A Quasi Gamma Distribution

Transcription

1 08; 3(4): 08-7 ISSN: Maths 08; 3(4): Stats & Maths Received: Accepted: Rama Shanker Department of Statistics, College of Science, Eritrea Institute of Technology, Asmara, Eritrea Kamlesh Kumar Shukla Department of Statistics, College of Science, Eritrea Institute of Technology, Asmara, Eritrea Shambhu Sharma Department of Mathematics, Dayalbagh Educational Institute, Dayalbagh, Agra, Uttar Pradesh, India Ravi Shanker Department of Mathematics, G.L.A. College, N.P University, Daltonganj, Jharkhand, India A Quasi Gamma Distribution Rama Shanker, Kamlesh Kumar Shukla, Shambhu Sharma and Ravi Shanker Abstract In this paper, a two-parameter quasi gamma distribution which includes one parameter quasi eponential distribution has been proposed and studied. Its moments have been obtained and it has been found that the first four even order moments about origin of the proposed distribution are the first four moments about origin of the gamma distribution. Its survival function, hazard rate function, mean residual life function and stochastic ordering have been discussed. Estimation of parameters has been discussed using both the method of moments and the method of maimum likelihood. Application of the distribution has been eplained with a real lifetime data from engineering. Keywords: Gamma distribution, Moments, Hazard rate function, Mean residual life function, Stochastic ordering, Parameter Estimation, Applications Introduction The eponential distribution having scale parameter is defined by its probability density function (pdf) and cumulative distribution function (cdf) f ; e ; 0, 0 F ; e ; 0, 0 (.) (.) Gupta and Kundu (00) [3] have introduced eponentiated eponential distribution and discussed its properties and applications. Although eponential distribution was the first one parameter lifetime distribution for modeling lifetime data from engineering and biomedical sciences, but due to one parameter it is not always a suitable lifetime distribution. Also the hazard rate function and the mean residual life function of eponential distribution are always constant and function of the parameter only. Recently Shanker et al (08) [4] proposed a new one parameter lifetime distribution named quasi eponential distribution (QED) having scale parameter and defined by its pdf and cdf Correspondence Kamlesh Kumar Shukla Department of Statistics, College of Science, Eritrea Institute of Technology, Asmara, Eritrea f ; e ; 0, 0 (.3), F ; ; 0, 0, (.4) where the function, z is the upper incomplete gamma function defined as ~08~

2 y, z e y dy; 0, z 0, (.5) z Shanker et al (08) [4] have discussed its various statistical properties including its moments, hazard rate function, mean residual life function, stochastic ordering, and estimation of parameter and established its superiority over eponential distribution through a lifetime dataset. A weighted version of eponential distribution with weight function, where 0 is the two-parameter gamma distribution. A two-parameter gamma distribution (GD) having parameters and is defined by its pdf and cdf f e 3 ;, ; 0, 0, 0, F3 ;, ; 0, 0, 0, (.6), (.7) where is a scale parameter and is a shape parameter. The first four moments about origin of gamma distribution are given by,, 3, It can be easily shown that the two-parameter gamma distribution reduces to classical eponential distribution at. Stacy (96) [0] obtained a generalization of gamma distribution using power transformation of gamma distribution and Stacy and Mihram (965) [] discussed the parameter estimation for the generalized gamma distribution. Detailed discussions about works done by different researchers on gamma distributions and their properties and applications are available in chapter 7 of Johnson et al (994) [5]. Nadarajah and Kotz (006) [8] proposed several eponentiated distributions including eponentiated gamma distribution and discussed their mathematical and statistical properties. Nadarajah and Gupta (007) [9] proposed another generalization of gamma distribution and discussed its application to drought data. Cordeiro et al (03) [] obtained the eponentiated generalized gamma distribution and discussed its application to lifetime data. The two-parameter Weibull distribution, introduced by Weibull (95) [5], is defined by its pdf and cdf f4 ;, e ; 0, 0, 0 (.8) F4 ;, e ; 0, 0, 0 (.9) It should be noted that Weibull distribution is the power transformation of eponential distribution. Due to additional parameter Weibull distribution proved to be better lifetime distribution than eponential distribution. Most of the research works done on Weibull distribution and its generalization and their applications are available in Murthy et al (994) and Lai (04) [6]. Gupta and Groll (96) [4] discussed in detail the gamma distribution as a lifetime model. Although gamma distribution is used as a lifetime model, it is not as much as the Weibull distribution. The main reason for the wide popularity of the Weibull distribution over the gamma distribution is that the survivor and the hazard functions of the gamma distribution involves upper incomplete gamma function and hence is more difficult to work with than that of Weibull distribution. Note that eponential distribution is the particular case of both gamma and Weibull distributions. Recently Shanker et al (06) [3] have detailed discussion about the comparative study of two-parameter gamma and Weibull distribution for modeling lifetime data from engineering and biomedical sciences and found that both are competing each other in many lifetime datasets, Weibull distribution gives better fit than gamma distribution. The statistical modeling and analysis of lifetime data are crucial in almost all branches of knowledge especially engineering and biomedical sciences. The one parameter eponential distribution and the two-parameter Weibull and gamma distributions are popular in statistics literature for modeling lifetime data. It has been observed that these lifetime distributions are not always a suitable model either due to theoretical or applied point of view. An attempt has been made to find a two-parameter lifetime distribution which competes well with gamma and Weibull distributions. In this paper a two-parameter quasi gamma distribution of which one parameter quasi eponential distribution is a particular case has been proposed. Its statistical properties including shapes of pdf for varying values of parameters, moments, hazard rate function, mean residual life function, stochastic ordering have been discussed. Method of moments and the method of maimum likelihood have been discussed for estimating its parameters. Finally, an application to a real lifetime data from engineering has been presented to test its goodness of fit over one parameter eponential and quasi eponential distributions and two-parameter gamma, Weibull and Gompertz distributions. A Quasi gamma distribution A two-parameter quasi gamma distribution (QGD) with parameters and is defined by its pdf ~09~

