Applied Mathematical Sciences, Vol. 8, 2014, no. 2 - PDF document

Applied Mathematical Sciences, Vol. 8, 2014, no. 2 7 , 12 97 - 1 310 HIKARI Ltd, www.m - hikari.com http://dx.doi.org/10.12988/ams.2014.4 2105 Transmuted Exponentiated Gamma Distribution: A Generalization of the Exponentiated Gamma Probability Distribution Mohamed A. Hussian Departm ent of Mathematical Statistics Institute of S tatistical Studies and Research Cairo University, Giza, Egypt Copyright © 2014 Mohamed A. Hussian . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The exponentiated gamma (EG) distribution is one of the important families of distributions in lifetime tests. In this paper, a new generalized version of this distribution which is called Transmuted exponentiated gamma (TEG) distribution is introduced. A new distribution is more flexible and has some interesting properties. A comprehensive mathematical treatment of the TEG distribution is provided. We derive the rth moment and moment generating function this distribution. Moreover, we discuss the maximum likelihood estimat ion of this distribution. Keywords: gamma distribution; Hazard function; exponentiated gamma distribution; Maximum likelihood estimation; Moments 1. Introduction and Motivation One of the important families of distributions in lifetime tests is the exponentiated gamma (EG) distribution. The exponentiated gamma (EG) distribution has been introduced by Gupta et al. (1998), they proposed the use of the exponentiated gamma distribution as an alternative to gamma and weibull distributions . The cumulative distribution function (c.d.f.) and a probability density function (p.d.f.) of the form, respectively; 1298 Mohamed A. Hussian (1.1) where and are scale and shape parameters respectively. The corresponding probability density function (pdf) is given by (1.2) Shawky and Bakoban (2008) discussed the exponentiated gamma distribution as an important model of life time models and derived Bayesian and non - Bayesian estimators of the shape parameter, reliability and failure rate functions in the case of complete and type - II censored samples. Also order statistics from exponentiated gamma distribution and associ ated inference was discussed by Shawky and Bakoban (2009). Ghanizadeh, et al. (2011), dealt with the estimation of parameters of the Exponentiated Gamma (EG) distribution with presence of outliers. The maximum likelihood and moment of the estimators were derived. These estimators are compared empirically using Monte Carlo simulation. Singh et al.(2011) proposed Bayes estimators of the parameter of the Exponentiated gamma distribution and a ssociated reliability function under General Entropy loss function for a censored sample. The proposed estimators were compared with the corresponding Bayes estimators obtained under squared error loss function and maximum likelihood estimators through the ir simulated risks. Khan and Kumar (2011) established the explicit expressions and some recurrence relations for single and product moments of lower generalized order statistics from exponentiated gamma distribution. Sanjay et el. (2011) where proposed Bay es estimators of the parameter of the exponentiated gamma distribution and associated reliability function under General Entropy loss function for a censored sample. Navid and Muhammad (2012) introduced Bayesian Analysis of Exponentiated Gamma Distribution under Type II Censored Samples. Recently, Parviz et al. (2013) discussed Classical and Bayesian estimation of parameters on the generalized exponentiated gamma distribution. 1.1. Transmutation Map In this subsection we demonstrate transmuted probability distr ibution. Let and be the cumulative distribution functions, of two distributions with a common sample space. The general rank transmutation as given in (2007) is defined as Note that t he inverse cumulative distribution function also known as quantile function is defined as The functions and both map the unit interval into itself, and Transmuted exponentiated gamma distribution 1299 under suitable assumptions are mutual inverses and they satisfy and A quadratic Rank Transmutation Map (QRTM) is defined as (1.3) from which it follows that the cdf's satisfy the relationship (1.4) which on differentiation yields, (1.5) where and are the corresponding pdfs associated with cdf and respectively. An extensive information about the quadratic rank transmutation map is given in Shaw et al. (200 7). Observe that at we have the distribution of the base random variable. The following Lemma proved that the function in given (1.5) satisfies the property of probability density function. Many authors dealing with the generalization of some well - known distributions . Aryal and Tsokos (2009) defined the transmuted generalized extreme value distribution and they studied some