OJMSiOpen Journal of Modelling and Simulation2327-4018Scientific Research Publishing10.4236/ojmsi.2021.94021OJMSi-111569ArticlesPhysics&Mathematics Generalized Power Akshaya Distribution and Its Applications AhmedT. Ramadan1*AhlamH. Tolba2*BeihS. El-Desouky2*Department of Basic Sciences, High Raya Institute, New Damietta, EgyptDepartment of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt27082021090432333830, June 202127, August 2021 30, August 2021© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

In this paper, a new two-parameter distribution called generalized power Akshaya distribution extended from Akshaya distribution is introduced. This distribution is proposed to model lifetime data. Statistical properties like density, hazard, survival and moments are derived. Two parameters estimation is introduced using maximum likelihood and Bayesian techniques. Finally, an application of real data and a simulation study are introduced to illustrate the usefulness of the proposed distribution.

Akshaya Distribution Generalized Distributions Moments Estimation
1. Introduction

According to Shanker , the probability density function (pdf) of Akshaya distribution is given by

f ( x ; θ ) = θ 4 θ 3 + 3 θ 2 + 6 θ + 6 ( 1 + x ) 3 e − θ x ,   x , θ > 0 , (1)

the cumulative distribution function (CDF) is given by

F ( x ; θ ) = 1 − { 1 + θ 3 x 3 + 3 θ 2 ( θ + 1 ) x 2 + 3 θ ( θ 2 + 2 θ + 2 ) x θ 3 + 3 θ 2 + 6 θ + 6 } e − θ x ,   x , θ > 0 , (2)

and the hazard rate function is given by

h ( x ; θ ) = θ 4 ( 1 + x ) 3 θ 3 x 3 + 3 θ 2 ( θ + 1 ) x 2 + 3 θ ( θ 2 + 2 θ + 2 ) x + ( θ 3 + 3 θ 2 + 6 θ + 6 ) ,   x , θ > 0. (3)

The hazard rate function given in Equation (3) is increasing function of x and θ . However, Akshaya distribution is not suitable for many situations from a theoretical point of view. So, a more flexible extension of Akshaya distribution is introduced in this paper.

Ghitany et al.  used the transformation X = Y 1 α to generate a new distribution called power Lindley distribution. By using this transformation, a new generalized power Akshaya distribution can be introduced.

Let y = x α → x = y 1 α → d x = 1 α y 1 α − 1 d y

F ( y ) = F 0 ( x α ) , f ( y ) = α x α − 1 f 0 ( x α ) . (4)

The aim of this paper is to study some properties of the generalized power Akshaya distribution including the density and hazard functions as in Section 2. Section 3 studied some statistical properties like moments of the distribution, incomplete moments, mean residual lifetime and mean time to failure. Two methods of parameter estimation are given in Section 4. Application of two types of data, real data and simulation study are presented in Section 5 to show the flexleibility of the distribution.

2. Generalized Power Akshaya DistributionSome Basic Functions

According to Equations (1), (2) and (4), the cumulative distribution function (CDF) and the probability density function (pdf) of power Akshaya distribution are given respectively as

F ( x ; θ , α ) = 1 − { 1 + θ 3 x 3 α + 3 θ 2 ( θ + 1 ) x 2 α + 3 θ ( θ 2 + 2 θ + 2 ) x α θ 3 + 3 θ 2 + 6 θ + 6 } e − θ x α ,   x , θ , α > 0 , (5)

f ( x ; θ , α ) = α x α − 1 θ 4 θ 3 + 3 θ 2 + 6 θ + 6 ( 1 + x α ) 3 e − θ x α ,   θ , α > 0. (6)

Akshaya distribution function is obtained from Equation (5) when α = 1 .

From Equation (6), we can notice the behavior of f ( x ; θ , α ) at x = 0 and x = ∞ as the following

f ( 0 ) = ( ∞ ,   α < 1 , θ 4 θ 3 + 3 θ 2 + 6 θ + 6 ,   α = 1 , 0 ,   α > 1 ,     f ( ∞ ) = 0.

