INVERSE EXPONENTIATED EXPONENTIAL POISSON DISTRIBUTION WITH THEORY AND APPLICATIONS

This paper is based on new extension of the exponential distribution named “Exponentiated Exponential Poisson Inverse (IEEP) distribution”. The distribution is based on lifetime issues containing three parameters. Likelihood method is used to estimate the parameters of the distribution. Explicit expressions for reliability/survival function, the hazard rate function, reversed hazard rate, the quantile function and mode are introduced. Maximum Likelihood estimates as well as asymptotic confidence interval are obtained using theory of the Maximum likelihood. For illustration and application, a real data set is analyzed and compared with three other model of literature. Model fitted here is better compared to other models for data considered. All the graphical and computation analysis is performed using R programming


INTRODUCTION
Statistical distributions are the basic aspects of all parametric statistical techniques including inference, survival analysis, modelling and reliability etc.In recent years, new generate families of continuous models are derived.Lifetime problems can be solved by using the existing probability models but to get more precise result, we need some flexible probability models.Due to constant failure rate, exponential model cannot explain data with variable failure rate.Hence, misuse of exponential lifetime model will not be suitable.Here, our aim is to introduce a new three parameter lifetime distribution with strong physical motivations.At Inverse Exponentiated Exponential Poisson Distribution with Theory and Applications International Journal of Engineering Science Technologies 18 present many of new models are proposed by modifying, merging, and adding or removing some parameters in existing models Marshall and Olkin (2007).That is existing models can be defined in new family of distributions.Several techniques can be defined to form new family of distribution adding some extra parameters to the existing distributions Rinne (2009).and Pham and Lai (2007).
CDF of the continuous random variable X following exponential distribution having constant θ is given as, ( ; ) 1 ; 0, 0. Some alternative generalizations of exponential distribution have been proposed to give some flexibility.
Although there are several generalizations of the exponential distributions, following two distributions have received more attention in literature with respect to others.
• Marshall & Olkin (1997) introduced inventive general method by adding some parameters to family of distributions which states that X have the Marshall-Olkin extended exponential (MOEE) distribution, say • Generalize: exponential (GE) distribution Gupta and Kundu (1999) can have decreasing and right skewed with single mode value.Let X follows GE distribution.That is ( , ) X GE α λ  . The CDF of X is given by { } ( ; , ) 1 ;x > 0 Above distribution has expression of the survival function like Weibull distribution and properties similar to Gamma distribution.
To derive this model, we have used exponential distribution and the Poisson distribution.Let us consider N system-based plant working independently where N is truncated Poisson rv.Suppose each of the system contains α independent and identically distributed units arranged in parallel.Suppose X is a random variable defining time to fail the system Ristić & Nadarajah (2014) and Kus (2007) then probability mass function of the N will be.

( )
Lal Babu Sah Telee, and Vijay Kumar International Journal of Engineering Science Technologies

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where n = 1, 2, 3… Unconditional CDF of X having three parameters was introduced and was named as EEP Ristić & Nadarajah (2014) We have defined here three parameters Inverse Exponentiated Exponential Poisson distribution (IEEP) taking inverse of random variable X with CDF ( )

MODEL ANALYSIS 2.1. INVERSE EXPONENTIATED EXPONENTIAL POISSON (IEEP) DISTRIBUTION
Let X follows new extended exponential distribution then CDF of the proposed model having three parameters is, ( (1)

PROBABILITY DENSITY FUNCTION
We defined model with pdf in expression as (2)

THE RELIABILITY FUNCTION
It is defined as probability of not failing an event before given time t.Reliability function of IEEP is

THE HAZARD RATE (HRF)
Hazard function rate is defined as the instantaneous failure rate at time t.HRF of proposed model is   Behaviour of the hazard rate shows high flexibility for different values of parameters.The HRF curve shows that the function is unimodal.Function is monotonically increasing along with monotonically decreasing.It is inverted bathtub hazard rates not showing constant hazard rates.We know that many of the lifetime's distribution does not show upside-down bathtub hazard rates, but it exhibits in case of proposed model.

STATISTICAL PROPERTIES
Major characteristics such as quantile function, skewness, and kurtosis etc of the proposed model IEEP are derived in this section.

QUANTILE FUNCTION
To study the theoretical aspect of probability model, quantile function is used.Statistical measures like, partition values, skewness as well as kurtosis of the probability models can be studied using quantile function.Generating function of random variable can be expressed in terms of quantile function.Quantile function can be used as the alternative function of PDF and CDF for finding the nature of the distributions.Quantile function of function can be obtained by using the relation U is uniform variate U (0, 1).If we put u = 0.5 in (5) then median will be obtained of the model can be obtained.

