We propose a two parameter Inverted Generalized Exponential (IGE) and a three parameter Generalized Inverted

Generalized Exponential (GIGE) probability models as generalizations of the one-parameter Exponential distribution and some other

distributions in the literature. We explore the statistical properties of the GIGE distribution and its parameters were estimated for both

censored and uncensored cases using the method of maximum likelihood estimation (MLE). An application to a real data set is also

provided to assess the flexibility of the GIGE distribution over some of its sub-models

Adejumo, A. O.

Oguntunde, P.E.

Covenant University Repository

J. Stat. Appl. Pro. 4, No. 2, 223-230 (2015) 223Journal of Statistics Applications & ProbabilityAn International Journalhttp://dx.doi.org/10.12785/jsap/040204The Generalized Inverted Generalized ExponentialDistribution with an Application to a Censored DataP. E Oguntunde1,∗ and A. O. Adejumo21 Department of Mathematics, Covenant University, Ogun State, Nigeria2 Department of Statistics, University of Ilorin, Kwara State, NigeriaReceived: 12 Jan. 2015, Revised: 24 Apr. 2015, Accepted: 23 Apr. 2015Published online: 1 Jul. 2015Abstract: We propose a two parameter Inverted Generalized Exponential (IGE) and a three parameter Generalized InvertedGeneralized Exponential (GIGE) probability models as generalizations of the one-parameter Exponential distribution and some otherdistributions in the literature. We explore the statistical properties of the GIGE distribution and its parameters were estimated for bothcensored and uncensored cases using the method of maximum likelihood estimation (MLE). An application to a real data set is alsoprovided to assess the flexibility of the GIGE distribution over some of its sub-models.Keywords: Censored, Exponential distribution, Generalizations, Inverted Generalized Exponential, Probability models, Uncensored.1 Introduction([1]) and ([2]) proposed and studied a generalization of the Exponential distribution called the Generalized Exponential(GE) distribution by introducing a shape parameter to the Exponential distribution. The cumulative density function (cdf)of the Generalized Exponential (GE) distribution is given by;FGE(x) = (1− e−λ x)α ;x > 0, α > 0, λ > 0 (1)The corresponding probability density function (pdf) is given by;fGE(x) = αλ e−λ x(1− e−λ x)α−1 ;x > 0, α > 0, λ > 0 (2)where;α is a shape parameterλ is the scale parameterBeside, inverted distributions have been studied by a number of researchers. For instance, the Inverted Exponentialdistribution was introduced by ([4]) and its applicability as a lifetime model has been identified by ([6]). Practically, if arandom variable X has an exponential distribution, the variable Y = 1X will have an Inverted Exponential (IE)distribution.Motivated by the work of ([3]), we shall first introduce a two parameter probability model known as the InvertedGeneralized Exponential (IGE) distribution which is practically the inverse or reciprocal of the Generalized Exponential(GE) distribution. With this understanding, we shall further propose a three parameter model named the GeneralizedInverted Generalized Exponential (GIGE) distribution.However, the rest of this article is organized as follows; In Section 2, we present both the IGE and GIGE distributions,Section 3 deals with some basic statistical properties of the proposed models coupled with the estimation of model∗ Corresponding author e-mail: peluemman@yahoo.comc© 2015 NSPNatural Sciences Publishing Cor.224 P. E Oguntunde, A. O. Adejumo: The Generalized Inverted Generalized Exponential...parameters; Section 4 provides the estimation of the model parameters using the method of maximum likelihoodestimation (MLE) for both the Censored and Uncensored cases, Section 5 discusses the application of the GIGEdistribution using a