MODIFIED INVERSE GENERALIZED EXPONENTIAL DISTRIBUTION: MODEL AND PROPERTIES

A three parameter continuous probability distribution Modified Inverse Generalized Exponential Distribution: Model and Properties, is introduced in this article. To study the properties of the introduced model, probability distribution, density, survival and hazard rate functions are introduced. A data of real life is used for checking the application. Some important methods of estimation are used for estimation of the constants. Model validation is checked using Akakie’s information, Bayesian Information, Corrected Akaike’s information and Hannan Qiunan Information Criteria as well as by plotting the P-P and Q-Q plots. For testing the goodness of fit Kolmogrov Smirnov test, Anderson darling test and Cramer-von Mises test are used. All the data analysis is performed using R-language programming.


INTRODUCTION
Use of probability model is not limited to statistics only.Statistics has broad application in all the fields of study like applied science, management and economics etc.Since probability distribution has numerous use and applications, it is being modified and generalizing day by day.Probability distribution helps to simulate the real life conditions and to analyze, interpret, and summarize the real life data precisely and effectively and economically in very less time.In literature many new probability models are available that are formulated based on our present Modified Inverse Generalized Exponential Distribution: Model and Properties International Journal of Research -GRANTHAALAYAH 97 requirement of the data analysis.These techniques of introducing new distribution may be by adding some extra parameters to distribution, merging the distribution or inverting the variables etc.These methods make new distribution more flexible and useful than the existing distributions.Literature contains various models for studying the nature and potentiality of the data; still we need new models to explain the new emerging data more precisely.That is in all the cases, classical techniques are not effective as the new distributions.New family of distributions play important role to generalize different models by compounding to well known distribution for introducing suitable models having extra features and properties to handle the variety of data used in theory as well as in practical life Usman et al. (2017) Availability of statistical models for studying statistical data is not limited.It is getting introduced new model frequently that can explain various type of data with more precise results.This research is focused on construction of new parametric statistical model.One of methods of getting new model is by introducing extra parameters to existing distribution such as Weibull and exponential family of distribution Marhall & Olkin (1997).Extension of Lomax distribution applying family of Marshall and Olkin model Ghitany et al. (2007) is available in literature.
McDonald Lomax model Lemonte & Cordeiro (2013) has been obtained by Lomax distribution.Power Lomax distribution containing three constants is more flexible than existing Lomax distribution.This model has inverted bathtub as well as increasing and decreasing bathtub hazard rate function Rady et al. (2016).Model defined has increasing, decreasing and bathtub shaped hazard curve.Exponentiated Weibull Lomax distribution is formulated using exponentiated Weibull-G-family Hassan & Abd-Allah (2018).By taking alpha as an exponent, a new distribution called alpha power inverted exponential was introduced by Ceren et al. (2018) by use of inverted exponential model.Lomax random variable was used as generator by Ogunsanya et al. (2019) Maxwell et al. (2019).Lomax exponential distribution has increasing and decreasing hazard rate given by Ijaz & Asim (2019) which was formulated using Lomax distribution.Similarly, inverse Lomax-exponentiated G-family Falgore & Doguwa (2020) is based on Inverse Lomax distribution as generator.
In real life, numerous life time variables are available that may have shape of bathtub hazard rate function.There are many models in literature having bathtub shaped hazard rate curve also.We can get modification of Weibull distribution to get many modified models.Expression below is Weibull distribution having two parameters.
We can add an extra parameter α to modify Inverse generalized exponential distribution as to get new distribution called Modified Inverse Generalized Exponential (MIGE) Model.The cdf and pdf of MIGE model can be given as The whole study is studied dividing in different section.First section is introductory where introduction and literature review is mentioned.In second section model formulation and some important properties are mentioned.Next section contains the parameter estimation techniques, Application to real data set and Model comparison.Final section is the conclusion of the study.

Reverse hazard function:
The reverse hazard function of MIGE is, (5)

Cumulative hazard rate:
Cumulative hazard rate of the MIGE ( ) The Quantile function: The Quantile function is given by Lal Babu Sah Telee, and Vijay Kumar

Generation of random deviate:
Let u follows uniform distribution then generation of random deviate of MIGE ( ) We have also defined skewness as well as kurtosis based on quantiles Al-saiary et al. ( 2019) as, and Coefficient of kurtosis Moors (1988) and Al-saiary et al. ( 2019) is

METHODS OF ESTIMATION
This section includes some methods of parameter estimation of the proposed model.