3 f5 e ;, ; 0, 0, 0 where is a scale parameter and is a shape parameter. It can be easily verified that at (.), the pdf (.) reduces to quasi eponential distribution (QED) introduced by Shanker et al (08) [4]. Note that like gamma distribution, QGD is the weighted version of QED with weight function, 0. The survival (reliability) function of QGD (.) can be obtained as t S ;, P X e t dt. Taking u, we have u and, u d du u. Thus S ;, e u du, (.) where, z is the upper incomplete gamma function defined as y, z e y dy ; y 0, 0 (.3) z Thus the cdf of QGD (.) can be defined as, F5 ;, S ;, ; 0, 0, 0 (.4) The behavior of pdf and cdf of QGD for varying values of parameters have been shown in figures and respectively. ~0~

4 Fig : Behavior of the pdf of QGD for varying values of parameters and Fig : Behavior of the cdf of QGD for varying values of parameters and 3. Moments The r th moment about origin of QGD can be obtained as r r E X e d r r r ;,,3,... (3.) r 0 Taking r,, 3, 4, 5, 6, 7 and 8 in (3.), the first eight moments about origin of QGD are obtained as 3, 3 3, 5 4, 5, ~~

5 7 7, 7 The variance of QGD can be obtained as It is obvious that the first four even order moments about origin of QGD correspond to the first four moments about origin of the gamma distribution. Also for, the first four even order moments about origin of QGD reduce to the first four moments about origin of eponential distribution. And for QED., moments about origin of QGD reduces to the corresponding moments of 4. Hazard rate function and mean residual life function Let X be a continuous random variable with pdf and cdf f F. The hazard rate function rate function) and the mean residual life function of X are respectively defined as h P X X f lim 0 F (also known as the failure (4.) and m E X X F t dt t f tdt F S (4.) The corresponding hazard rate function, h and the mean residual life function, m of QGD (.) are obtained as h e, (4.3) and m t f5 t;, dt S;,, t t e t dt t e t dt,,, (4.4) The behavior of h and m of QGD for varying values of parameters are shown in figures 3 and 4, respectively. ~~

6 Fig 3: Behavior of the hazard rate function of QGD for varying values of parameters and Fig. 4: Behavior of the mean residual life function of QGD for varying values of parameters and 5. Stochastic Ordering Stochastic ordering of positive continuous random variables is an important tool for judging their comparative behavior. A random variable X is said to be smaller than a random variable in the (i) stochastic order X st if FX F for all (ii) hazard rate order X hr if hx h for all (iii) mean residual life order X if m m for all (iv) likelihood ratio order X lr mrl if f f X X decreases in. The following results due to Shaked and Shanthikumar (994) [] are well known for establishing stochastic ordering of distributions ~3~

7 X lr X hr X mrl Xst The QGD is ordered with respect to the strongest likelihood ratio ordering as established in the following theorem: Theorem: Let X lr X and hence QGD X hr, and, X mrl and Proof: We have f ;, X ;, QGD X e ; 0 f Now st, f ;, X ln ; ln ln f,. This gives d d f ;, X ln ; f.,. If and or and, then. d fx ;, Thus if and or and, ln ;, 0. This means that and hence f d X hr, X mrl and. This shows fleibility of QGD over eponential, quasi eponential and gamma distributions. X st 6. Estimation of parameters In this section, the estimation of parameters of gamma distribution has been discussed using both the method of moments and the method of maimum likelihood. 6.. Method of Moments Estimation of Parameters Since QGD has two parameters to be estimated, taking the first two even order moments of QGD, we have 4 Again from and this gives 4, we get. 4. Replacing and 4 by their respective sample moments in the above epressions, method of moments estimate (MOME), of parameters, can be obtained. 6.. Maimum Likelihood Estimation of Parameters Let,,..., n be a random sample of size n from the QGD (.). The natural log likelihood function of QGD (.) can be given by i ln L n ln ln ln ln n i i n The maimum likelihood estimates (MLE s) ˆ, ˆ of, of QGD (.) is the solutions of the following log likelihood equations ~4~ i X lr