The following theorem shows that there are three shapes for the density function of the generalized power Akshaya distribution according to the values of the parameters θ and α .

Theorem 1. The density function of the generalized power Akshaya distribution given in Equation (6) is

(a) decreasing if 0 < α ≤ 1 ,   θ ≥ η 1 ,

(b) uni-modal if α ≥ 1 ,

(c) decreasing-increasing-decreasing if 0 < α ≤ 1 ,   0 < θ < η 1 ,

Where η 1 = 1 + 2 α − 2 3 α ( 1 − α ) α .

Proof. The first derivative of f ( x ; θ , α ) of generalized power Akshaya distribution is

f ′ ( x ) = α θ 4 θ 3 + 3 θ 2 + 6 θ + 6 x α − 2 ( 1 + x α ) 2 e − θ x α Ψ 1 ( x α ) , x > 0 ,

where Ψ 1 ( y ) = a 1 y 2 + b 1 y + c 1 ,   y = x α > 0 , and a 1 = − θ α ,   b 1 = 4 α − 1 − θ α ,   c 1 = α − 1 .

It is obvious that f ′ ( x ) and Ψ 1 ( y ) have the same sign. The function Ψ 1 ( y ) is:

(a) decreasing if Ψ 1 has one or no real roots,

b 1 2 − 4 a 1 c 1 ≤ 0 , then ( 4 α − 1 − θ α ) 2 + 4 θ α ( α − 1 ) ≤ 0

which implies to 0 < α ≤ 1 ,   θ ≥ 1 + 2 α − 2 3 α ( 1 − α ) α ,

with Ψ 1 ( 0 ) = c 1 , Ψ 1 ( ∞ ) = − ∞ .

(b) uni-modal with maximum value at the point y 1 = − b 1 2 a 1 if c 1 > 0 , i.e. α > 1 .

(c) combining stated conditions in (a) and (b) we note that the function Ψ 1 ( y ) changes its sign from negative to positive to negative and this completes the proof.

Figure 1 and Figure 2 show the CDF and pdf functions of generalized power Akshaya distribution for different values of θ , α .

The survival function, S ( x ) and the hazard function, H ( x ) of generalized power Akshaya distribution, are given respectively as

S ( x ; θ , α ) = 1 − F ( x ; θ , α ) = { 1 + θ 3 x 3 α + 3 θ 2 ( θ + 1 ) x 2 α + 3 θ ( θ 2 + 2 θ + 2 ) x α θ 3 + 3 θ 2 + 6 θ + 6 } e − θ x α ,   x , θ , α > 0 , (7)

H ( x ; θ , α ) = f ( x ; θ , α ) S ( x ; θ , α ) = α θ 4 x α − 1 ( 1 + x α ) 3 θ 3 ( 1 + x 3 α ) + 3 θ 2 ( 1 + ( θ + 1 ) x 2 α ) + 3 θ ( 2 + ( θ 2 + 2 θ + 2 ) x α ) + 6 ,   x , θ , α > 0. (8)

From Equation (8), we can notice that the behavior of H ( x ; θ , α ) at x = 0 is the same as the behavior of f ( x ; θ , α ) at x = 0 , so that

H ( 0 ) = ( ∞ ,   α < 1 , θ 4 θ 3 + 3 θ 2 + 6 θ + 6 ,   α = 1 , 0 ,   α > 1 ,

and

H ( ∞ ) = ( 0 ,     α < 1 , θ ,     α = 1 , ∞ ,     α > 1.

The analytic analysis of the hazard function shape is very complicated, so according to , Glaser’s theorem is applied here. Now,

η ( x ) = − f ′ ( x ) f ( x ) = α θ x 2 α + x α ( α ( θ − 4 ) + 1 ) + 1 − α x ( 1 + x α ) ,

η ′ ( x ) = Ψ 2 ( x ) x 2 ( 1 + x α ) 2 ,

where

Ψ 2 ( x ) = α θ ( α − 1 ) x 3 α + x 2 α ( 2 θ α 2 − 2 α ( θ − 2 ) − 1 )       + x α ( α − 1 ) ( α ( θ − 3 ) + 2 ) + α − 1.

