A Generalized Form of Power Transformation on Exponential Family of Distribution with Properties and Application

In this paper, we proposed a new generalized family of distribution namely new alpha power Exponential (NAPE) distribution based on the new alpha power transformation (NAPT) method by Elbatal et al. (2019). Various statistical properties of the proposed distribution are obtained including moment, incomplete moment, conditional moment, probability weighted moments (PWMs), quantile function, residual and reversed residual lifetime function, stress-strength parameter, entropy and order statistics. The percentage point of NAPE distribution for some specific values of the parameters is also obtained. The method of maximum likelihood estimation (MLE) has been used for estimating the parameters of NAPE distribution. A simulation study has been performed to evaluate and execute the behavior of the estimated parameters for mean square errors (MSEs) and bias. Finally, the efficiency and flexibility of the new proposed model are illustrated by analyzing three real-life data sets.


Introduction
From the past decades, there has been an increased interest in developing new generalized distributions by adding one or more additional parameters (shape) to an existing family of distributions.Adding extra parameter tends to bring more flexibility in the distribution and it is also useful to incorporate skewness into a family of distribution, Pescim et al. (2010).
Since the late 1980s, the method of adding parameters to an existing distribution or combining existing distributions has been used for generating new distribution.For instance, Azzalini (1985) proposed the skew normal distribution by introducing an additional parameter to the normal distribution.This additional parameter incorporates skewness and brings more flexibility to the symmetric normal distribution.Mudholkar & Srivastava (1993) proposed the exponentiated Weibull model with two shape parameters and one scale parameter.Due to the presence of an additional shape parameter, the proposed exponentiated Weibull model is more flexible than the two-parameter Weibull model.Marshall & Olkin (1997) proposed a new method for generating distributions by introducing an additional parameter to any distribution function.Gupta & Kundu (1999) introduce the generalized Exponential Pakistan Journal of Statistics and Operation Research distribution and discuss some of its recent developments.Eugene et al. (2002) proposed the Beta generated method that uses the Beta distribution to develop the Beta generated distributions.This Beta-generated approach was further generalized by Jones (2004).Alzaatreh et al. (2003) introduced a new method for generating families of continuous distributions called the T-X family.A detail regarding the various methods for generating new distributions has been given by Lee et al. (2013) and Tahir & Nadarajah (2015).
In the recent past, Mahdavi & Kundu (2017) proposed the alpha power transformation (APT) method.The proposed APT method is quite easy to apply by simply raising the cumulative distribution function (CDF) of an existing distribution to a power of an additional parameter "".They used the APT method and introduced alpha power Exponential (APE) distribution.Unal et al. (2018) proposed alpha power inverted Exponential distribution using the concept of the APT method.Hassan et al. (2019) proposed a three-parameter lifetime distribution namely alpha power transformed extended Exponential distribution (APTEE) motivated by the APT method.In recent past, Elbatal et al. (2019) used a new scheme to add an extra parameter to introduce a new class of distributions.The proposed method is the new alpha power transformation (NAPT) method.
According to Elbatal et al. (2019), the new alpha power transformation method is defined as follows: Let () be the CDF of a continuous random variable R X  , thus the CDF of the NAPT method is defined as, () And the corresponding probability distribution function (PDF) is defined as, ( ) They have used the proposed model to study a special class of distribution function namely new alpha power transformed Weibull (NAPTW) distribution.
The main aim of this paper is to introduce and study a new lifetime distribution by using the concept of the NAPT method proposed by Elbatal et al. (2019).We define the new distribution as new alpha power Exponential (NAPE) distribution.The NAPE distribution is a very versatile distribution which is effective in modeling various lifetime data having monotonic and non-monotonic hazard rate functions.The rest of the paper is organized as follows: In Section 2, we introduce the NAPE distribution and we provide a mixture representation to study the importance of the NAPE distribution.In Section 3, basic statistical properties of NAPE distribution including moments, entropy, order statistics and quantile function are derived.In section 4, the maximum likelihood estimation (MLE) method is applied for estimating the value of the parameters.In section 5, simulation study is performed to outline the performance of the parameters.In section 6, the analyses of three real-life data sets are presented to illustrate the usefulness and flexibility of the NAPE distribution.Finally, in section 7, we conclude the findings of the paper.