basic mathematical characteristics of transmuted Gumbel probability distribution and it has been observed that the transmuted Gumbel can be used to model climate data. Also Aryal and Tsokos (2011) presented a new generalization of Weibull distribution called the transmuted Weibull distribution . Mahmoud and Alam (2010) in troduced new generalization of the linear exponential distribution is generalized linear exponential distribution. This distribution is important since it contains as special sub - models some widely well known distributions. It also provides more flexibilit y to analyze complex real data sets. Recently, Aryal (2013) proposed and studied the various structural properties of the transmuted Log - Logistic distribution, and Muhammad khan and king (2013) introduced the transmuted modified Weibull distribution which extends recent development on transmuted Weibull distribution by Aryal et al. (2011). and they studied the mathematical properties and maximum likelihood estimation of the unknown parameters. Elbatal (2013) proposed a functional composition of the cumulat ive distribution function of one probability distribution with the inverse cumulative distribution function of another is called the transmutation map.He used the quadratic rank transmutation map (QRTM) in order to generate a flexible family of probability distributions taking modified inverse weibull distribution as the base value distribution by introducing a new parameter that would offer more distributional flexibility. It will be shown that the analytical results are applicable to model real world data . Elbatal and Aryal (2013) presented the transmuted additive Weibull distribution, that extends the additive Weibull distribution and some other distributions they used the quadratic rank transmutation map (QRTM) proposed by Shaw & Buckley( 2007) in order to generate the 1300 Mohamed A. Hussian transmuted additive Weibull distribution. The rest of the paper is organized as follows. In Section 2 we demonstrate transmuted probability distribution, the hazard rate and reliability functions of TEG distribution. In Section 3 we studied the statistical properties include quantile functions, moments, moment generating function. The minimum , maximum and median order statistics models are discussed in Sectio n 4. In section 5 we studied the least square estimation. Finally, In Section 6 we demonstrate the maximum likelihood estimates of the unknown parameters . 2. Transmuted Exponentiated Gamma Distribution In this section we studied the transmuted exponentiated gamma (TEG) distribution . Now using (1.1) and (1.2) we have the cdf of transmuted exponentiated gamma distribution (2.1) where and are the shape parameters representing the d ifferent patterns of the transmuted exponentiated gamma distribution and are positive and is the transmuted parameter. The restrictions in equation (2.1) on the values of and are always the same. The probability density function (pdf) of the transmuted exponentiated gamma distribution is given by (2.2) Figures (1) and (2) illustrate the graphical behavior of the pdf and cdf of transmuted exponentiated gamma distribution for selected values of the parameters. Figure 1: the pdf function for different values of the parameters Transmuted exponentiated gam ma distribution 1 301 Figure 2: cdf function for different values of the parameters The reliability function of the transmuted exponentiated gamma distribution is denoted by also known as the survivor function and is defined as (2.3) It is important to note that . One of the characteristic in reliability analysis is the hazard rate function (HF) defined by (2.4) It is important to note that the units for is the probability of failure per unit of time, distance or cycles. These failure rates are defined with different choices of parameters. The cumulative hazard function of the transmuted exponentiated gamma distribution is denoted by and is defined as (2.5) It is important to note that the units for is the cumulative probability of failure per unit of time, distance or cycles. we can show that . For all choice of parameters the distribu tion has the decreasing patterns of cumulative instantaneous failure rates. Figures (3) and (4) illustrate the graphical behavior of the reliability function and hazard rate function the transmuted exponentiated gamma distribution for selected values of th e parameters. 