ASYMPTOTIC PROPERTIES
This property of pdf the density function follows condition of 0 limit ( ) limit ( ) with the resulting value as 0. That is, if both the limits converge to zero the proposed model satisfies the asymptotic behavior indicating that model value exists.
Since both the limits exist and have the limiting values as zero, we confirm that the proposed modal has unique mode.The necessary and sufficient conditions for mode are ( ) 0 df x dx = and

SKEWNESS AND KURTOSIS
These are the measures that describe the nature like consistency of data and the normality of probability distribution.Bowley's skewness Al-Saiary et al. ( 2019) based on quartiles can be calculated using expression as (1988) and Al-Saiary et al. (2019) introduced kurtosis using Octiles given by the relation.showing that distribution is skewed and not normal in nature.

SOME EXPANSIONS
Following distribution is derived for studying the various characteristic of the model by application of generalized binomial series.
The power series expansion of corresponding to an exponential function is; Using above two binomial series and exponential expansion in given pdf equation, the proposed model in series form is.

CALCULATION OF MOMENTS
Quantitative measurements of the distribution in form of function that describes characteristics of the probability distributions can be explained using moments.The X IEEP α β λ is given as Integrating equation ( 9), we can get th r raw moments of the IEEP can be obtained as 0 ' 1 0 ( ( 1)) (1 ) When r =1 then mean of the IEEP will be as Second order raw moment of IEEP can be obtained taking r as 2. That is.1)) (1 , ) Inverse Exponentiated Exponential Poisson Distribution with Theory and Applications International Journal of Engineering Science Technologies

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The conditional moments is Using density function and upper incomplete gamma function in equation ( 11), we can get conditional moment as.
Similarly, MGF of the proposed model is given as; Hence, we can get MGF as

RESIDUAL LIFE FUNCTION
Here, th n moment of the residual life of random variable X of the IEEP can be defined by − can expanded using binomial series expansions as, 0 ( ) ( 1) th n moment of residual life of X of the distribution becomes.
Lal Babu Sah Telee, and Vijay Kumar International Journal of Engineering Science Technologies

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(1 ) 2 Using upper incomplete gamma function in (13), we have Applying binomial expansion and substituting pdf, we can get following expression.

THE PROBABILITY WEIGHTED MOMENT
The probability weighted moment can be obtained using relation Where, Now, using equations ( 14) and (15), we can write Integrating equation ( 17), we get is order statistics of any sample of size n from IEEP.
The PDF of th m order statistics David & Nagaraja (2004) is defined as, ) Where (.,.) B denotes the beta function.Substituting the values of PDF and equation ( 16) replacing s by Where, Now moment of the order statistics is Using equations ( 18) and ( 19) the th r moment of the order statistics become

R'ENYI AND Q-ENTROPIES
The entropy is used in many fields such as in statistics, mathematics, engineering, physics, thermodynamics etc.It can be used to measures the variation of uncertainty of the random variable R'enyi entropy is defined as; ( ) ( ) where, 20), we can write Similarly, we can define q-entropy as; ( ) ( ) ∫ Thus, we can define the q-entropy is as.

PARAMETER ESTIMATION TECHNIQUE 4.1. ESTIMATION USING MAXIMUM LIKELIHOOD
Here we found the MLE of the parameters for constructed model.The MLE of the parameters are based on the observed sample x1, x2,…,xn.The likelihood function of parameters ( , , ) L α β λ is given by Log likelihood function ( , , ) L α β λ is given by Here, differentiating the log likelihood function ( 23) and maximum likelihood estimators were obtained by equating the differentiated equations to zero.That is ( ) Estimation of unknown parameters , , α β λ is done by solving nonlinear equations ( 24), ( 25) and ( 26).It will be difficult in solving these equations analytically so Newton-Rapson's iteration techniques is applied in log likelihood function of equation ( 23) using . The fisher's information matrix ( ) I δ given by; ( ) The maximum likelihood estimates (MLE) was obtained by solving three nonlinear equations analytically and using statistical software.
The ( ) x e x e ) is the parameter vector and ˆˆ( , , ) − ) as asymptotic normal where ( ) I δ is fishers information matrix given by It will be worthless that MLE gives asymptotic variance ( ) where H is the hessian.
We can obtain the observed information matrix maximizing the likelihood.For this Newton Rapshon algorithm is used.We have also find expression of variance covariance matrix as Hence from asymptotic normality of MLE approximate 100(1-γ ) % CI for the parameters are constructed using upper percentile standard normal variate as, / 2 / 2 / 2 ˆˆˆv ar( ) , var( ) and var( ) where /2 z γ is the upper percentile of standard normal variate.