real data set followed by a concluding remark.2 The Generalized Inverted Generalized Exponential DistributionWe shall start by proposing the Inverted Generalized Exponential (IGE) distribution. Let X denote a non-negativecontinuous random variable and given that the Generalized Exponential distribution is as defined in Equation (1) andEquation (2), therefore, the cdf and the pdf of the Inverted Generalized Exponential distribution are respectively givenby;FIGE(x) = 1− (1− e−(λx ))α ;x,α,λ > 0 (3)fIGE (x) = αλ e−( λx )x−2(1− e−( λx ))α−1 ;x,α,λ > 0 (4)where;α is a shape parameterλ is the scale parameterWe can otherwise name the Inverted Generalized Exponential distribution as the Complementary or ReciprocalGeneralized Exponential distribution.We present the Reliability function and the Failure rate respectively as;SIGE(x) = (1− e−(λx ))α (5)For x > 0, α > 0 and λ > 0hIGE(x) = αλ e−(λx )x−2(1− e−(λx ))−1 (6)For x > 0, α > 0 and λ > 0With this understanding, we can confidently propose the Generalized Inverted Generalized Exponential (GIGE)distribution by generalizing the IGE distribution as follows;FGIGE(x) = 1− (1− e−γ(λx ))α ;x,α,λ ,γ > 0 (7)Differentiating Equation (7) with respect to x gives the pdf of the propose GIGE distribution as;fGIGE (x) = αλ γx−2e−γ( λx )(1− e−γ( λx ))α−1 ;x,α,λ ,γ > 0 (8)where;α and γ are shape parametersλ is the scale parameterFor notation purposes, we write; X ∼ GIGE(λ ,α,γ)The plot for the pdf and cdf of the GIGE distribution at various parameter values are given in Figure 1 and 2 respectively.As shown in Figure 1, the proposed GIGE distribution is positively skewed and the shape of the model is unimodal.Special Cases:Some known distributions in the literature are found to be sub-models of the proposed GIGE distribution. For instance,–For γ = 1, we get the Inverted Generalized Exponential (IGE) distribution.–For α = γ = 1, we get the Inverse Exponential distribution.Observations from the GIGE distribution with parameters α,λ ,γ can be simulated using the following transformation;X = λ γ[log(1− (1−U) 1α )]−1 (9)where U is a random variable uniformly distributed on (0,1).c© 2015 NSPNatural Sciences Publishing Cor.J. Stat. Appl. Pro. 4, No. 2, 223-230 (2015) / www.naturalspublishing.com/Journals.asp 225Fig. 1: Plot for the pdf of the GIGE distribution at α = 1, λ = 3, γ = 2Fig. 2: Plot for the cdf of the GIGE distribution at α = 1, λ = 3, γ = 23 Statistical Properties of the GIGE DistributionThis section provides some basic statistical properties of the proposed Generalized Inverted Generalized ExponentialDistribution.3.1 Reliability AnalysisThe reliability (survival) function is given by;S(x) = 1−F(x) (10)Therefore, the reliability function for the GIGE distribution is given by;SGIGE(x) = (1− e−γ(λx ))α (11)For x > 0,α > 0,λ > 0,γ > 0The corresponding plot for the reliability function of the GIGE distribution is as shown in Figure 3;The probability that a system having age ′x′ units of time will survive up to ′x+ t ′ units of time for x > 0,α > 0,λ >0,γ > 0 and t > 0 is given by;SGIGE(t|x) =SGIGE(x+ t)SGIGE(x)(12)c© 2015 NSPNatural Sciences Publishing Cor.226 P. E Oguntunde, A. O. Adejumo: The Generalized Inverted Generalized Exponential...Fig. 3: Plot for the Survival function of the GIGE distribution at α = 1, λ = 3, γ = 2Fig. 4: Plot for the Failure rate of the GIGE distribution at α = 1, λ = 3, γ = 2SGIGE(t|x) =(1− e−γ(λx+t ))α(1− e−γ( λx ))α(13)Hazard function is given by;h(x) = f (x)1−F(x)(14)Therefore, the corresponding failure rate (hazard function) is given by;hGIGE(x) = αλ γx−2e−γ(λx )(1− e−γ(λx ))−1 (15)x > 0,α > 0,λ > 0,γ > 0The plot for the failure rate at different parameter values is provided in Figure 4.3.2 MomentsThe r-th moment of a continuous random variable X is given by;µr = E[X r] =∫∞0xr f (x)dx (16)c© 2015 NSPNatural Sciences Publishing Cor.J. Stat. Appl. Pro. 4, No. 2, 223-230 (2015) / www.naturalspublishing.com/Journals.asp 227If a continuous random variable X is such that; X ∼ GIGE(α,λ ,γ),the rth moment is given by;E[X r] =∫∞0αλ γx−2e−γ( λx )(1− e−γ( λx ))α−1dx (17)Let θ = γ(λx) in Equation (17) and by following ([3]), then;E[X r] = αλ rγr∫∞0θ−re−θ (1− e−θ)α−1dθ (18)= αλ rγr∫∞0θ−re−θ (1+∞∑i=1aie−θ i)dθwhere;ai =((−1)i(α − 1)(α − 2)...