1302 Mohamed A. Hussian Figure 3: the reliability function ( RF ) for different values of the parameters 3. Statistical Properties This section is devoted to studying statistical properties of the distribution, specifically quantile function , moments and moment generating function 3.1. Quantile Function The qth quantile of the transmuted exponentiated gamma distribution can be obtained from (2.2) as we simulate the distribution by solving the nonlinear equation (3.1) where has the uniform U(0;1) distribution. Transmuted exponentiated gamma distribution 1 303 Figure 4: the hazard rate ( HF ) function for different values of the parameters 3.2. Moments In this subsection we discuss the moment and moment generating function for distribution. Moments are necessary and important in any statistical analysis, especially in applications. It can be used to study the most important features and chara cteristics of a distribution (e.g., tendency, dispersion, skewness and kurtosis). If has then the moment of is given by the following (3.2) Based on the first four moments of the distribution, the measures of skewness and kurtosis of the distribution can obtained as and 1304 Mohamed A. Hussian and the moment generating function of X, has the following form (3.3) 4. Order Statistics In fact, the order statistics have many applications in reliability and life testing. The order statistics arise in the study of reliability of a system. Let be a simple random sample from with cumulative distribution function and probability density function as in (2.1) and (2.2), respectively. Let denote the order statistics obtained from this sample. In reliability literature, denote the lifetime of an out - of - system which consists of independent and identically components. Then the pdf of is given by (4.1) where . Substituting (2.1) and (2.2) into (4.1) we get (4.2) where also, the joint pdf of , and is (4.3) where Transmuted exponentiated gamma distribution 1 305 We defined the first order statistics , the last order statistics as . thus, the Distribution of the m inimum , and the m aximum and 5. Least Squares and Weighted Least Squares Estimators In this section , we provide the regression based method estimators of the unknown parameters of the transmuted exponentiated gamma distribution, which was originally suggested by Swain, Venkatraman and Wilson (1988) to estimate the parameters of beta distributions. It can be used some other cases also. Suppose is a random sample of size from a distribution function and suppose ; denotes the ordered sample. The proposed method uses the distribution of . For a sample of size , we have and see Johnson, Kotz and Balakrishnan (1995). Using the expectations and the variances, two variants of the least squares methods can be used. 1306 Mohamed A. Hussian Method 1 (Least Squares Estimators) . Obtain the estimators by minimizing with respect to the unknown parameters. Therefore in case of distribution the least squares estimators of and , say and respectively, can be obtained by minimizing with respect to and . Method 2 (Weighted Least Squares Estimators). The weighted least squares estimators can be obtained by minimizing with respect to the unknown parameters, where Therefore, in case of distribution the weighted least squares estimators and of and , can be obtained by minimizing with respect to the unknown parameters only. 6. Maximum Likelihood Estimation In this section, we determine the maximum likelihood estimates (MLEs) of the parameters of the distribution from complete samples only. Let be a random sample of size from .The likelihood function for the vector of parameters can be written as Transmuted exponentiated gamma distribution 1 307 (5.1) Taking the log - likelihood function for the vector of parameters we get (5.2) The log - likelihood can be maximized either directly or by solving the nonlinear likelihood equations obtained by differentiating (5.2). The components of the score vector are given by (5.3) (5.4) and (5.5) We can find the estimates of the unknown parameters by maximum likelihood method by setting these above non - linear equations (5.3) - (5.5) to zero and solve them simultaneously. Therefore, we use mathematical package to get the MLE of the unknown parameters. 1308 Mohamed A. Hussian Table 1 : The mean square errors of the MLEs. of the TEG for different sample sizes (  ,  ,  ) n MSE(  ) MSE(  ) MSE(  ) (0.45,0.25,0.35) 25 0.0513 0.0729 0.0053 50 0.0376 0.0280 0.003 75 0.0335 0.0147 0.0026 100 0.0308 0.0102 0.001 1.0,0.75,0.8 25 0.6965 0.3679 0.3241 50 0.4127 0.1255 0.1759 75 0.3564 0.107 0.1459 100 0.3043 0.0651 0.1394 (1.5,1.0,0.5) 25 0.0196 0.1367 0.1054 50 0.0087 0.0301 0.0821 75 0.0080 0.0255 0.0500 100 0.0073 0.0179 0.0273 We noticed from the above Table 1 that all MSEs decrease as the sample size increases, while they increase with increasing of the true parameter. REFERENCES [1] G. R. Aryal. and C. P. Tsokos , On the transmuted extreme value distribution with applications. Nonlinear Analysis: Theory , Methods and applicat ions, 71 , (2009), 1401 - 1407. [2] G. R. Aryal and C. P. Tsokos, Transmuted Weibull distribution: A Generalization of the Weibull Probability Distribution. 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