Consequently, the hazard function, H ( x ) , and the function Ψ 2 ( x ) have the same sign and the function Ψ 2 ( x ) is:

(a) increasing if α ≥ 1 ,   θ > 0 ,

(b) decreasing if 0 < α ≤ 0.25 ,   θ > 0 ,

(c) decreasing-increasing-decreasing if 0.25 < α < 1 ,   θ > 0 .

Figure 3 and Figure 4 show the survival and hazard functions of generalized power Akshaya distribution for different values of θ , α .

3. Statistical Properties3.1. Moments

In this subsection, the first four moments about zero and about mean and the incomplete moments of generalized power Akshaya distribution are derived. The general form of the rth moment about zero is given by

μ ′ r = E ( x r ) = ∫ 0 ∞     x r f ( x ; θ , α ) d x . (9)

According to Equation (9) and using Equation (6), the rth moment about zero of generalized power Akshaya distribution is given by

μ ′ r = c α ∫ 0 ∞     x r x α − 1 ( 1 + x α ) 3 e − θ x α d x , c = θ 4 θ 3 + 3 θ 2 + 6 θ + 6 .

Let t = θ x α → x = ( t θ ) 1 α → d x = 1 θ α ( t θ ) 1 α − 1 d t ,

μ ′ r = c α ∫ 0 ∞ ( t θ ) r α ( t θ ) 1 − 1 α ( 1 + ( t θ ) ) 3 e − θ ( t θ ) 1 θ α ( t θ ) 1 α − 1 d t .

Finally, the rth moment about zero of generalized power Akshaya distribution is given by

μ ′ r = r θ − r α ( r 3 + 3 r 2 α ( 2 + θ ) + r α 2 ( 11 + 3 θ ( 3 + θ ) ) + α 3 ( 6 + θ ( 6 + θ ( 3 + θ ) ) ) ) Γ ( r α ) α 4 ( 6 + θ ( 6 + θ ( 3 + θ ) ) ) . (10)

Substituting in Equation (10) with r = 1 , 2 , 3 and 4 we get

μ ′ 1 = θ − 1 α ( 1 + α ( 3 ( 2 + θ ) + α ( 11 + 3 θ ( 3 + θ ) + α ( 6 + θ ( 6 + θ ( 3 + θ ) ) ) ) ) ) Γ ( 1 α ) α 4 ( 6 + θ ( 6 + θ ( 3 + θ ) ) ) ,

μ ′ 2 = 2 θ − 2 α ( 8 + α ( 12 ( 2 + θ ) + α ( 22 + 6 θ ( 3 + θ ) + α ( 6 + θ ( 6 + θ ( 3 + θ ) ) ) ) ) ) Γ ( 2 α ) α 4 ( 6 + θ ( 6 + θ ( 3 + θ ) ) ) ,

μ ′ 3 = 3 θ − 3 α ( 27 + α ( 27 ( 2 + θ ) + α ( 33 + 9 θ ( 3 + θ ) + α ( 6 + θ ( 6 + θ ( 3 + θ ) ) ) ) ) ) Γ ( 3 α ) α 4 ( 6 + θ ( 6 + θ ( 3 + θ ) ) ) ,

and

μ ′ 4 = 4 θ − 4 α ( 64 + 48 α ( 2 + θ ) + 4 α 2 ( 11 + 3 θ ( 3 + θ ) ) + α 3 ( 6 + θ ( 6 + θ ( 3 + θ ) ) ) ) Γ ( 4 α ) α 4 ( 6 + θ ( 6 + θ ( 3 + θ ) ) ) .

Also, the first four moments about mean can be deduced from moments about zero as follows

μ 1 = 0 ,

μ 2 = μ ′ 2 − ( μ ′ 1 ) 2 ,

μ 3 = μ ′ 3 − 3 μ ′ 2 μ ′ 1 + 2 ( μ ′ 1 ) 3 and

μ 4 = μ ′ 4 − 4 μ ′ 3 μ ′ 1 + 6 μ ′ 2 ( μ ′ 1 ) 2 − 3 ( μ ′ 1 ) 4 .