(α − i))i!(19)Therefore;E[X r] = αλ rγrΓ (1− r)(1+∞∑i=1ai(i+ 1)1−r)(20)The general expression for the moment generating function (mgf) of the GIGE distribution is given by;MX(t) = αn∑r=0(trr!λ rγrΓ (1− r)(1+∞∑i=1ai(i+ 1)1−r))(21)4 Parameter EstimationThis section provides the estimation of the proposed GIGE distribution both for the Censored and Uncensored cases usingthe method of maximum likelihood estimation (MLE) as follows;4.1 For Censored Case:Censoring is a situation in which the value of an observation is only partially known. It is a form of missing data problemand it is common in survival analysis.Following ([3]), let Xi and Ci be independent random variables, where Xi denotes the lifetime of ith individual and Cidenotes the censoring time and ti = min(Xi,Ci) for i = 1,2,3, ....,n. Each Xi is distributed according to Equation (8) withparameters α,λ and γ .Let m denote the number of failures and let C and L denote the sets of censored and uncensored observationsrespectively.The likelihood function for the censored case is given by;L = ∏i⊂Ff (ti)∗∏i⊂CS(ti)where; f (ti) and S(ti) are the pdf and reliability function of the GIGE distribution as given in Equations (8) and (11)respectively.Let l = logL;l = m(logα + logγ + logλ )− 2 ∑i⊂Flog ti− γλ ∑i⊂Ft−1i +(α − 1) ∑i⊂Flog(1− e−γ(λti))+α ∑i⊂Clog(1− e−γ(λti))(22)c© 2015 NSPNatural Sciences Publishing Cor.228 P. E Oguntunde, A. O. Adejumo: The Generalized Inverted Generalized Exponential...Differentiating l with respect to α , λ and γ respectively gives;∂ l∂α =mα+ ∑i⊂Flog(1− e−γ(λti))∂ l∂λ =mλ − γ ∑i⊂F ti +(α − 1)γ ∑i⊂F t−1i(e−γ( λti )(1− e−γ(λti)))+αγ ∑i⊂Ct−1i(e−γ( λti )(1− e−γ(λti)))∂ l∂γ =mγ −λ ∑i⊂F ti +(α − 1)λ ∑i⊂F t−1i(e−γ( λti )(1− e−γ(λti)))The 3× 3 observed information matrix for hypothesis testing and interval estimation for parameters α , λ and γ is givenby;J(θ ) =− Jλ ,λ Jλ ,α Jλ ,γJα ,α Jα ,γJγ,γwhere;Jλ ,λ =∂ 2l∂λ 2 =−mλ 2 +(α − 1)γλ−1× ∑i⊂Ft−1i +(α − 1)(e−γ( λti )(1− e−γ(λti)))×(−γ(1− e−γ(λti))(λti))+αγλ−1× ∑i⊂Ct−1i(e−γ( λti )(1− e−γ(λti)))(−γ(1− e−γ(λti))(λti))(23)Jλ ,α =∂ 2l∂λ ∂α = γ ∑i⊂F t−1i(e−γ( λti )(1− e−γ(λti)))+ γ ∑i⊂Ct−1i(e−γ( λti )(1− e−γ(λti)))(24)Jλ ,γ =∂ 2l∂λ ∂γ =− ∑i⊂F t−1i +(α − 1)× ∑i⊂Ft−1i(e−γ( λti )(1− e−γ(λti)))×(1− γ(1− e−γ(λti))(λti))+α ∑i⊂Ct−1i(e−γ( λti )(1− e−γ(λti)))(1− γ(1− e−γ(λti))(λti))(25)Jα ,α =∂ 2l∂α2 =−mα2(26)Jγ,γ =∂ 2l∂γ2 =−mγ2 − (α − 1)λ2 ∑i⊂Ft−2i(e−γ( λti )(1− e−γ(λti))2)−αλ 2 ∑i⊂Ct−2i(e−γ( λti )(1− e−γ(λti))2)(27)4.2 For Uncensored Case:Let X1,X2,...Xn be a random sample of ’n’ independently and identically distributed random variables each having aGeneralized Inverted Generalized Exponential distribution defined in Equation (8), the likelihood function L is given by;L(X˜ |α,λ ,γ)=n∏i=1(αλ γx−2e−γ( λx )(1− e−γ( λx ))α−1)(28)c© 2015 NSPNatural Sciences Publishing Cor.J. Stat. Appl. Pro. 4, No. 2, 223-230 (2015) / www.naturalspublishing.com/Journals.asp 229Following Oguntunde et al (2014),Let l = logL(X˜ |α,λ ,γ),l = n logα + n logγ + n logλ − 2n∑i=1log(xi)−n∑i=1γ(λxi)+ (α − 1)n∑i=1log(1− e−γ(λxi)) (29)Differentiating l with respect to α , λ and γ respectively gives;∂ l∂α =nα+n∑i=1log(1− e−γ(λxi)) (30)∂ l∂λ =nλ −n∑i=1(λxi)+ (α − 1)n∑i=1( λxi)e−γ( λxi )(1− e−γ(λxi)) (31)∂ l∂γ =nγ −n∑i=1(λxi)+ (α − 1)n∑i=1( λxi)e−γ( λxi )(1− e−γ(λxi)) (32)Setting ∂ lα = 0,∂ lλ = 0,∂ lγ = 0 and solving the resulting nonlinear equations gives the maximum likelihood estimates ofparameters α , λ and γ .5 ApplicationTo assess the