The Transmuted Geometric-Inverse Weibull Distribution : Properties , Characterizations and Application

In this paper, a four parameters flexible life time distribution called the transmuted geometric-inverse Weibull (TG-IW) distribution is obtained from mixture of inverse Weibull distribution, geometric distribution and transmuted distribution. Some structural and mathematical properties including descriptive measures on the basis of quantiles,moments, factorial moments, incomplete moments, inequality measures, residual life functions and some other properties are theoretically taken up.TheTGIW distribution is characterized via different techniques.The estimates of parametersfortheTG-IW distribution are being obtained from maximum likelihood method. The significance and flexibility of the TG-IW distribution is tested through different measures by application to physical data set.


Introduction
The generalizations of probability distribution are more flexible and suitable for many real data sets compared to classical distributions.Azzalini (1985) derived Skewed Family with additional skewing parameter.Gupta et al. (1998) developed exponentiated family.Marshal and Olkin (1997) introduced a parameter to a family of distributions.Eugene et al. (2002) established family formed from Beta distribution.Jones (2004) also presented a family generated from Beta distribution.The transmuted family was presented by Shaw and Buckley (2007).Zografos Balakrishnan (2009) established family made from gamma distribution.Cordeiro and Castro (2011) developed family produced from Kumaraswamy The cumulative distribution function (cdf) for TG-G family mixture of continuous probability distribution, geometric distribution and transmuted distribution is The popular lifetime distribution applicable for useful life, failure time spans, mortality, maintenance spans and maintenance cost in the fields like survival and reliability analysis are called inverse Weibull distributions.The cdf and pdf of inverse Weibull distributions are This research article is composed as follows.In Section 2, the TG-IW distribution is introduced.In Section 3, the TG-IW distribution is studied in terms of various structural properties, plots,sub-models and descriptive measures on the basis of quantiles are taken up.In Section 4, moments about origin, negative moments, fractional moments, moments about mean, moment generating function, cumulants generating function, incomplete moments, residual life functions, mean inactivity life and mean residual life function and inequality measures are and some other properties are theoretically derived.In Section 5, order statistics for the TG-IW distribution are proposed.In Section 6, the TG-IW distribution is characterized through (i) ratio of truncated moments; (ii) reverse hazard rate function and (iii) elasticity function.In Section 7, estimates of parameters for the TG-IW distribution are obtained from maximum likelihood method.Goodness of fit of the TG-IW distribution is checked through different methods is studied.Conclusion is given at the end.

Structural Properties of the TG-IW Distribution
The survival, hazard, reverse hazard and cumulative hazard functions of a random variable X with the TG-IW distribution are given, respectively, by x e e fx e e x e e e hx e e e e and The Mills ratio of the TG-IW distribution is x e e e The elasticity of the TG-IW distribution shows the behavior of accumulation of probability in the domain of the random variable.

Shapes of the TG-IW Density and Hazard Rate Function
The TG-IW density is arc and positively skewed (Fig. 1).The TG-IW hazard is increasing or decreasing and inverted bathtub hazard rate function (Fig. 2).

3.2Sub Models of the TG-IW Distribution
The TG-IW distribution has wide applications in life testing, survival analysis, and reliability theory.The TG-IW distribution has the following sub models.

Descriptive Measures Based On Quantiles
In this sub-section, descriptive measures on the basis of quantiles are taken up.
The quantile function of the TG-IW distribution is Median of the TG-IW distribution is the random number generator of the TG-IW distribution is where the random variable Z has uniform distribution on   . The measures based on quantile exist for the distributions whose moments does not exist.The measures based on quantiles are rarer sensitive to the outliers.

Median Inactivity Time Function
The MDIT function in terms of (5) for the TG-IW distribution is

Moments and Inequality Measures
In this section, moments about origin, negative moments, fractional moments, moment about mean, moment generating function, cumulants generating function, incomplete moments, inequality measures, residual life functions, mean inactivity life, mean residual life function and some other properties are theoretically derived.

Moments about origin
The r th moments of the random variable X with the TG-IW distribution about the origin are given by x dx e e Mean and Variance for the TG-IW distribution are given, respectively An important measure of variability of a random variable is Fisher index of dispersion For FI=1, the TG-IW distribution is equidispersed, for FI<1, the TG-IW distribution is under dispersed and for FI>1, the TG-IW distribution is over dispersed.
The fractional positive moments of X with the TG-IW distribution about the origin are given by The q th central moments about mean of X for the TG-IW distribution are determined from , 17


The negative moments are used to determine to harmonic mean and many other measures.The r th negative moments about origin of X for the TG-IW distribution are The fractional negative moments about origin of X for the TG-IW distribution are given as The factorial moments for the TG-IW distribution are The Mellin transform helps to determine moments for a probability distribution.By definition, the Mellin transform is The Mellin transform of X for the TG-IW distribution is The moments generating function of random variable X with the TG-IW The cumulant moments generating function   Ktfor X with the TG-IW distribution is The cumulants for X with the TG-IW distribution are obtained from relation

Incomplete Moments
Incomplete moments are used to study mean inactivity life, mean residual life function and other inequality measures.
The lower incomplete moments for the random variable X with the TG-IW distribution are The upper incomplete moments for the random variable X with the TG-IW distribution are The mean deviation about mean is and the mean deviation about median is   where   q Q p  .