8 n ln L n i 0 i n ln L n ln n ln i 0, i d where is the sample mean and ln is the digamma function. d These two log likelihood equations do not seem to be solved directly because they can not be epressed in closed forms. The MLE s of can be computed directly by solving the log likelihood equation using Newton-Raphson method iteratively using R-software till sufficiently close values of ˆ and ˆ are obtained. Note that the initial values of the parameters for using Newton-Raphson method are the MOME of the parameters., ˆ, ˆ,, 7. Goodness of Fit The goodness of fit of the QGD has been eplained with a real dataset from engineering. The following data represent the tensile strength, measured in GPa, of 69 carbon fibers tested under tension at gauge lengths of 0mm, available in Bader and Priest (98) [] For this data set, QGD has been fitted along with one parameter eponential and quasi eponential distributions, and two - parameter Gompertz, Weibull and Gamma distributions. The pdf and cdf of Gompertz distribution are given by f ;, e e and F, e ;, e for 0, 0, 0. The ML estimates, values of ln L, Akaike Information criteria (AIC), K-S statistics and p-value of the fitted distributions are presented in table. The AIC and K-S Statistics are computed using the following formulae: AIC ln L k and 0 K-S Sup F n F, where cumulative distribution function, and k F0 = the number of parameters, n = the sample size, Fn is the empirical (sample) is the theoretical cumulative distribution function. The best distribution is the distribution corresponding to lower values of ln L, AIC, and K-S statistics. It is clear from the goodness of fit in table that QGD gives better fit among the considered distributions and hence it can be considered an important lifetime distribution to model lifetime data from engineering. Table : Summary of ML estimates, ln L, AIC, K-S and p-value of the fitted distributions AIC K-S p-value Distributions ML Estimates Standard Errors ln L Quasi gamma ˆ ˆ Gamma ˆ ˆ Weibull ˆ ˆ Gompertz ˆ ˆ Quasi eponential ˆ Eponential ˆ ~5~

9 The variance-covariance matri and 95% confidence intervals (C.Is) for the parameters presented in table. Table : The variance-covariance matri for the parameters Parameters ˆ ˆ, ˆ Variance-Covariance Matri ˆ ˆ ˆ, ˆ of QGD for given dataset of QGD for the given dataset are ˆ The fitted plots of the considered distributions for the given dataset are presented in figure 5. Fig 5: Fitted plots of the considered distributions for the given dataset. 8. Concluding Remarks A two-parameter quasi gamma distribution (QGD) has been proposed and studied. Its moments have been obtained and it has been found that the first four even order moments about origin of QGD are the first four moments about origin of the gamma distribution. Its survival function, hazard rate function, mean residual life function and stochastic ordering have been discussed. Estimation of parameters has been discussed using both the method of moments and the method of maimum likelihood. Application of the distribution has been eplained with a real lifetime data from engineering. The goodness of fit of QGD shows satisfactory fit over one parameter eponential and quasi eponential and two-parameter Gompertz, Weibull and gamma distributions. Since QGD competing well with two-parameter Gompertz, Weibull and gamma distributions, it can be considered an important two-parameter lifetime distribution for modeling lifetime data from engineering. 9. References. Bader MG, Priest AM. Statistical aspects of fiber and bundle strength in hybrid composites, In; hayashi, T., Kawata, K. Umekawa, S. (Eds), Progress in Science in Engineering Composites, ICCM-IV, Tokyo, 98, Cordeiro GM, Ortega EMM, Silva GO. The eponentiated generalized gamma distribution with application to lifetime data. Journal of Statistical Computation and Simulation. 03; 8: Gupta RD, Kundu D. Eponentiated Eponential family-an alternative to gamma and Weibull distributions. Biometrical Journal. 00; 43: Gupta SS, Groll PA. Gamma distribution in acceptance sampling based on life tests. Journal of the American Statistical Association. 96; 56: Johnson NL, Kotz S, Balakrishanan N. Continuous Univariate Distributions, Second edition, John Wiley and Sons, New ork, 994,. 6. Lai CD. Generalized Weibull Distributions, Springer Briefs in Statistics, Murthy DNP, Xie M, Jiang R. Weibull Models, John Wiley & Sons, New ork, Nadarajah S, Kotz S. The eponentiated type distributions, Acta Applicandae Mathematicae. 006; 9: Nadarajah S, Gupta AK. A generalized gamma distribution with application to drought data, Mathematics and Computers in simulation. 007; 74: Stacy EW. A generalization of the gamma distribution. Annals of Mathematical Statistics. 96; 33: Stacy EW, Mihram GA. Parameter estimation for a generalized gamma distribution, Technometrics, 965; 7: ~6~

10 . Shaked M, Shanthikumar JG. Stochastic Orders and Their Applications, Academic Press, New ork, Shanker R, Shukla KK, Shanker R, Tekie AL. On modeling of Lifetime data using two-parameter gamma and Weibull distributions. Biometrics & Biostatistics International journal. 06; 4(5): Shanker R, Shukla KK, Sharma S, Shanker R. A quasi eponential distribution, to appear in Biometrics & Biostatistics International journal Weibull W. A statistical distribution of wide applicability. Journal of Applied Mathematics. 95; 8: ~7~