Transmuted New Generalized Inverse Weibull Distribution

This paper introduces the transmuted new generalized inverse Weibull distribution by using the quadratic rank transmutation map (QRTM) scheme studied by Shaw et al. (2007). The proposed model contains twenty three lifetime distributions as special sub-models. Some mathematical properties of the new distribution are formulated, such as quantile function, Rényi entropy, mean deviations, moments, moment generating function and order statistics. The method of maximum likelihood is used for estimating the model parameters. We illustrate the flexibility and potential usefulness of the new distribution by means of two real data sets.


Introduction
In the theory of life testing, many different families of lifetime distribution have been developed for describing the reliability behaviour of the components or process.These new families of lifetime distributions can be obtained by adding parameters to the well established distributions for obtaining more flexibility in the new extended lifetime distribution.Because of this motivation, we introducing a new lifetime distribution called the transmuted new generalized inverse weibull distribution by using quadratic rank transmutation map (QRTM) technique studied by Shaw et al. (2007).Historically speaking, the Inverse Weibull distribution (also known as type 2 extreme value or the Fréchet distribution) is a very flexible lifetime distribution having the inverse Rayleigh and inverse exponential distributions as special sub-models commonly used for modelling reliability data.The Inverse Weibull distribution has been applied in many areas of scientific disciplines, such as reliability engineering, aeronautics, hydrology, physics, biomedical sciences, agriculture, pharmacutical sciences, psychology, metrology, economics and actuarial sciences etc.More recently, Khan and King (2016) proposed the new generalized inverse Weibull (NGIW) distribution and investigated many structural properties for modeling reliability engineering application.
The cdf of the NGIW distribution is given by where,  > 0 are the shape parameters and ,  > 0 are the scale parameters.The probability density function corresponding to (1) is given by The CDF given in equation ( 1) approches to the eleven lifetime distributions when its parameters change.Khan and King (2012) proposed the modified Inverse Weibull distribution and presented a comprehensive description of the mathematical properties of this model along with its reliability behavior.Using quadratic rank transmutation map (QRTM) technique, we introduce the transmuted the new generalized inverse Weibull (NGIW) distribution by introducing a new parameter λ that would offer more flexibility in the proposed model.Several distributions have been proposed under this methodology such as transmuted extreme value distribution (Gokarna and Chris, 2009) studied with application to climate data, the transmuted Weibull distribution (Gokarna and Chris, 2011) proposed with two applications, Gokarna (2013) proposed the transmuted Log-Logistic distribution and studied its various structural properties.Khan and King (2013) proposed the transmuted modified Weibull distribution as an important competitive model with eleven lifetime distributions as sub-models along with its theoretical properties.Khan and King (2013) and where () is the cdf of the baseline distribution.It is important to note that at λ = 0 we have the distribution of the baseline random variable.
The rest of this article is organized as follows, In Section 2, we present the analytical shapes of the probability density and distribution function of the proposed model.Some mathematical properties are formulated in Section 3, such as expressions for the moment estimation and moment generating function.Maximum likelihood estimates (MLEs) of the unknown parameters are discussed in Section 4. We derive expressions for the Rényi entropy and mean deviations in section 5.The order statistics are formulated in Section 6.In Section 7, we compare the proposed model with three other lifetime distributions by means of two real data sets to illustrate its usefulness.In Section 8, we offer some Concluding remarks.

Transmuted New Generalized Inverse Weibull Distribution
A random variable  is said to have transmuted new generalized inverse Weibull (TNGIW) distribution with parameters , , ,  >  and || ≤ ,  > .If the probability density function is given by 5) The CDF corresponding to equation ( 5) is given by Figure 1 shows the visualizations of the transmuted new generalized inverse Weibull PDF with some selected choice of parameters.Some useful characterizations of the TNGIW distribution are formulated as reliability function (RF), hazard function and reversed hazard function defined as     The cumulative hazard function (CHF) of the TNGIW distribution is defined as The quantile function of the TNGIW distribution is the real solution of the following equation A random variable  with density ( 5) is denoted by ~(; , , , , ).When the transmuting parameter  = 0, we obtain the new generalized inverse Weibull distribution.Figure 2 illustrates the hazard function of the TNGIW distribution with different choice of parameters.These visualizations of the failure rates show that the proposed model has upside down hazard rate function for some selected choice of parameters.Table 1 listed twenty three lifetime distributions as special sub-models of the transmuted new generalized inverse Weibull distribution.

Statistical Properties
This section formulates the  ℎ moment and the moment generating function of the transmuted new generalized inverse Weibull distribution.Proof: The  ℎ moment of  can be obtained from (5) as By using equation ( 6) we can write the above integral as the above integral reduces to By using the Binomial expansion, the above integral reduces to Hence, we obtain the final result Proof: By definition themoment generating function of  can be obtained from (5) as By using equation ( 6) the above integral reduces to Finally we obtain the moment generating function of the TNGIW distribution as

