243 8.167 0.236 0.165 0.286 0.271 0.223 0.584 − − − − − Coefficients Pearson System 4.620 4.189 3.648 2.096 4.977 6.259 Total RMSE 0.4 0.37 1.07 0.682 1.971 4.386 0.567 1.338 4.047 0.466 0.979 3.565 2.044 0.243 0.665 4.522 4.861 3.795 = 3) , c = 2 , β 9 1.640 1.466 1.750 2.352 1.175 0.503 . = 0 α Table A2: ( 0.063 0.008 0.013 0.045 0.503 0.164 1.669 1.834 0.437 1.407 1.625 0.094 0.207 1.466 0.034 0.023 1.174 0.501 Biases Total Bias RMSE − − − − Set(2): 2.937 2.525 2.094 3.207 2.366 2.034 3.023 2.074 1.991 1.401 1.987 3.503 2.955 2.164 2.569 4.834 2.437 2.307 estimators Mean of 9 9 9 9 9 9 ...... = 3 = 3 = 3 = 3 = 3 = 2 = 2 = 2 = 2 = 2 = 2 = 3 = 0 = 0 = 0 = 0 = 0 = 0 c β β β β β β α α α α α α α α α α α Parameters 30 50 20 10 100 300 Size Sample

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 308 Mohamed Ali Ahmed Sample Size 300 100 10 20 50 30 Parameters α α α α α α α α α α α α β β β β β β 1 = 1 = 1 = 1 = 1 = 1 = 2 = 2 = 2 = 2 = 2 = 2 = 3 = 3 = 3 = 3 = 3 = 3 = ...... 5 5 5 5 5 5 enof Mean estimators 3.939 2.566 4.841 3.676 1.987 2.018 2.256 2.608 2.915 3.417 2.967 3.385 2.081 2.169 3.446 3.586 2.95 3.11 Set(3): − − − Biases − 0.018 1.108 1.917 0.081 0.169 1.946 2.086 2.439 0.566 1.841 2.176 0.256 0.385 0.11 0.013 0.085 0.033 0.05 ( al A3: Table α 1 = Total . 5 3.107 2.224 1.109 1.918 1.947 2.095 Bias β , 2 = θ , 3) = RMSE 1.214 6.983 0.324 0.294 4.286 0.426 5.766 0.869 0.774 6.004 1.816 5.973 1.202 0.653 6.074 2.85 0.62 0.53 RMSE Total 7.639 4.308 5.814 6.320 6.059 6.226 ero System Pearson Coefficients − − − − 0.191 0.272 0.151 0.004 0.211 0.024 0.201 0.527 0.134 0.365 0.78 0.29 0.27 0.28 0.341 0.088 0.264 0.522 Pearson Type IV IV IV IV IV IV IV IV IV IV IV IV IV IV I I I I

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 On the Alpha Power Kumaraswamy Distribution 309 I I I I I I IV IV IV VI IV IV IV IV IV IV IV VI Type Pearson 0.016 0.591 0.319 0.228 1.091 0.296 0.241 0.225 0.107 1.336 0.237 0.015 0.277 0.533 0.423 0.289 0.316 2.282 − − − − − − Coefficients Pearson System 1.049 3.257 3.067 2.398 3.792 5.375 Total RMSE 2.81 0.47 0.875 0.297 0.822 0.498 2.815 0.723 1.472 0.606 1.071 2.204 2.156 0.887 2.991 2.995 1.389 4.242 = 3) , θ = 3 , β 1.00 5 0.254 1.176 0.682 1.424 2.392 . = 0 α Table A4: ( 0.033 0.036 0.021 0.015 0.036 0.03 0.46 1.137 0.678 0.139 1.245 1.944 1.316 0.251 0.302 1.005 0.098 0.681 Biases Total Bias RMSE − − − − − Set(4): 3.03 3.46 2.967 2.964 0.751 2.979 3.302 1.505 2.985 3.098 2.964 1.181 1.637 3.678 3.139 1.745 4.944 1.816 estimators Mean of 5 5 5 5 5 5 ...... = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 3 = 0 = 0 = 0 = 0 = 0 = 0 β β β β β β α α α α α α α α α α α α Parameters 50 30 20 10 300 100 Size Sample

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 310 Mohamed Ali Ahmed Sample Size 300 100 10 20 50 30 Parameters α α α α α α α α α α α α β β β β β β 0 = 0 = 0 = 0 = 0 = 0 = 4 = 4 = 4 = 4 = 4 = 4 = 3 = 3 = 3 = 3 = 3 = 3 = ...... 5 5 5 5 5 5 enof Mean estimators 3.895 2.772 3.957 3.959 0.743 2.967 1.171 3.097 1.785 3.968 4.182 1.526 3.333 3.717 4.052 1.678 2.98 4.62 Set(5): − − − − issTtlBa RMSE Bias Total Biases − 0.243 0.671 0.097 1.026 0.333 0.052 1.178 0.895 2.272 1.285 0.182 0.717 0.62 0.043 0.041 0.033 0.032 0.02 ( al A5: Table α 0 = . 2.365 0.247 0.673 1.482 1.031 1.225 5 β , 4 = θ , 3) = 0.496 0.394 0.626 3.166 1.871 0.827 2.129 1.094 3.982 0.811 2.777 1.526 3.053 2.228 3.041 1.18 0.98 0.8 RMSE Total 1.020 2.368 5.420 3.092 3.959 3.540 ero System Pearson Coefficients − − − − − 0.509 2.572 0.323 0.064 0.251 0.241 0.019 0.229 0.283 0.215 0.167 0.269 0.211 0.145 0.314 0.228 0.492 0.02 Pearson Type IV VI IV IV IV IV IV IV IV IV IV IV IV I I I I I