flexibility of the proposed GIGE distribution, we make use of a censored data given by ([5]), ([8]) and ([3]).The data consist of death times (in weeks) of patients with cancer of tongue with aneuploid DNA profile. The observationsare; 1, 3, 3, 4, 10, 13, 13, 16, 16, 24, 26, 27, 28, 30, 30, 32, 41, 51, 61, 65, 67, 70, 72, 73, 74, 77, 79, 80, 81, 87, 87, 88,89, 91, 93, 93, 96, 97, 100, 101, 104, 104, 108, 109, 120, 131, 150, 157, 167, 231, 240, and 400, where the twenty-oneobservations written in bold denote censored observations. The analysis in Table 1 is performed with the aid of R software.The Log-likelihood and Akaike Information Criteria (AIC) for the GIGE, IGE and IE distributions are provided.Table 1: Descriptive Statistics on Death TimesMin Q1 Q2 Mean Q3 Max Var Skewness Kurtosis1.00 30.00 78.00 81.76 100.20 400.00 4774.898 2.193221 10.35995Table 2: Performance of the GIGE, IGE and IE distributionsDistribution Parameters LOG-Likelihood AIC Rank by AICGIGE (Proposed) α̂ = 5.7711e−01 -300.856 607.712 2γ̂ = 4.0605e−03λ̂ = 2.7875e+03IGE (Proposed) α̂ = 0.5771 -300.856 605.712 1λ̂ = 11.3186IE ([4]) λ̂ = 17.3795 -306.107 614.213 3It is good to note that the model with the lowest AIC is ranked the best.c© 2015 NSPNatural Sciences Publishing Cor.230 P. E Oguntunde, A. O. Adejumo: The Generalized Inverted Generalized Exponential...6 ConclusionA two-parameter Inverted Generalized Exponential (IGE) distribution and a three-parameter Generalized InvertedGeneralized Exponential (GIGE) distribution are defined in this article. The basic statistical properties of the GIGEdistribution are rigorously studied. The model is positively skewed and its shape is unimodal. A number of distributionsare found to be sub-models of the GIGE distribution. We estimated the parameters of the GIGE distribution for both theCensored and Uncensored cases using the method of maximum likelihood estimation (MLE). The result based on thedata used, shows that the two-parameter Inverted Generalized Exponential (IGE) distribution is better than thethree-parameter Generalized Inverted Generalized Exponential (GIGE) distribution. This in turn means that, generalizingthe IGE distribution by introducing another shape parameter may not be needful except if the data set is more skewed.AcknowledgementThe authors are grateful to the anonymous referee for a careful checking of the details and for helpful comments thatimproved this paper.References[1] Gupta, R. D and Kundu, D., Generalized Exponential Distributions, Australian and New Zealand Journal of Statistics, 41 (2),173-188, 1999.[2] Gupta, R. D and Kundu, D., Generalized Exponential Distribution: Existing results and some recent developments, Journal ofStatistical Planning and Inference, Vol. 137, Issue 11, 3537-3547, 2007.[3] Jain, K., Singla N. and Sharma S. K., The Generalized Inverse Generalized Weibull Distribution and Its Properties, Journal ofProbability, Volume 2014, Article ID 736101, 11 pages, http://dx.doi.org/10.1155/2014/736101[4] Keller, A. Z and Kamath, A. R., Reliability analysis of CNC Machine Tools. Reliability Engineering. Vol. 3, pp. 449-473, 1982.[5] Klein J. P. and Moeschberger, M. L., SurvivalAnalysis: Techniques for Censoredand Truncated Data, Springer, New York,NY,USA,2003.[6] Lin, C. T, Duran, B. S and Lewis, T. O., Inverted Gamma as life distribution. Microelectron Reliability, Vol. 29 (4), 619-626, 1989.[7] Oguntunde, P. E, Adejumo A. O & Balogun O. S., Statistical Properties of the Exponentiated Generalized Inverted ExponentialDistribution, Applied Mathematics, 4 (2): 47-55, 2014.[8] Sickle-Santanello, B. J., Farrar, W. B., Keyhani-Rofagha, S., A reproducible System of flow cytometric DNA analysis of paraffinembedded solid tumors: technical improvements and statistical analysis, Cytometry, vol. 9, pp. 594-599, 1988.c© 2015 NSPNatural Sciences Publishing Cor.