3.2. Incomplete Moments and Related Measures

In this subsection, we introduce the rth incomplete moment, m r ( y ) and some related measures like mean deviation about mean and median and Bonferroni and Lorenz curves.

3.2.1. Incomplete Moments

The rth incomplete moments is given by

m r ( y ) = ∫ 0 y     x r f ( x ) d x = y r ( θ y α ) − r α 6 + 6 θ + 3 θ 2 + θ 3 ( θ 3 Γ ( 1 + r α ) + 3 θ 2 Γ ( 2 + r α ) + 3 θ   Γ ( 3 + r α )       + Γ ( 4 + r α ) − θ 3 Γ ( 1 + r α , θ y α ) − 3 θ 2 Γ ( 2 + r α , θ y α )       − 3 θ   Γ ( 3 + r α , θ y α ) − Γ ( 4 + r α , θ y α ) ) . (11)

3.2.2. Mean Deviation about Mean and Median

Mean deviation about mean of a parameter X~ power Akshaya distribution, δ 1 ( x ) can be given as follows

δ 1 ( x ) = ∫ 0 ∞ | x − μ ′ 1 | f ( x ) d x = 2 μ ′ 1 F ( μ ′ 1 ) − 2 ∫ 0 μ ′ 1     x f ( x ) d x = 2 μ ′ 1 F ( μ ′ 1 ) − 2 m 1 ( μ ′ 1 ) , (12)

where m 1 ( . ) is the incomplete moment given in Equation (11) when r = 1 .

The mean deviation about median (M) of a parameter X~ power Akshaya distribution, δ 2 ( x ) can be given as follows

δ 2 ( x ) = ∫ 0 ∞ | x − M | f ( x ) d x = μ ′ 1 − 2 ∫ 0 M     x f ( x ) d x = μ ′ 1 − 2 m 1 ( M ) . (13)

3.2.3. Bonferroni and Lorenz Curves

The Bonferroni and Lorenz curves have large applications in economy to study income and poverty and other fields. Bonferroni and Lorenz curves are defined as

B ( p ) = 1 p μ ∫ 0 q     x f ( x ) d x = m 1 ( q ) p μ , (14)

L ( p ) = 1 μ ∫ 0 q     x f ( x ) d x = m 1 ( q ) μ . (15)

3.3. Quantile Function, Bowley Skewness and Moors Kurtosis

For any q ∈ ( 0,1 ) , the qth quantile function (Q(q)) is the solution of F ( Q ( q ) ) = q ; Q ( q ) > 0 , in other words, q = F − 1 ( x ; θ , α ) .

It’s obvious if we set q = 0.5 we get the median (M). Bowley skewness  and Moors kurtosis  can be obtained as

Bowleyskewness = Q ( 3 4 ) + Q ( 1 4 ) − 2 Q ( 1 2 ) Q ( 3 4 ) − Q ( 1 4 ) ,

and

Moorskurtosis = Q ( 3 8 ) − Q ( 1 8 ) + Q ( 7 8 ) − Q ( 5 8 ) Q ( 6 8 ) − Q ( 2 8 ) .

Mean residual lifetime (m) is a reliability term based on lifetime of the product. It is a way to give a numeric value based on the residual lifetime of the product. Mean residual lifetime (m) can be given as follows

