Three Parameters Quasi Gamma Distribution and with Properties and Applications

This paper introduced a new life time data analysis distribution name three parameters quasi gamma distribution discussed about its some properties including moment generating function, rth moment about origin and mean, mean deviations, reliability measurements, Bonferroni and Lorenz curve, Order statistics, Renyi entropy, also discussed about maximum likelihood method and real-life data applications.


Introduction
The gamma distribution playing important role in life time data applications, It used in field of sciences like engineering, meteorology, demography, business etc. The gamma function, a generalization of the factorial function to non-integral values, Milton and Stegun (1972) discussed Euler gamma function which in the 18th century was introduced by Swiss mathematician Leonhard Euler . For values of > 0, the it is known as gamma function is defined using an integral formula as Γ = ∫ If ≤ 1 (1.1) is a J-shaped function, if > 1 it is uni-model maximum at = − 1; for = 1 is an exponential distribution. Gupta and Kundu (2006) studied the closeness between the gamma distribution and the generalized exponential distribution and observed that if the shape parameter of the gamma distribution is not very high then a gamma distribution approximately very well by a generalized exponential distribution. This closeness between the two distributions studied by Kolmogorov discrepancy measure and by Kullback-Leibler discrepancy measure, it is observed that for shape parameter close to one the two distributions are almost indistinguishable.  introduced a new quasi exponential distribution and discussed its properties, maximum likelihood method and applications. The probability density function pdf and cumulative density function cdf of quasi exponential distribution defined below; ( ; ) = The quasi exponential is better fitted than exponential distribution and least standard error, showed using biomedical data application.
The pdf of Quasi gamma distribution defined as; In chapter 17 the detailed discussions are available on gamma distributions with their properties and applications done by different researchers that described by Johnson et al (1994). Another generalization of gamma distribution and discussed its application to drought data by Nadarajah and Gupta (2007). The exponentiated generalized gamma distribution proposed by Cordeiro et al (2013) and discussed its application to lifetime data. The Weibull distribution is the power transformation of exponential distribution developed by Weibull (1951) and proved to be better lifetime distribution than exponential distribution due to an additional parameter. The statistical modeling and analysis of lifetime data are crucial in almost all branches of physical, technical, engineering and biomedical sciences. For modeling lifetime data analyzing the one parameter exponential distribution, the two-parameter Weibull and gamma distributions are common in statistics literature. Due to theoretical or applied point of view it has been noted that these lifetime distributions are not always a suitable model either. In this respect an attempt has been made to determined three parameters lifetime distribution which strives well with exponential, quasi exponential, gamma and quasi gamma distributions.
In this paper a three parameters quasi gamma distribution (TPQGD) of which one parameter quasi exponential as well as quasi gamma distribution is a particular case has been proposed. Its statistical properties including shapes of pdf for varying values of parameters, moment generating function, rth moment about origin and mean, mean deviations, reliability measurements, Bonferroni and Lorenz curve, Order statistics, Renyi entropy, also discussed about maximum likelihood method Finally, applications related to a real lifetime data from physical sciences, engineering and biomedical has been presented to test its goodness of fit over one parameter exponential and quasi exponential distributions and two-parameter gamma and quasi gamma distributions, the new three parameters Quasi Gamma distribution have discussed in following.

New Three Parameters Quasi Gamma Distribution (TPQGD)
The quasi gamma distribution (TPQGD) probability density function (pdf) and cumulative density function (cdf) as a weighted version of quasi exponential distribution (QED) with weight function 2 −1 ; > 0, > 0; scale parameter and shape parameters is defined following as; ( ; , , ) =

Moment generating function:
The moment generating function of TPQGD defined following as;

Rth moments:
The rth moment function of TPQGD defined following such that; From (1.10) 1 st four moment about origin are: By using above moments about origin; the moments about mean are obtain as: Using moments about origin in (3.3), (3.4) and (3.5) obtain moments about mean following; The mean, variance, coefficient of variation, coefficient of skewness, coefficient of kurtosis, index of dispersion, mean deviation about mean and median of TPQGD are obtained as: Index of Dispersion: By using 2 nd and 3 rd moments about mean in (3.9) have;

Coefficient of Kurtosis:
By using 2 nd and 4 th moments about mean in (3.11); Mean Deviations: The amount of variation in a population is generally measured to some extent by the totality of deviations usually either from the mean or the median. These are known as the mean deviation about the mean and the mean deviation about the median and are defined; ( ) The measure of 1 ( ) and 2 ( ) can be calculated as: So by using pdf (2.1) found that:    Also note that put y=0 in (4.9) and get 1 st moment about origin such as:

Bonferroni and Lorenz Curves
The Bonferroni and Lorenz curves Bonferroni, (1930), and Bonferroni and Gini indices have applications not only in economics to study the variation in income and poverty, but also in other fields like reliability, demography, insurance and medicine. The Bonferroni and Lorenz curves are defined as;

Renyi Entropy Measure
A popular entropy measure is Renyi entropy (1961), an entropy of a random variable Y is a measure of the variation of uncertainty. Let Y is a continuous random variable having probability density function (TPQGD) ( ; , , ), then Renyi entropy is defined as ( ; , , ) pdf of (TPQGD) the Renyi entropy such that:

Maximum Likelihood Method
Let be a random sample 1  The iterative solution of the equations (8.5) to (8.10) using matrix given following will be the MLEs ̂, ̂ ̂o f parameters , and k of TPQGD. Here 0 , 0 and 0 initial values of parameters of , and k of TPQGD.

Applications
F is the cumulative distribution function of the specified distribution and are ordered data. Cramer Von Mises Statistics: Here (1) ≤ (2) ≤ ⋯ ≤ ( ) is the varied series based on the sample 1 , 2 , … .

Conclusion
The new proposed Quasi Gamma lifetime data distribution have three parameters preforming as shape parameters and is scale parameter, if in eq. (2.1) put =