The Generalized Inverted Generalized Exponential

Distribution with an Application to a Censored Data

Abstract: We propose a two parameter Inverted Generalized Exponential (IGE) and a three parameter Generalized Inverted Generalized Exponential (GIGE) probability models as generalizations of the one-parameter Exponential distribution and some other distributions in the literature. We explore the statistical properties of the GIGE distribution and its parameters were estimated for both censored and uncensored cases using the method of maximum likelihood estimation (MLE). An application to a real data set is also provided to assess the flexibility of the GIGE distribution over some of its sub-models

P E Oguntunde

A O Adejumo

CiteSeerX

The Generalized Inverted Generalized Exponential Distribution with an Application to a Censored Data

We propose a two parameter Inverted Generalized Exponential (IGE) and a three parameter Generalized Inverted Generalized Exponential (GIGE) probability models as generalizations of the one-parameter Exponential distribution and some other distributions in the literature. We explore the statistical properties of the GIGE distribution and its parameters were estimated for both censored and uncensored cases using the method of maximum likelihood estimation (MLE). An application to a real data set is also provided to assess the flexibility of the GIGE distribution over some of its sub-models

E Oguntunde, P.

O. Adejumo, A.

Arab Journals Platform

J. Stat. Appl. Pro.4, No. 2, 223-230 (2015) 223Journal of Statistics Applications & ProbabilityAn International Journalhttp://dx.doi.org/10.12785/jsap/040204The Generalized Inverted Generalized ExponentialDistribution with an Application to a Censored DataP. E Oguntunde1,∗ and A. O. Adejumo21 Department of Mathematics, Covenant University, Ogun State, Nigeria2 Department of Statistics, University of Ilorin, Kwara State, NigeriaReceived: 12 Jan. 2015, Revised: 24 Apr. 2015, Accepted: 23 Apr. 2015Published online: 1 Jul. 2015Abstract: We propose a two parameter Inverted Generalized Exponential (IGE) and a three parameter Generalized InvertedGeneralized Exponential (GIGE) probability models as generalizations of the one-parameter Exponential distribution and some otherdistributions in the literature. We explore the statistical properties of the GIGE distribution and its parameters were estimated for bothcensored and uncensored cases using the method of maximum likelihood estimation (MLE). An application to a real data seti alsoprovided to assess the flexibility of the GIGE distribution over some of its sub-models.Keywords: Censored, Exponential distribution, Generalizations, Inverted Generalized Exponential, Probability models, Uncensored.1 Introduction([1]) and ([2]) proposed and studied a generalization of the Exponentialdistribution called the Generalized Exponential(GE) distribution by introducing a shape parameter to the Exponential distribution. The cumulative density function (cdf)of the Generalized Exponential (GE) distribution is given by;FGE(x) = (1− e−λ x)α ;x > 0, α > 0, λ > 0 (1)The corresponding probability density function (pdf) is given by;fGE(x) = αλ e−λ x(1− e−λ x)α−1 ;x > 0, α > 0, λ > 0 (2)where;α is a shape parameterλ is the scale parameterBeside, inverted distributions have been studied by a number of researchers. For instance, the Inverted Exponentialdistribution was introduced by ([4]) and its applicability as a lifetime model has been identified by ([6]). Practically, if arandom variable X has an exponential distribution, the variable Y = 1X will have an Inverted Exponential (IE)distribution.Motivated by the work of ([3]), we shall first introduce a two parameter probability model known as the InvertedGeneralized Exponential (IGE) distribution which is practically the inverse or reciprocal of the Generalized Exponential(GE) distribution. With this understanding, we shall further propose a three parameter model named the GeneralizedInverted Generalized Exponential (GIGE) distribution.However, the rest of this article is organized as follows; InSection 