m ( x ) = E [ X − x / X > x ] = 1 1 − F ( x ) ∫ x ∞ ( 1 − F ( t ) ) d t = [ e θ x α θ − 1 α ( θ x α ) − 1 α ( − x θ 1 α + ( θ x α ) 1 α ) ( 1 + 3 α ( 2 + θ ) + α 2 ( 11 + 9 θ + 3 θ 2 )       + α 3 ( 6 + 6 θ + 3 θ 2 + θ 3 ) ) Γ ( 1 α ) + x α 3 θ 1 α ( 3 ( 2 + 2 θ + θ 2 ) Γ ( 1 + 1 α , θ x α )       + 3 ( 1 + θ ) Γ ( 2 + 1 α , θ x α ) + Γ ( 3 + 1 α , θ x α ) + 6 θ   Γ ( 1 α , θ x α )       + 3 θ 2 Γ ( 1 α , θ x α ) + θ 3 Γ ( 1 α , θ x α ) ) ] / [ α 4 ( θ x α ( 6 + 3 θ ( 2 + x α )       + θ 2 ( 3 + 3 x α + x 2 α ) ) + e θ x α ( 6 + 6 θ + 3 θ 2 + θ 3 ) ) ] . (16)

3.5. Mean Time to Failure

Mean time to failure (MTTF) is also a reliability term based on lifetime of the product. It gives a numeric value based on a compilation of data to quantify a failure rate of the product. MTTF can be given as follows

MTTF = θ − 1 α ( 1 + α ( 3 ( 2 + θ ) + α ( 11 + 3 θ ( 3 + θ ) + α ( 6 + θ ( 6 + θ ( 3 + θ ) ) ) ) ) ) Γ ( 1 α ) α 4 ( 6 + 6 θ + 3 θ 2 + θ 3 ) . (17)

4. Parameters Estimation

In this section, two techniques including maximum likelihood estimation (MLE) method and Bayesian estimation method are used to estimate the parameters of generalized power Akshaya distribution.

4.1. Maximum Likelihood Estimation Method

Let ( x 1 , x 2 , ⋯ , x n ) be a random sample from generalized power Akshaya distribution, then the likelihood estimation function, L can be given as follows

L = ∏ i = 1 n     f ( x ; θ , α ) = α n θ 4 n ( θ 3 + 3 θ 2 + 6 θ + 6 ) n e − θ ∑ i = 1 n x i α ∏ i = 1 n     x i α − 1 ( 1 + x i α ) 3 , (18)

and the natural log likelihood function is given by

ln ( L ) = n { ln ( α ) + 4 ln ( θ ) − ln ( θ 3 + 3 θ 2 + 6 θ + 6 ) }       − θ ∑ i = 1 n     x i α + ∑ i = 1 n { ( α − 1 ) ln ( x i ) + 3 ln ( 1 + x i α ) } . (19)

The first derivatives of the natural log likelihood function with respect to θ , α are given by

∂ ∂ θ ln ( L ) = 4 n θ − 3 θ 2 + 6 θ + 6 θ 3 + 3 θ 2 + 6 θ + 6 − ∑ i = 1 n     x i α , (20)

∂ ∂ α ln ( L ) = n α − θ ∑ i = 1 n     x i α ln ( x i ) + ∑ i = 1 n { ln ( x i ) ( 1 + 3 x i α 1 + x i α ) } . (21)

Equations (20) and (21) have no analytic closed form when equating by zero, so numerical methods are used to give solutions. The second derivatives of the natural log likelihood function with respect to θ , α can be given by

∂ 2 ∂ θ 2 ln ( L ) = − 4 n θ 2 − 6 ( θ + 1 ) ( θ 3 + 3 θ 2 + 6 θ + 6 ) − ( 3 θ 2 + 6 θ + 6 ) 2 ( θ 3 + 3 θ 2 + 6 θ + 6 ) 2 , (22)

∂ 2 ∂ θ ∂ α ln ( L ) = ∑ i = 1 n     x α ln ( x i ) , (23)

∂ 2 ∂ α ∂ θ ln ( L ) = ∑ i = 1 n     x α ln ( x i ) , (24)

∂ 2 ∂ α 2 ln ( L ) = − n α 2 − θ ∑ i = 1 n     x i α ( ln ( x i ) ) 2 + ∑ i = 1 n { ln ( x i ) ( 1 + 3 x i α ln ( x i ) ( 1 + x i α ) 2 ) } . (25)

The ( 1 − ζ ) 100 % confidence interval for the parameters θ and α can be written as

( θ ^ L , θ ^ U ) = θ ^ ∓ z 1 − ζ 2 var ( θ ^ ) ,   ( α ^ L , α ^ U ) = α ^ ∓ z 1 − ζ 2 var ( α ^ ) ,

where θ ^ and α ^ are the maximum likelihood estimates of θ and α , z 1 − ζ 2 is the percentile of the standard normal distribution and var ( θ ^ ) , var ( α ^ ) are the asymptotic variances of maximum likelihood estimates calculated using the inverse of the information matrix as follows

F − 1 = [ − ∂ 2 ∂ θ 2 ln L − ∂ 2 ∂ θ ∂ α ln L − ∂ 2 ∂ α 2 ln L − ∂ 2 ∂ α ∂ θ ln L ] − 1 = [ var ( θ ^ ) cov ( θ ^ , α ^ ) cov ( α ^ , θ ^ ) var ( α ^ ) ] − 1 . (26)

4.2. Bayesian Estimation Method

In this subsection, Bayesian estimation (BE) approach is used to estimate the parameters θ and α which are assumed to be independent and follow gamma prior distribution with parameters a and b.

The gamma prior density function has the form

g ( u ; a , b ) = b a Γ ( a ) u a − 1 e − u b ,   u , a , b > 0. (27)

Then, the joint prior density of θ and α is given by

g ( θ , α ) = ∏ i = 1 n     g ( θ ) g ( α ) ∝ ( θ α ) a − 1 e − ( θ + α ) b . (28)

The joint posterior distribution function according to Bayesian procedure is given by

g ( θ , α | x _ ) = g ( θ , α ) L ( x _ ) ∫ g ( θ , α ) L ( x _ ) ∝ g ( θ , α ) L ( x _ ) . (29)

Substituting from Equations (28) and (18) into Equation (29) we get

g ( θ , α | x _ ) ∝ ( θ α ) a − 1 e − ( θ ( ∑ i = 1 n x i α + b ) + α b ) ∏ i = 1 n     x i α − 1 ( 1 + x i α ) 3 . (30)

Markov Chain Monte Carlo method (MCMC)  is used to summarize the posterior distribution numerically without calculating the normalized constant.

5. Applications and Goodness of Fit

In this section, the goodness of fit of generalized power Akshaya distribution to real lifetime data is proposed and compared with some one parameter and two parameters distributions.

The data set represents the waiting times (in minutes) before service of 100 bank customers and analyzed and examined by Ghitany et al.  for fitting the lindley distribution. The data set is given as follows

Some statistics like, − 2 ln ( L ) , Akaike Information Criterion (AIC), Kolmogorov-Samirnov Statistics (K-S) and Bayesian estimate (BE) for this data are computed to compare between various lifetime distributions. These statistics are shown in Table 1.

The best distribution fitting the data is the distribution with least − 2 ln ( L ) , AIC, and least K-S statistics and Table 1 showed that power Akshaya distribution is better than others. The inverse of the information matrix of power Akshaya distribution using the estimated parameters, θ ^ and α ^ according to the MLE method can be given by

F − 1 = [ var ( θ ^ ) cov ( θ ^ , α ^ ) cov ( α ^ , θ ^ ) var ( α ^ ) ] − 1 = [ 0.004 − v e − v e 0.002 ] − 1 , (31)

and the 95% confidence interval for the parameters θ and α can be given as

The MLE, BE estimates, − 2 ln ( L ) , AIC, K-S and P-value statistics
DistributionsMLEBE− 2 ln ( L )AICK-SP-value
θ ^α ^θ ^α ^
power Akshaya0.5560.8140.5590.813635.73639.730.040.9876
power Lindley0.1531.0830.1601.072636.64640.640.502.2e16
Akshaya0.368-0.368-649.72651.720.130.074
Akash0.295-0.296-641.93643.910.102.2e16
Lindley0.187-0.187-638.07640.070.0582.2e16
Exponential0.101-0.102-658.04660.040.1632.2e16

( θ ^ L , θ ^ U ) = ( 0.431 , 0.680 ) ,   ( α ^ L , α ^ U ) = ( 0.719 , 0.908 ) .