Maximum Likelihood Estimation
Consider the random samples  1 ,  2 , … ,   consisting of  observations from the TNGIW(; , , , , ) distribution.The log-likelihood function ℒ = lnL of the density (5) for the parameter vector Θ = (, , , , ) is given by The components of score vector can be obtained by differentiating (15) with respect to , , ,  and λ then equating it to zero, we obtain the estimating equations are The log-likelihood function can be maximized by using the BFGS method in R or SAS languages.These nonlinear system of equations cannot be solved analytically and statistical software can be used to solve them numerically such as R-Package (Adequacy Model), SAS (PROC NLMIXED) by using iterative methods such as Limited-Memory quasi-Newton algorithm for Bound-constrained optimization (L-BFGS-B), these solutions will yield the ML estimators  ̂, , ̂ ̂,  ̂ and  ̂.For the five parameters TNGIW distribution pdf all the second order derivatives exist.Thus we have the inverse dispersion matrix as Equation ( 16) is the variance covariance matrix of the TNGIW(; , , , , ) distribution.The asymptotic multivariate normal  5 (0, () −1 ) distribution can be used to construct the approximate confidence intervals and confidence region of individual parameters for the transmuted new generalized inverse Weibull distribution.By using the observed information matrix an approximately 100(1 − )% confidence intervals for , , ,  and  can be determined as where  2 is the upper ℎ percentile of the standard normal distribution.

Entropy and Mean Deviation
The entropy is the measure of variation or the uncertainty of a random variable  for the probability density function from the lifetime distribution.The Rényi entropy for the random variable  with () is defined as Finally, we obtain the Rényi entropy as The extent of dissemination in a population is measured by the totality of deviations from the mean and the median.If  has the TNGIW(; , , , , ), then we can derive the mean deviation about mean and about the median M can be obtain from the following equations The mean is obtained from (13) with  = 1 and the median M is the solution of the nonlinear equation is obtained from (12), where () can be attained from ( 5) where Hence, the measure in ( 21) can be obtained from (22).The quantity () can also be used to determine the Bonferroni and the Lorenz curves which have applications in econometrics and finance.They are given by Where is calculated from (12) for a given probability P.

Applications
In this section, we illustrate the usefulness of the TNGIW distribution to two real data sets.

Application 1: Ball bearings data
The first subsection provides the data analysis in order to assess the goodness-of-fit of the proposed model with failure times.We consider the ball bearings data for the number of revolution before failure, each of 23  The data set is reported by Lawless (1982).Transmuted new generalized inverse Weibull (TNGIW), new generalized inverse Weibull (NGIW), Kumaraswamy modified inverse Weibull (KMIW), Exponentiated Kumaraswamy inverse Weibull (EKIW), Kumaraswamy inverse weibull (KIW) and modified inverse Weibull (MIW) distributions are fitted to the ball bearings data.
(1) New generalized inverse Weibull (NGIW) distribution with the pdf where ,  > 0 are the scale parameters and ,  > 0 is the shape parameter of the NGIW distribution.where ,  > 0 are the shape parameters and ,  > 0 is the scale parameter of the KIW distribution.(Shahbaz, et al. 2012) (5) Modified inverse Weibull (MIW) distribution with the pdf where ,  > 0 are the scale parameters and  > 0 is the shape parameter of the MIW distribution.(Khan and King, 2012)  Table 2 listed the MLEs of the unknown parameter(s) and the corresponding standard errors for the model parameters.In order to evaluate the performance of the TNGIW distribution and can be consider as a superior lifetime model, we shall compare the goodness of fit with five other lifetime distributions recently proposed in the literature.The visualization of the estimated densities with histogram displayed in Figure 3 indicate that the transmuted new generalized inverse Weibull distribution has the better estimates comparing with other five distributions.Hence the data points from the TNGIW distribution has better relationship and can be consider as the virtuous model for life time data.For the goodness of fit statistics, we use the Kolmogorov-Smirnov (K-S) test to see which model provides the better estimates and results are displayed in Table 2.In order to assess if the model is appropriate, Figure 4 plots the empirical and estimated survival functions of the TNGIW distribution.

Application 2: Fatigue life of aluminium data
The second data set is prearranged by Birnbaum and Saunders (1969)     We examine the use of the use of the Transmuted new generalized inverse Weibull (TNGIW) distribution for modelling the fatigue fracture life of aluminium data.We fitted the TNGIW, NGIW, KMIW, EKIW, KIW and MIW densities are displayed in Table 4 and goodness of fit measures are listed in Table 5.The histogram of the fatigue fracture life of aluminium data is shown in Figure 5 along with the estimated densities of the TNGIW, NGIW models.The fitted model suggest that the TNGIW distribution is reasonable.These goodness of fit results indicate that the TNGIW distribution has the lowest values of the Kolmogorov-Smirnov (K-S) test, Cramér-von Mises and Anderson-Darling goodness of-fit statistics among the all fitted distributions.We conclude that the TNGIW distribution provides a good fit to these data sets.

Figure 1 :
Figure 1: Plots of the TNGIW pdf for some parameter values.

Figure 2 :
Figure 2: Plots of the TNGIW hf for some parameter values.
By using the Binomial expansion, the above integral can be written as= ∑ ,,,, ∫ ( +  (

Figure 3 :Figure 4 :
Figure 3: Fitted Models for failure of ball bearings data

Figure 5 :
Figure 5: Fitted Models for failure of fatigue life aluminium data

Table 3 .
The smaller values of these statistics indicate the better fit.We detect from Tables2 and 3that the TNGIW distribution has the lowest values for the Kolmogorov-Smirnov (K-S) test, Cramér-von Mises and Anderson-Darling goodness of-fit statistics among the all fitted distributions recently proposed in the literature.Therefore the TNGIW distribution can be consider as a good model for the failure times of ball bearings data.Figures 4 displays the estimated Survival function of the TNGIW distribution with better relationship for the ball bearings data.