2, we present both the IGE and GIGE distributions,Section 3 deals with some basic statistical properties of the proposed models coupled with the estimation of model∗ Corresponding author e-mail:peluemman@yahoo.comc© 2015 NSPNatural Sciences Publishing Cor.224 P. E Oguntunde, A. O. Adejumo: The Generalized Inverted Generaliz d Exponential...parameters; Section 4 provides the estimation of the model parameters using the method of maximum likelihoodestimation (MLE) for both the Censored and Uncensored cases, S ction 5 discusses the application of the GIGEdistribution using a real data set followed by a concluding remark.2 The Generalized Inverted Generalized Exponential DistributionWe shall start by proposing the Inverted Generalized Exponential (IGE) distribution. Let X denote a non-negativecontinuous random variable and given that the Generalized Exponential distribution is as defined in Equation (1) andEquation (2), therefore, the cdf and the pdf of the Inverted GeneralizedExponential distribution are respectively givenby;FIGE(x) = 1− (1− e−( λx ))α ;x,α,λ > 0 (3)fIGE (x) = αλ e−(λx )x−2(1− e−(λx ))α−1 ;x,α,λ > 0 (4)where;α is a shape parameterλ is the scale parameterWe can otherwise name the Inverted Generalized Exponentialdistribution as the Complementary or ReciprocalGeneralized Exponential distribution.We present the Reliability function and the Failure rate respectively as;SIGE(x) = (1− e−( λx ))α (5)For x > 0, α > 0 andλ > 0hIGE(x) = αλ e−(λx )x−2(1− e−(λx ))−1 (6)For x > 0, α > 0 andλ > 0With this understanding, we can confidently propose the Generaliz d Inverted Generalized Exponential (GIGE)distribution by generalizing the IGE distribution as follows;FGIGE(x) = 1− (1− e−γ( λx ))α ;x,α,λ ,γ > 0 (7)Differentiating Equation (7) with respect to x gives the pdf of the propose GIGE distribution as;fGIGE (x) = αλ γx−2e−γ(λx )(1− e−γ(λx ))α−1 ;x,α,λ ,γ > 0 (8)where;α andγ are shape parametersλ is the scale parameterFor notation purposes, we write;X ∼ GIGE(λ ,α,γ)The plot for the pdf and cdf of the GIGE distribution at various parameter values are given in Figure 1 and 2 respectively.As shown in Figure 1, the proposed GIGE distribution is positively skewed and the shape of the model is unimodal.Special Cases:Some known distributions in the literature are found to be sub-models of the proposed GIGE distribution. For instance,–For γ = 1, we get the Inverted Generalized Exponential (IGE) distribu ion.–For α = γ = 1, we get the Inverse Exponential distribution.Observations from the GIGE distribution with parametersα,λ ,γ can be simulated using the following transformation;X = λ γ[log(1− (1−U)1α )]−1 (9)where U is a random variable uniformly distributed on(0,1).c© 2015 NSPNatural Sciences Publishing Cor.J. Stat. Appl. Pro.4, No. 2, 223-230 (2015) /www.naturalspublishing.com/Journals.asp 225Fig. 1: Plot for the pdf of the GIGE distribution atα = 1, λ = 3, γ = 2Fig. 2: Plot for the cdf of the GIGE distribution atα = 1, λ = 3, γ = 23 Statistical Properties of the GIGE DistributionThis section provides some basic statistical properties ofthe proposed Generalized Inverted Generalized ExponentialDistribution.3.1 Reliability AnalysisThe reliability (survival) function is given by;S(x) = 1−F(x) (10)Therefore, the reliability function for the GIGE distribution is given by;SGIGE(x) = (1− e−γ( λx ))α (11)For x > 0,α > 0,λ > 0,γ > 0The corresponding plot for the reliability function of the GIGE distribution is as shown in Figure 3;The probability that a system having age′x′ units of time will survive up to′x+ t ′ units of time forx > 0,α > 0,λ >0,γ > 0 andt > 0 is given by;SGIGE(t|x) =SGIGE(x+ t)SGIGE(x)(12)c© 2015 NSPNatural Sciences Publishing Cor.226 P. E Oguntunde, A. O. Adejumo: The Generalized Inverted Generaliz d Exponential...Fig. 3: Plot for the Survival function of the GIGE distribution atα = 1, λ = 3, γ = 2Fig. 4: Plot for the Failure rate of the GIGE distribution atα = 1, λ = 3, γ = 2SGIGE(t|x) =(1− e−γ(λx+t ))α(1− e−γ(λx ))α(13)Hazard function is given by;h(x) =f (x)1−F(x)(14)Therefore, the corresponding failure rate (hazard functio) is given by;hGIGE(x) = αλ γx−2e−γ(λx )(1− e−γ(λx ))−1 (15)x > 0,α > 0,λ > 0,γ > 0The plot for the failure rate at different parameter values is provided in Figure 4.3.2 MomentsThe r-th moment of a continuous random variable X is given by;µr = E[X r] =∫ ∞0xr f (x)dx (16)c© 2015 NSPNatural Sciences Publishing Cor.J. Stat. Appl. Pro.4, No. 2, 223-230 (2015) /www.naturalspublishing.com/Journals.asp 227If a continuous random variable X is such that;X ∼ GIGE(α,λ ,γ),the rth moment is given by;E[X r] =∫ ∞0αλ γx−2e−γ(λx )(1− e−γ(λx ))α−1dx (17)Let θ = γ(λx ) in Equation (17) and by following ([3]), then;E[X r] = αλ rγr∫ ∞0θ−re−θ (1− e−θ)α−1dθ (18)= αλ rγr∫ ∞0θ−re−θ (1+∞∑i=1aie−θ i)dθwhere;ai =((−1)i(α −1)(α −2)...(α − i))i!(19)Therefore;E[X r] = αλ rγrΓ (1− r)(1+∞∑i=1ai(i+1)1−r)(20)The general expression for the moment generating function (mgf) of the GIGE distribution is given by;MX(t) = αn∑r=0(trr!λ rγrΓ (1− r)(1+∞∑i=1ai(i+1)1−r))(21)4 Parameter EstimationThis section provides the estimation of the proposed GIGE distribution both for the Censored and Uncensored cases usingthe method of maximum likelihood estimation (MLE) as follows;4.1 For Censored Case:Censoring is a situation in which the value of an observationis only partially known. It is a form of missing data problemand it is common in survival analysis.Following ([3]), let Xi andCi be independent random variables, whereXi denotes the lifetime ofith individual andCidenotes the censoring time andti = min(Xi,Ci) for i = 1,2,3, ....,n. EachXi is distributed according to Equation (8) withparametersα,λ andγ.Let m denote the number of failures and letC and L denote the sets of censored and uncensored observationsrespectively.The likelihood function for the censored case is given by;L = ∏i⊂Ff (ti)∗∏i⊂CS(ti)where; f (ti) andS(ti) are the pdf and reliability function of the GIGE distribution as given in Equations (8) and (11)respectively.Let l = logL;l = m(logα + logγ + logλ )−2 ∑i⊂Flogti − γλ ∑i⊂Ft−1i +(α −1) ∑i⊂Flog(1− e−γ(λti))+α ∑i⊂Clog(1− e−γ(λti))(22)c© 2015 NSPNatural Sciences Publishing Cor.228 P. E Oguntunde, A. O. Adejumo: The Generalized Inverted Generaliz d Exponential...Differentiatingl with respect toα, λ andγ respectively gives;∂ l∂α=mα+ ∑i⊂Flog(1− e−γ(λti))∂ l∂λ=mλ− γ ∑i⊂Fti +(α −1)γ ∑i⊂Ft−1i(e−γ( λti )(1− e−γ(λti)))+αγ ∑i⊂Ct−1i(e−γ( λti )(1− e−γ(λti)))∂ l∂γ=mγ−λ ∑i⊂Fti +(α −1)λ ∑i⊂Ft−1i(e−γ(λti)(1− e−γ(λti)))The 3×3 observed information matrix for hypothesis testing and interval estimation for parametersα, λ andγ is givenby;J(θ ) =−Jλ ,λ Jλ ,α Jλ ,γJα ,α Jα ,γJγ,γwhere;Jλ ,λ =∂ 2l∂λ 2=−mλ 2+(α −1)γλ−1× ∑i⊂Ft−1i +(α −1)(e−γ( λti )(1− e−γ(λti)))×(−γ(1− e−γ(λti))(λti))+αγλ−1× ∑i⊂Ct−1i(e−γ(λti)(1− e−γ(λti)))(−γ(1− e−γ(λti))(λti))(23)Jλ ,α =∂ 2l∂λ ∂α= γ ∑i⊂Ft−1i(e−γ( λti )(1− e−γ(λti)))+ γ ∑i⊂Ct−1i(e−γ( λti )(1− e−γ(λti)))(24)Jλ ,γ =∂ 2l∂λ ∂γ=− ∑i⊂Ft−1i +(α −1)× ∑i⊂Ft−1i(e−γ( λti )(1− e−γ(λti)))×(1−γ(1− e−γ(λti))(λti))+α ∑i⊂Ct−1i(e−γ( λti )(1− e−γ(λti)))(1−γ(1− e−γ(λti))(λti))(25)Jα ,α =∂ 2l∂α2=−mα2(26)Jγ,γ =∂ 2l∂γ2=−mγ2− (α −1)λ 2 ∑i⊂Ft−2i(e−γ( λti )(1− e−γ(λti))2)−αλ 2 ∑i⊂Ct−2i(e−γ( λti )(1− e−γ(λti))2)(27)4.2 For Uncensored Case:Let X1,X2,...Xn be a random sample of ’n’ independently and identically distribu ed random variables each having aGeneralized Inverted Generalized Exponential distribution defined in Equation (8), the likelihood functionL is given by;L(X̃ |α,λ ,γ)=n∏i=1(αλ γx−2e−γ(λx )(1− e−γ(λx ))α−1)(28)c© 2015 NSPNatural Sciences Publishing Cor.J. Stat. Appl. Pro.4, No. 2, 223-230 (2015) /www.naturalspublishing.com/Journals.asp 229Following Oguntunde et al(2014),Let l = logL(X̃ |α,λ ,γ),l = n logα + n logγ + n logλ −2n∑i=1log(xi)−n∑i=1γ(λxi)+ (α −1)n∑i=1log(1− e−γ(λxi)) (29)Differentiatingl with respect toα, λ andγ respectively gives;∂ l∂α=nα+n∑i=1log(1− e−γ(λxi)) (30)∂ l∂λ=nλ−n∑i=1(λxi)+ (α −1)n∑i=1( λxi )e−γ( λxi )(1− e−γ(λxi)) (31)∂ l∂γ=nγ−n∑i=1(λxi)+ (α −1)n∑i=1( λxi )e−γ( λxi )(1− e−γ(λxi)) (32)Setting ∂ lα = 0,∂ lλ = 0,∂ lγ = 0 and solving the resulting nonlinear equations gives the maxi um likelihood estimates ofparametersα, λ andγ.5 ApplicationTo assess the flexibility of the proposed GIGE