DESCRIPTIVE DATA ANALYSIS
Exploratory data analysis is a collection of different statistical analysis that explains and to summarize the data set used in research.Objective of this is to gain detailed idea of data set used.It may include some descriptive statistics as well as the graphical plots of the data.Following are main measures that can be included in descriptive data analysis.
• The boxplot, histogram, density curve etc.These are a graphical plot that help to find the pattern of the data and also helps to detect if there is any unusual pattern and observations in data.• Measures of location, measures of scatters, skewness, and kurtosis etc gives some specific aspect and nature of the data.R programming language is used to find summary of the data and the values obtained are tabulated below in table

PARAMETER ESTIMATION
There are various methods and tools for optimization of the function.Forgetting maximum likelihood estimates (MLE) of the defined model log-likelihood function defined in expression ( 23) is maximized.The maximum likelihood estimates their standard error along with 95% confidence Interval (CI) for parameters , α β and λ is obtained using R programming [R Development core   Team, (2016)].We have also used the quasi-Newton-Raphson algorithm in R [Rizzo, 2008] for maximum likelihood estimation.The profile plots of negative log-likelihood function of proposed model for , α β and λ are plotted separately and are shown in Figure 3.  4.

MODEL COMPARISON
Here, log-likelihood as well as the information criteria values like (i) Akaike's information criteria (AIC), (ii) Bayesian information criteria(BIC) (iii) Corrected Akaike's information criteria(CAIC), and (iv) Hannan-Quinn Information Criteria(HQIC)are calculated and tabulated in Table 2 and Table 5 Following relations are used to find the values of AIC, BIC, CAIC and HQIC.

( ) 2
where n and k are total number of samples and total number of constants respectively.Since IEEP has least information criteria values with respect to the other competing, it is considered that IEEP fits data well.

MODEL VALIDATION
For model validation, we can use different statistical techniques.Here we have used two types of graphical plots; probability versus probability (P-P) plots and quantile versus quantile (Q-Q) plots are drawn and are shown in

SUMMARY AND CONCLUSION
This article is based on derivation, study and application of newly introduced probability model having three parameters.It is name as Inverse Exponentiated Exponential Poisson Inverse along with some statistical and mathematical properties, probability, weighted moments, order statistics, skewness, kurtosis, residual life time, entropy and survival functions etc. Different information criteria values are obtained for both the IEEP and the considering model and are compared.Study also showed that the goodness of fit statistics has least test statistics and higher p value respective to the other considering model.We have also plotted the empirical cdf versus the fitted cdf as well as the histogram versus the fitted density plot of the models.All the statistical computations and the graphical measures are performed using R language programming.
and scale constants.MOEE model reduces to exponential distribution for α equal to 1.
4) Some of density function curves and hazard curves of IEEP using some of values of α and β are plotted in Figure 1 at constant value of λ = 2 indicating that density curve of the IEEP is of different shapes at different parameters values.

Figure 1 Figure 1
Figure 1 th r raw moment ' r µ new model ~( , , ) Poisson Distribution with Theory and Applications International Journal of Engineering Science Technologies 28 asymptotic variance can be done by taking estimated values of the parameters.For this fisher's information matrix ( ) O δ  which is given as;

Figure 2
Figure 2 represents the boxplot and the histogram & and density fit of the proposed model IEEP.

Figure 3 Figure 3
Figure 3 Figure 5. PP and QQ plots show the theoretical distribution versus distribution.A P-

Figure 4 Figure 4
Figure 4 statistics.Concave shape of Curve of the TTT plot shows that hazard rate curve is increasing.

Figure 6 Figure 6
Figure 6 Inverse Exponentiated Exponential Poisson Distribution with Theory and Applications for model IEEP is, Statistical measures of this new model are obtained.For this, 100 random samples from the quantile function mentioned in expression (5).Here, we have taken initial values of the parameters as .By using the generated values different basic statistics of the proposed model are calculated.Table1Mean, Median, Mode, Sd, Skewness and Kurtosis of IEEP contains summaries for some set of parameters.
Inverse Exponentiated Exponential Poisson Distribution with Theory and ApplicationsInternational Journal of Engineering Science Technologies 22 Q (.) is quantile functions of the model.

Table 1 Table 1 Mean, Median, Mode, Sd, Skewness and Kurtosis of IEEP
Standard deviation is decreasing when values of and α β are increasing.Also values of λ are decreasing.Values of skewness as well kurtosis is not unique variance can be obtained.Mean median and others measures of the proposed model are given above in Table 1 Mean, Median, Mode, Sd, Skewness and Kurtosis of IEEP.Lower incomplete moments ( )

Table 3 Table 3 MLE, Standard Error and 95 Percent C. I
Inverse Exponentiated Exponential Poisson Distribution with Theory and ApplicationsInternational Journal of Engineering Science Technologies 32