New Alpha Power Exponential (NAPE) distribution
In this section, we apply the new alpha power transformation (NAPT) method to a specific class of distribution, namely the Exponential distribution and we refer the new distribution as new alpha power Exponential (NAPE) distribution with shape parameter  and scale parameter  .
and the corresponding PDF is, The survival function ) The main motivation for using the NAPT method on Exponential distribution is as follows: a) The NAPT method is a very simple and efficient method of introducing only one additional parameter i.e., .b) The NAPT method provides greater flexibility to a family of distribution functions.c) The NAPT method makes the distribution richer and flexible which is capable of modeling monotonically increasing, monotonically decreasing, increasing-decreasing and bathtub shape hazard rate function.d) The NAPT method provides to be a better fit than other existing models.e) When the additional parameter  = 1, we get the original baseline distribution which in this case is the Exponential distribution.  .Also, from figure (2), it can be clearly observed that the hazard rate function is increasing, decreasing, constant, upside-down bathtub and bathtub shapes for different values of the parameters.

Useful expansion of NAPE distribution
In this section, the useful expansion of the mixture representation of the PDF and CDF is presented.Using the series representation, ( ) The PDF of the NAPE distribution given in equation ( 4) can be written as, ( ( ) Furthermore, another form of the PDF given in (10) which provides the following infinite combination  Also, the CDF of the NAPE distribution given in (3) can be written as,

(
) ( ) Let u be an integer, then the expression of (; , )(; , ) is derived as, The different functions derived from (10) to ( 14) can be used for deriving various statistical properties of the NAPE distribution.

Statistical Properties
In this section, some basic statistical properties of the NAPE distribution have been derived and discussed.

Moment and Moment Generating Function
In statistical probability theory, the moment generating function is used to determine the moments of a distribution, i.e., first moment (mean), second moment (variance), third moment (skewness) and fourth moment (kurtosis).

 
and the moment generating function of NAPE distribution is Substituting ( 4) in (15) we get, using the series representation given in equation ( 9) and binomial expansion in equation ( 16), the n th moment of NAPE distribution is obtained as, Also, the moment generating function of NAPE distribution can be derived by using 0 ( ) ( ; , ) Thus the moment generating function of NAPE distribution is obtained as, Hence proved.

Incomplete Moment
The incomplete moment is used for measuring inequality, for instance, the Lorentz curve and Gini measures of inequality all rely upon the incomplete moments (Butler & McDonald, (1989)).

Theorem 2:
The th r incomplete moment for the density function ) , ; ( Substituting ( 4) in (19) we get, ( ) using the series representation given in equation ( 9) and binomial expansion in equation ( 20), the th r incomplete moment of NAPE distribution is obtained as, ) , ( b a  is the lower incomplete Gamma function.

The Conditional Moment
Theorem 3: Let X ~) , (   NAPE , then the conditional moments for the random variable The conditional moment of a random variable X is defined as, Substituting ( 4) and ( 5) in ( 22) we obtained, ( ) Using the series representation given in equation ( 9) and binomial expansion in equation ( 23), the conditional moment of NAPE distribution is obtained as, ( ( ) where ) , ( b a  is the lower incomplete Gamma function.

Probability Weighted Moments(PWMs)
The PWMs are used for estimating the parameters of a probability distribution.
Proof: For a random variable X , the PWMs represented by Substituting equation ( 3) and (4) in equation ( 25 Using the series representation given in equation ( 9) and binomial expansion in the above equation, the PWMs of NAPE distribution is obtained as, (26) Hence proved.

Residual and Reversed Residual Life
The residual lifetime of the random variable

Renyi and q-entropy
In statistical theory, entropy is defined as a statistical tool for measuring the variation of the uncertainty of a random variable X .
Let the random variable Using the generalized binomial expansion series in the following form ( ) Solving the above equation, the Renyi entropy is derived as, ( ) Furthermore, the q-entropy say Using the generalized binomial expansion and solving the above expression, the q-entropy is derived as, ( , 0 0

Quantile function
The quantile function is used for simulation study and to measure the percentile.The quantile function is defined as the inverse of the cumulative distribution function F(x) for a random variable X .

