Further Developments on the ( EG ) Exponential-MIR Class of Distributions

The Modified Inverse Rayleigh (MIR) distribution appeared in [Khan, M. S. (2014).Modified inverse Rayleigh distribution. International Journal of Computer Applications, 87(13):28–33] who got some theoretical properties of this distribution, and in[Nasiru, S., Mwita, P. N. and Ngesa, O. (2017). Exponentiated Generalized Exponential Dagum Distribution. Journal of King Saud UniversityScience, In Press] they introduced the (EG) Exponential-X class of distributions and obtained some theoretical properties with application. By assuming the random variable X follows the MIR distribution, some theoretical properties with application of the (EG) Exponential-MIR Class of distributions appeared in [Nasiru, S., Mwita, P. N. and Ngesa, O. (2018). Discussion on Generalized Modified Inverse Rayleigh Distribution. Applied Mathematics and Information Sciences, 12(1):113-124]. In the present paper we propose some extensions of the (EG) Exponential-MIR class of distributions. The (EG) ExponentialMIR class of distributions is part of Chapter 5 [Nasiru, S. (2018). A New Generalization of Transformed-Transformer Family of Distributions. Doctor of Philosophy thesis in Mathematics (Statistics Option). Pan African University, Institute for Basic Sciences, Technology and Innovation, Kenya], where the naming convention “NEGMIR” is used


Introduction T-X (W) Family of Distributions
This family of distributions is a generalization of the betagenerated family of distribu-tions first proposed by Eugene et.al [Eugene, N, Lee, C, Famoye, F: The beta-normal dis-tribution and its applications.Communications in Statistics-Theory and Methods 31 (4), 497-512 (2002)].In particular, let r(t) be the PDF of the random variable T ∈ [a, b], −∞ ≤ a < b ≤ ∞, and let W (F (x)) be a monotonic and absolutely continuous function of the CDF F (x) of any random variable X.The CDF of a new family of distributions defined by Alzaatreh et.al [ The PDF of the T-X (W) family of distributions is obtained by differentiating the CDF above.

Remark 1.2
When we set W(F(x)) : ln(1 F(x)) = − − then we use the term "T-X Family of Distributions" to describe all sub-classes of the T-X (W) family of distributions induced by the weight function    A random variable Y1 will be called T − X distributed of type I if the weight function is given by

Definition 2.2
A random variable Y2 will be called T − X distributed of type II if the weight function is given by

Definition 2.3
A random variable Y3 will be called T − X distributed of type III if the weight function is given by For some A random variable Y4 will be called T − X distributed of type IV if the weight function is given by For some The CDF of the (EG) Exponential-MIR class of distributions of type III has the following integral representation for a 0 ≥ and implies the following from Definition 2.8

Corollary 2.9
The CDF of the exponentiated Exponential-MIR class of distributions of type III is given by

Corollary 2.13
The CDF of the Exponential-MIR class of distributions of type IV is given by Where and

Application
In this section we compare the Exponential-Weibull class of distributions of type II and the Exponential-Weibull class of distributions of type IV in modeling the aircraft data,  The CDF of the Exponential-Weibull class of distributions of type IV is given by In order to compare the two distribution models, we used the following criteria: -2(Log likelihood) and AIC (Akaike information criterion) , AICC (corrected Akaike information criterion), and BIC (Bayesian information criterion) for the data set.The better distribution corresponds to the smaller -2(Log-likelihood) AIC, AICC, and BIC values: Where k is the number of parameters in the statistical model, n is the sample size, and l is the maximized value of the loglikelihood function under the considered model.From

Concluding Remarks
Our hope is that the researchers will further develop the properties and applications of the new class of distributions presented in this paper.Finally we hope the new developments have practical significance in modeling biological data, health data, etc.
Alzaatreh, A, Lee, C, Famoye, F: A new method for generating families of continuous distributions.Metron 71(1), 63-79 (2013b)] is given by W(F(x )) a G(x) r(t)dt R{W(F(x))} = = ∫ Where R (•) is the CDF of the random variable T and Remark 1.1 (x) is the CDF of a base distribution, then L(x) gives the CDF of the exponentiated generalized class of distributions [G.M. Cordeiro, E.M.M. Ortega and C.C.D. da Cunha, The exponentiated generalized class of distributions, Journal of Data Science,11(1), (2013),1-27] Here we assume the random variable X follows the MIR distribution with CDF And the random variable T follows the exponential distribution with PDF and CDF Now put And observe the CDF of the (EG) Exponential-MIR family of distributions as proposed in [Nasiru, S, Mwita, P N and Ngesa, O.(2018) Discussion on Generalized Modified Inverse Rayleigh Distribution.Applied Mathematics and Information Sciences, 12(1):113-124; Nasiru, S. (2018).A New Generalization of Transformed-Transformer Family of Distributions.Doctor of Philosophy thesis in Mathematics (Statistics Option) Pan African University, Institute for Basic Sciences, Technology and Innovation, Kenya] is given by Further Developments In this section we present some new generalizations of the (EG) Exponential-MIR family of distributions which are induced by the other weight functions introduced in [Alzaatreh, A, Lee, C, Famoye, F: A new method for generating families of continuous distributions.Metron 71(1), 63-79 (2013b)], when the random variable T in the T-X(W) class of distributions has support [a, ) ∞ , where a 0 ≥ Definition 2.1

The
The CDF of the (EG) Exponential-MIR class of distributions of type II has the following integral representation for a 0 ≥ Note that ( ,, ,1,c )The CDF of the exponentiated Exponential-MIR class of distributions of type II is given by CDF of the Exponential-MIR class of distributions of type The CDF of the Exponential-MIR class of distributions of type III is the (EG) Exponential-MIR class of distributions of type IV has the following integral representation for 0

Remark 3. 1 . 2
When a random variable X follows the Exponential-Weibull class of distributions of type II we writeRemark 3When a random variable X follows the Exponential-Weibull class of distributions of type IV we write X EWIV(a, b, c, d)  In this section we assume the CDF of the Weibull distribution is given by and the CDF of the Exponential distribution is given by Theorem 3.3The CDF of the Exponential-Weibull class of distributions of type II is given by Proof in Corollary 2.7, let : a λ = and Theorem 3.4

Table 1
above, it is clear the EWII (3975.82,1.0863,121474) distribution has the smallest AICC and BIC values, whilst the EWIV (216660, 1.47149, 1.21237_107, 0.681738) distribution has the smallest -2(Log-likelihood) and AIC values.When we compared the CDF's of the two distributions we obtained the following On the other hand when we compared the PDF's of the two distributions we obtained the Following The results fromTable 1 and the Figures above, suggest the EWIV (216660, 1.47149, 1.21237_107, 0.681738) distribution is slightly better than the EWII (3975.82,1.0863, 121474) distribution in modeling the aircraft data, Table 3 [Suleman Nasiru, Peter N. Mwita and Oscar Ngesa, Discussion on Generalized Modified Inverse Rayleigh, Appl.Math.Inf.Sci. 12, No. 1, 113-124 (2018)]

Table 1 :
Criteria for Comparison