distribution,we make use of a censored data given by ([5]), ([8]) and ([3]).The data consist of death times (in weeks) of patients with cancer of tongue with aneuploid DNA profile. The observationsare; 1, 3, 3, 4, 10, 13, 13, 16, 16, 24, 26, 27, 28, 30, 30, 32, 41,51,61, 65, 67, 70, 72, 73,74, 77,79, 80, 81, 87, 87, 88,89, 91, 93,93, 96,97, 100,101, 104,104, 108, 109, 120, 131, 150, 157, 167,231, 240, and400, where the twenty-oneobservations written in bold denote censored observations. The analysis in Table 1 is performed with the aid of R software.The Log-likelihood and Akaike Information Criteria (AIC) for the GIGE, IGE and IE distributions are provided.Table 1: Descriptive Statistics on Death TimesMin Q1 Q2 Mean Q3 Max Var Skewness Kurtosis1.00 30.00 78.00 81.76 100.20 400.00 4774.898 2.193221 10.35995Table 2: Performance of the GIGE, IGE and IE distributionsDistribution Parameters LOG-Likelihood AIC Rank by AICGIGE (Proposed) α̂ = 5.7711e−01 -300.856 607.712 2γ̂ = 4.0605e−03λ̂ = 2.7875e+03IGE (Proposed) α̂ = 0.5771 -300.856 605.712 1λ̂ = 11.3186IE ([4]) λ̂ = 17.3795 -306.107 614.213 3It is good to note that the model with the lowest AIC is ranked the best.c© 2015 NSPNatural Sciences Publishing Cor.230 P. E Oguntunde, A. O. Adejumo: The Generalized Inverted Generaliz d Exponential...6 ConclusionA two-parameter Inverted Generalized Exponential (IGE) distribution and a three-parameter Generalized InvertedGeneralized Exponential (GIGE) distribution are defined inthis article. The basic statistical properties of the GIGEdistribution are rigorously studied. The model is positively skewed and its shape is unimodal. A number of distributionsare found to be sub-models of the GIGE distribution. We estimated the parameters of the GIGE distribution for both theCensored and Uncensored cases using the method of maximum likelihood estimation (MLE). The result based on thedata used, shows that the two-parameter Inverted Generalized Exponential (IGE) distribution is better than thethree-parameter Generalized Inverted Generalized Exponential (GIGE) distribution. This in turn means that, generalizingthe IGE distribution by introducing another shape parameter may not be needful except if the data set is more skewed.AcknowledgementThe authors are grateful to the anonymous referee for a careful checking of the details and for helpful comments thatimproved this paper.References[1] Gupta, R. D and Kundu, D., Generalized Exponential Distributions, Australian and New Zealand Journal of Statistics, 41 (2),173-188, 1999.[2] Gupta, R. D and Kundu, D., Generalized Exponential Distribution: Existing results and some recent developments, Journal ofStatistical Planning and Inference, Vol.137, Issue 11, 3537-3547, 2007.[3] Jain, K., Singla N. and Sharma S. K., The Generalized Inverse Generalized Weibull Distribution and Its Properties, Journal ofProbability, Volume 2014, Article ID 736101, 11 pages, http://dx.doi.org/10.1155/2014/736101[4] Keller, A. Z and Kamath, A. R., Reliability analysis of CNC Machine Tools. Reliability Engineering. Vol.3, pp. 449-473, 1982.[5] Klein J. P. and Moeschberger, M. L., SurvivalAnalysis: Techniques for Censoredand Truncated Data, Springer, New York,NY,USA,2003.[6] Lin, C. T, Duran, B. S and Lewis, T. O., Inverted Gamma as life distribution. Microelectron Reliability, Vol.29 (4), 619-626, 1989.[7] Oguntunde, P. E, Adejumo A. O & Balogun O. S., StatisticalProperties of the Exponentiated Generalized Inverted ExponentialDistribution, Applied Mathematics,4 (2): 47-55, 2014.[8] Sickle-Santanello, B. J., Farrar, W. B., Keyhani-Rofagh , S., A reproducible System of flow cytometric DNA analysisof paraffinembedded solid tumors: technical improvements and statistic l analysis, Cytometry, vol.9, pp. 594-599, 1988.c© 2015 NSPNatural Sciences Publishing Cor.

English

http://eprints.covenantuniversity.edu.ng/5491/1/NSP_Oguntunde%5B1%5D.pdf

The Generalized Inverted Generalized Exponential Distribution with an Application to a Censored Data

Abstract

Similar works

Full text

Available Versions

Covenant University Repository

CiteSeerX

Arab Journals Platform