6. Simulation Study

In this section, random data of generalized power Akshaya is generated using the inverse of cumulative distribution function numerically. Mathematica program is used to generate different samples of the distribution when the size is n = 20, 50, 70, 100, and 150. The experiment is repeated 5000 times with initial values θ = 1.5 and α = 0.5 . Five quantities are examined in this study

(a) Mean of the estimated values (ME) of ν ^ , ν ^ = θ ^ , α ^ which equals 1 5000 ∑ i = 1 5000     ν ^ i .

(b) Average bias of the MLE (AB) of ν ^ which equals 1 5000 ∑ i = 1 5000 ( ν ^ i − ν ) .

(c) The mean squared error (MSE) of the MLE of ν ^ which equals 1 5000 ∑ i = 1 5000 ( ν ^ i − ν ) 2 .

(d) Average width (AW) of 95% confidence intervals of parameter ν = θ , α which equals ( ν ^ + 1.96 ν ) − ( ν ^ − 1.96 ν ) .

(e) Coverage probability (CP) of 95% confidence intervals of parameter ν = θ , α , i.e. the percentage of intervals that contain true values of the parameter ν .

Table 2 shows that

· The absolute value of the average bias | A B | for the parameters θ , α decreases as the sample size (n) increases.

· The mean squared error (MSE) for the parameters θ and α decreases as the sample size (n) increases.

· The average width (AW) for the parameters θ and α decreases as the sample size (n) increases.

Figure 5 shows the scaled TTT-transform, and it found increasing, and the empirical pdf for the simulated data. Figure 6 shows the Q-Q plots for the simulated data and distributions mentioned in Table 1 and it shows that the generalized power Akshaya distribution is the best fit for the data. Figure 7 shows the Kaplan Meier curve for the simulated data and the survival functions of the

Some measures of the simulated data for various sample sizes
nθα
MEABMSEAWCPMEABMSEAWCP
201.478−0.0220.0530.8810.9430.5320.0320.0090.3200.946
501.478−0.0220.0200.5560.9410.5140.0140.0030.1950.953
701.487−0.0130.0140.4710.9510.5110.0110.0020.1650.955
1001.487−0.0130.0090.3940.9550.5100.0100.0010.1370.958
1501.488−0.0120.0070.3220.9510.5080.0080.0010.1120.946
Posterior summaries for the simulated data
PriorParametersMeanSDMC Error2.5%Median97.5%
Gammaα0.81290.058510.0020070.72010.81030.9091
Gammaθ0.5590.067510.0017580.44080.55890.6887

distributions mentioned in Table 1 and also shows that the generalized power Akshaya distribution is the best fit for the data.

Table 3 shows a summary of some measures for the joint posterior distribution for the simulated data. Figure 8 and Figure 9 show the density and trace plot of parameters θ and α to assess the convergence visually.

7. Conclusion

A new two parameters lifetime distribution named generalized power Akshaya distribution has been introduced for modeling lifetime data. Some statistical properties such as cumulative distribution, density, survival, hazard and moments functions. Also, maximum likelihood and Bayesian techniques are used to estimate distribution parameters. The goodness of fit using − 2 ln ( L ) , Akaike Information Criterion (AIC), Kolmogorov-Samirnov Statistics (K-S) and P-value for real lifetime data have been presented to show its applicability overpower Lindley, Akshaya, Akash, Lindley and exponential distributions. Finally, a simulation study is carried out to show the mean of the estimated values. The average bias and mean square error of the maximum likelihood estimators of the model parameters are discussed. In addition, the coverage probability and average width of the confidence intervals for the parameters are calculated.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Cite this paper

Ramadan, A.T., Tolba, A.H. and El-Desouky, B.S. (2021) Generalized Power Akshaya Distribution and Its Applications. Open Journal of Modelling and Simulation, 9, 323-338. https://doi.org/10.4236/ojmsi.2021.94021

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