4.3Residual Life Functions
The residual life says   n mz of X for the TG-IW distribution having the following The mean residual life function or life expectancy at a specified time z, say   1 mz, computes the expected left over lifetime of an individual of age z and is The reverse residual life, say   n Mz , of X for the TG-IW distribution has the following n th moment The mean waiting time (MWT) or mean inactivity time signifies the waiting time pass by since the failure of an item on condition that this failure had happened in the interval [0, z].The MWT of X, say

Order Statistics
The order statistics mostly appear in the problems of the estimation and testing.The application of extreme values is very common in reliability, meteorology, econometrics and various areas of research.
The pdf

 
: jn X fx of jth order statistic j: , n X from a cdf F with pdf f, is The pdf

 
: jn X fx of jth order statistic j:n X for the TG-IW distribution is given by The pdf

 
: nn X fx of nth order statistic n: , n X from a cdf F with pdf f, is The pdf

 
: nn X fx of nth order statistic n:n X for the TG-IW distribution is given by x e e e f x e e e e The pdf   1:n X fx of 1st order statistic 1: n X for the TG-IW distribution is given by x e e e f x e e e e

Characterizations
In this section, the TG-IW distribution is characterized through: (i) Ratio of the truncated moments; (ii) the reverse hazard rate function and (iii) Elasticity function.
We present our characterizations (i)-(iii) in three subsections.

Characterization of the TG-IW Distribution through Ratio of Truncated Moments
In this subsection, the TG-IW distribution is characterized using Theorem 1 (Glänzel; 1990) on the basis of two the truncated moments of X. Theorem 1 is given in Appendix A.
The pdf of X is (4) if and only if Proof.The pdf of X is (4).Now Now, in the light of theorem 1, X has pdf (4).
x e e e e p x h x h x The answer of the above differential equation is where D is constant.

Characterization via Reverse Hazard Function
Here we characterize the TG-IW distribution via reverse hazard function of X. Definition 6.2.1:Let   X: 0,    be a continuous random variable having absolutely continuous cdf

 
Let X: 0,    be continuous random variable .The pdf of X is (4) if and only if its reverse hazard function, F r fulfills the first order differential equation Proof: If the pdf of X is (4), then the above differential equation holds.Now, if the differential equation holds, then which is the reverse hazard function of the TG-IW distribution.

Characterization via Elasticity Function
Here we introduce characterization of the TG-IW distribution via elasticity.

 
Let X: 0,    be continuous random variable .The pdf of X is (4) if and only if its elasticity,   F ex fulfills the first order differential equation x e e x e e x e x e Proof: If the pdf of is (4), then the above differential equation holds.Now, if the differential equation holds, then which is the elasticity function of the TG-IW distribution.

Maximum Likelihood Estimation
The parameter estimates are derived with maximum likelihood method for TG-IW distribution.The log-likelihood function for the TG-IW the parameters vector In order to compute the estimates of parameters of TG-IW distribution, the following nonlinear equations must be solved simultaneously: n e e e e e  The TG-IW distribution is best fitted than T-IW, G-IW and IW distribution because the values of all criteria are smaller for the TG-IW distribution.

Conclusions
We have developed a more flexible the TG-IW distribution that is suitable for applications in survival analysis, reliability and actuarial science.The important properties of the TG-IW distribution like survival function, hazard function, reverse hazard function, cumulative hazard function, Mills ratio, elasticity, quantile function, moments about origin, negative moments, fractional moments, moment generating function, Cumulants, incomplete moments, inequality measures, residual and reversed residual life functions, order statistics and many other properties are presented.
The TG-IW distribution is characterized through ratio of truncated moments, reverse hazard rate function and elasticity function.Maximum likelihood estimates are computed.Goodness of fit shows that the TG-IW distribution is a better fit.An application of the TG-IW model to times of failure and running times for 30 units from eld-tracking is illustrated to show significance and flexibility of the TG-IW distribution.

Fxis
this article is to propose four parameters the TG-IW distribution from mixture of inverse Weibull distribution, geometric distribution and transmuted distribution by the application of Transmuted geometric-G family (TG-G).The pdf and cdf of a random variable X with the TG-IW distribution are obtained by inducting (3a) and (3b) in (1) and (2) as strictly increasing and differential in (0,  ).The cdf of the TG-IW also show that It means that   Fx is an absolutely continuous cdf.

Fig. 1 :Fig. 2 :
Fig. 1: Plots of pdf of the TG-IW Distribution  be a continuous random variable and

36) 7.1Application: Times of Failure and Running Times for 30 Units from Eld- Tracking Study
. The maximum likelihood estimates (MLEs) of unknown parameters and values of goodness of fit measures are computed for the TG-IW distribution and its sub-models.The MLEs and goodness-of-fit statistics like W and A are given in table2.Table3displays goodness-of-fit values.
The better fit corresponds to smaller W, A, K-S, AIC, CAIC, BIC, HQIC and  value