Method of Maximum Likelihood Estimation (MLE)
Here, ML estimators (MLE's) of the MGIE model are estimated by using MLE method.Let ( ) α β λ then the log density function can be written as, Modified Inverse Generalized Exponential Distribution: Model and Properties Finding first order derivatives of (8) with respect to constantans x x x e e x x x x x e x e e e x x x x x x e e e x x x  ( ) ( ) Since Θ may not be known practically, so it will be worthless that the MLE has an asymptotic variance ( ) ( ) Newton-Raphson method may be used for optimization that will give the observed information matrix.Expression (11) is the variance covariance matrix.
( )  ( ) is a random sample with n items from F (.) is taken The LSE of α. β, and λ respectively, can be determined by minimizing the function (13) as Performing partial derivative of ( 13) with respect to constants as, x e e e x n x e x To find the weighted LSE of the function we have minimized the expression below with the parameters to be estimated.
( )  ( ) Var X i n i By minimizing the function ( 14) we can get weighted least square estimation, ( )

Estimation using Cramer-von Mises method
This method of estimating constants α, β, and λ are determined using minimization of the function First order partial derivatives are, ( ) x n e e x x Solving above partial derivatives setting to zero we will get the CVM estimators of the proposed model MIGE.

MODEL COMPARISON
Applicability testing of MIGE is presented in this section.We have compared the potentiality of the proposed model by comparing this model with other four well known distributions.These distributions are Modified Weibull (MW) Distribution, Generalized Exponential Extension (GEE. ) distribution, Weibull Extension (WE) distribution, and Generalized Exponential (GE) distribution.Information criteria values are presents in Table 4. Results shows that defined model fit data better compare to model taken in consideration.Histogram and the fitted pdf for proposed model as well as the competing models are mentioned in Figure 5.It also includes the fitted cdf and the empirical cdf of the model.Goodness-of-fit of the MIGE distribution with other four competing model used earlier by other researchers are compared also.We have also tabulated the value of KS, AD and CVM statistic using R function in Table 5.It is found from calculation that the MIGE smaller value of the test statistic with maximum p-value compared to other These evidences helped us to finalize that the MIGE gets well fit and produce more regular & valid results from other distributions used for testing.

CONCLUSION
Study is based on formulation of new probability model called Modified inverse Generalized Exponential distribution.Some statistical properties and their expressions are derived here.Pdf curve shows that the model is skewed and non normal in nature.Hazard rate curve is monotonically increasing and inverted bathtub shaped.Parameters of the model are estimated using three methods of estimation and the applicability of model is checked using a real data set.For validity testing P-P, Q-Q and fitted versus empirical distribution curves are plotted.For model comparisons, four existing models are considered and some information criteria values are also mentioned.It is found that model fits data better compared to considered model.To test the goodness of fit three well known methods are used.All the computations and the graphical measurement are performed using r programming.The proposed model will play a significant role in studying the different data sets more precisely and will help researcher for the further study of the probability models.
the above model is not bathtub.Many modifications have been performed on this model resulting new model having bathtub hrf.Exponentiated Weibull model introduced byMudholkar & Srivastava (1993) is well known modification of Weibull distribution.Lai et al. (2016) introduced new lifetime distribution using suitable limits on beta integrated distribution as

Figure 1
Figure 1 displays hazard rate curve and pdf curve of MIGE ( ) , , α β λ with different parameters.From pdf plot it is clear that the density plot for different values of parameter are of different shape.Hazard rate curve is increasing and decreasing or inverted bathtub shaped based on set of parameters.

Figure 1 Figure 1
Figure 1 calculations for the α.β, and λ we get the ML estimators of the MIGE ( ) , , α β λ model.But normally, it is not possible to solve non-linear equations mentioned above.Much computer software is available for solving such equations.Let ( (1-b) % CI for parameters of MIGE ( ) , , α β λ is determined by taking /2 b Z as the upper percentile of the standard normal variate.Modified Inverse Generalized Exponential Distribution: Model and Properties International Journal of Research -GRANTHAALAYAH Constants α, β, and λ of MIGE distribution and can be determined by minimizing the function (12) also as 2 Exponential Distribution: Model and Properties International Journal of Research -GRANTHAALAYAH 105

Figure 4 Figure 4 Figure 4
Figure 4 displays histogram versus density function under fitted distributions It also shows fitted quantile versus sample quantile under estimation techniques.

Figure 5 Figure 5
Figure 5 curve with estimated fitted CDF curve of the model MIGE in Figure6

Figure 6 Figure 6
Figure 6 in formulating Type III Odd Lomax exponential model.Compounding of inverted Lomax model with odd generalized exponential model results a new distribution called Odd generalized exponentiated Inverse Lomax model given by Sah Telee, and Vijay Kumar