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 On the Alpha Power Kumaraswamy Distribution 311 I I I I I I IV IV IV IV IV IV IV IV IV IV VI IV Type Pearson 0.328 0.012 0.136 0.108 0.268 0.263 0.47 0.22 0.08 0.357 0.385 0.237 0.015 0.241 0.255 0.244 3.163 0.256 − − − − − − Coefficients Pearson System 3.00 3.599 2.386 1.014 4.308 6.762 Total RMSE 0.7 0.47 6.03 2.93 3.107 2.912 2.115 2.575 0.388 1.501 2.105 0.306 0.192 0.602 1.083 0.709 2.924 0.884 = 4) , θ = 2 , β 5 1.228 0.983 0.610 0.221 1.563 3.023 . = 0 α Table A6: ( 0.033 0.025 0.011 0.066 0.059 0.984 0.082 1.212 2.738 0.292 1.249 1.128 0.034 0.485 0.975 0.122 0.606 0.213 Biases Total Bias RMSE − − − − − Set(6): 1.99 4.984 1.628 2.034 4.485 1.475 1.988 4.122 1.967 1.106 3.934 0.733 3.941 2.082 6.738 2.292 1.749 1.1712 estimators Mean of 5 5 5 5 5 5 ...... = 4 = 4 = 4 = 4 = 4 = 4 = 2 = 2 = 2 = 2 = 2 = 2 = 0 = 0 = 0 = 0 = 0 = 0 β β β β β β α α α α α α α α α α α α Parameters 30 50 20 10 100 300 Size Sample

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 312 Mohamed Ali Ahmed Sample Size 300 100 10 20 50 30 Parameters α α α α α α α α α α α α β β β β β β 0 = 0 = 0 = 0 = 0 = 0 = 2 = 2 = 2 = 2 = 2 = 2 = 5 = 5 = 5 = 5 = 5 = 5 = ...... 5 5 5 5 5 5 enofl Mean estimators 1.989 2.290 4.944 0.765 4.926 8.815 1.688 1.976 1.146 5.159 1.627 2.094 1.415 5.628 6.363 2.039 1.559 1.99 Set(7): − − − − issTt isRMSE Bias Tota Biases − 0.200 0.600 0.159 0.915 0.628 0.030 1.059 0.290 3.815 1.188 1.127 0.094 1.363 0.056 0.074 0.024 0.009 0.01 ( al A7: Table α 0 = . 0.207 0.605 1.771 0.928 1.231 5 4.00 β , 2 = θ , 5) = 0.809 0.188 0.634 1.365 0.659 0.297 8.405 2.908 1.887 1.917 0.377 2.377 2.856 0.507 2.853 4.273 2.804 0.46 RMSE Total 1.044 8.918 2.347 3.076 5.162 4.028 ero System Pearson Coefficients − − − − − − 0.283 0.273 0.238 0.219 0.021 0.254 0.294 0.242 0.071 0.232 0.234 0.65 0.462 0.234 0.087 0.258 0.011 0.715 Pearson Type IV IV IV IV IV IV IV IV IV IV IV IV I I I I I I

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313 On the Alpha Power Kumaraswamy Distribution 313   Recibido: noviembre de 2019 — Aceptado: abril de 2020

References

Ali Ahmed, M. (2019), ‘The new form libby-novick distribution’, Communications in Statistics-Theory and Methods p. 1.

Arnold, B. C., Balakrishnan, N. & Nagaraja, H. N. (1992), A first course in order statistics, Siam. Garthwaite, P., Jolliffe, I. & Byron Jones, B. (2002), Statistical inference, Oxford University Press. Gradshteyn, I. S. & Ryzhik, I. M. (2000), Table of integrals, series, and products, Academic press, San Diego. Johnson, N. L., Kotz, S. & Balakrishnan, N. (1995), Continuous univariate distributions, John Wiley & Sons, Ltd, New York. Jones, M. (2009), ‘Kumaraswamys distribution: A beta-type distribution with some tractability advantages’, Statistical Methodology 6(1), 70–81. Kumaraswamy, P. (1980), ‘A generalized probability density function for double- bounded random processes’, Journal of Hydrology 46(1-2), 79–88. Mahdavi, A. & Kundu, D. (2017), ‘A new method for generating distributions with an application to exponential distribution’, Communications in Statistics- Theory and Methods 46(13), 6543–6557. McDonald, J. B. (2008), Some generalized functions for the size distribution of income, in ‘Modeling Income Distributions and Lorenz Curves’, Springer, pp. 37–55. Meeker, W. Q. & Escobar, L. A. (2014), Statistical methods for reliability data, John Wiley & Sons. Meniconi, M. & Barry, D. (1996), ‘The power function distribution: A useful and simple distribution to assess electrical component reliability’, Microelectronics Reliability 36(9), 1207–1212.

Merovci, F. & Puka, L. (2014), Transmuted pareto distribution, in ‘ProbStat Forum’, Vol. 7, pp. 1–11. Pearson, K. (1895), ‘X. Contributions to the mathematical theory of evolution .II. Skew variation in homogeneous material’, Philosophical Transactions of the Royal Society of London (A.) (186), 343–414.

Revista Colombiana de Estadística - Applied statistics 43 (2020) 285–313