   
Solving the above equation numerically, the quantile function of NAPE distribution is defined as,

Parameter Estimation
The method of maximum likelihood estimation method has been used for estimating the parameters of NAPE distribution. Let 1 be a random sample of size n from the NAPE distribution with PDF given in (4), then the loglikelihood function is For obtaining the partial derivatives, differentiating (36) for  and  we get, Setting ( 37) and (38) to zero and solving these equations simultaneously gives the MLE of  and  i.e., ˆand   .
However, solving these equations to get the estimates of the unknown parameter is quite difficult.Therefore, a numerical technique such as the newton-raphson method may be used to solve these non-linear equations.

Simulation Study
In this section, a simulation study has been performed to illustrate the behaviour of the estimates ˆ and  ˆ in terms of the sample size n.We generate 1000 random sample The bias and MSE are calculated by ( ) The average values of MSEs and Bias from NAPE distribution for different values of n are displayed in table 2.
From table 2, it can be observed that as the value of the sample size n increases i.e., n = (30, 50, 100, 150, 200), the MSEs and Bias decreases indicating the reliability and accuracy of the estimates.

Figure 1 :Figure 2 : 1 
Figure 1: Plot of the density function of NAPE distribution for different values of the parameters exponentiated generated (Exp-G) densities with power parameters a and b .
in the above equation, we derived the residual lifetime as, in the above equation, we derived the reversed residual lifetime as, Let us consider the identity, (30, 50, 100, 150, 200) from NAPE distribution using theorem 5.Then, considering the initial values of the parameters bias and MSE from NAPE distribution.

Figure 3 :
Figure 3: TTT plots of the first, second and third data sets The TTT plot of the first, second and third data sets are shown in figure 3. The TTT plot shown in figure 3(a) represents a decreasing hazard rate function, 3(b) represents an increasing hazard rate function and 3 (c) represents an increasing hazard rate function.We fit the proposed NAPE distribution to the above three data sets along with other competing distribution namely; alpha power Exponential (APE) (by Mahdavi & Kundu, 2017), alpha power inverted Exponential (APIE) (by Unal et al., 2018), Exponential (E) (by Gupta et al., 2010), Exponentiated Exponential (EE) (by Gupta & Kundu, 2001) and Generalized Inverted Exponential (GIE) (by Abouammoh & Alshangiti, 2009) distribution respectively.We have computed the maximum likelihood estimates (MLEs) along with its standard error (SE) and the associated log likelihood(-LogL) in all the cases.

Figure 4 :Figure 5 :
Figure 4: Plot of the estimated densities and CDFs of the fitted distributions for the first data set

Figure 6 :
Figure 6: PP-plots of the fitted distribution to first data set

Figure 7 :Figure 8 :
Figure 7: Plot of the estimated densities and CDFs of the fitted distributions for the second data set

Figure 10 :
Figure 10: Plot of the estimated densities and CDFs of the fitted distributions for the third data set

Figure 11 :Figure 12 :
Figure 11: Plot of the estimated density and CDF of NAPE distribution for the second data set

Table 1 :
The following table displays the percentage point for different values of the parameters

table 1 ,
it is observed that as the value of  increases, for a fixed value of  , the values of the percentage point increase.Also, as the value of  increases, for the fixed value of  , the value of the percentage point decreases.

Table 2 :
The average values of Bias and MSEs of NAPE distribution for different values of n.

Table 3 :
Descriptive statistic of the data sets

Table 4 ,
table 6 and table 8 provides the maximum likelihood estimates (MLEs) along with the standard error (SE), minus log-likelihood and p-values for the first, second and third data sets respectively.The analytical measures for the first, second and third data set are provided in table 5, table 7 and table 9 respectively.

Table 4 :
The MLEs (SE) of the parameter fitted to the first data set

Table 5 :
Analytical measures of the NAPE distribution and other competing distributions for the first data set

Table 6 :
The MLEs (SE) of the parameter fitted to the second data set

Table 7 :
The analytical measures of the NAPED and other competing distribution for the second data set From tables 6 and 7, it is clear that NAPE distribution provides the overall best fit as compared to the other wellknown probability distribution.Hence, we can say that NAPE distribution is more adequate as compared to the other competing distributions like Alpha Power Exponential (APE), Alpha Power Inverted Exponential (APIE), New Alpha Power Transformed Exponential (NAPTE), Exponential (E), Exponentiated Exponential (EE) and Generalized Inverted exponential (GIE) distribution for explaining the second data set.Figure

Table 8 :
The MLEs (SE) of the parameter fitted to the third data set

Table 9 :
The analytical measures of the NAPED and other competing distribution for the third data set