THE INVERSE WEIBULL GENERATOR OF DISTRIBUTIONS : PROPERTIES AND APPLICATIONS

In this paper, we introduce a new family of univariate distributions with two extra positive parameters generated from inverse Weibull random variable called the inverse Weibull generated (IW-G) family. The new family provides a lot of new models as well as contains two new families as special cases. We explore four special models for the new family. Some mathematical properties of the new family including quantile function, ordinary and incomplete moments, probability weighted moments, Rѐnyi entropy and order statistics are derived. The estimation of the model parameters is performed via maximum likelihood method. Applications show that the new family of distributions can provide a better fit than several existing lifetime models.


Introduction
The inverse Weibull (IW) distribution is an important probability distribution which can be used to analyze the life time data with some monotone failure rates.It is suitable model to describe degradation phenomena of mechanical components as mentioned by Keller and Kamath (1982).According to Nelson (1982), the IW distribution provides a good fit to several data sets such as the times to breakdown of an insulating fluid subject to the action of a constant tension.Many works have been made about the IW distribution; for example, Calabria andPulcini (1990, 1994) dealt with parameter estimation of the distribution.Some useful measures for the inverse Weibull distribution have been discussed by Jiang et al. (2001).Mahmoud et al. (2003) derived order statistics from inverse Weibull distribution.Based on lower record values, Sultan (2008) derived the Bayesian estimators and obtained the estimators of the reliability and hazard functions for the unknown parameters of the inverse Weibull distribution.Based on Type-II censored data, Kundu and Howlader (2010) studied the Bayes estimates of the unknown parameters of IW distribution under a squared error loss function.Hassan and Al-Thobety (2012) provided optimum simple failure step stress partially accelerated life tests for the model parameters and acceleration factor for inverse Weibull model.Hassan et al. (2015) discussed the constant-stress partially accelerated life test for inverse Weibull model based on multiple censored data.
The random variable  has an inverse Weibull distribution if its cumulative distribution function (cdf) takes the form () =  −   − ; , ,  > 0, where,  and  are the scale and shape parameters respectively.The corresponding probability density function (pdf) is given by () =    −−1  −   − ; , ,  > 0 (2) In recent years, some endeavors have been made to define new generators of continuous distributions from classic ones to provide great flexibility in modelling data in several applied areas.Several generated families have been suggested by several authors; see for examples, beta-G by Eugene et al. (2002), gamma-G by Zografos and Balakrishanan (2009), Kumaraswamy-G by Cordeiro  where, () is the pdf of a random variables  and [()] be a function of the cdf of any random variables .
In this article, we provide a new family of distributions using inverse Weibull as a generator with the hope that it will attract a wider application in some areas.This paper can be organized as follows.In Section 2, we provide a formation of the IW-G family.Four special models of IW-G family are defined in Section 3. Some useful expansions for the pdf and cdf of IW-G family are derived in Section 4. In the same section, explicit expressions for the moments, probability weighted moments, order statistics and quantile function are obtained.Estimation of the model parameters using maximum likelihood method is performed in Section 5. Section 6 provides two applications to two real data sets are presented to illustrate the potentially of the new family.Section 7 ends with some concluding remarks.

Inverse Weibull-G Family
In this section, we display the formation of the IW-G family of probability distributions.

Special Models
A number of new distributions can be deduced as special models from the IW-G family of distributions.Here, four special models, namely; the inverse Weibull Weibull (IWW); the inverse Weibull Pareto (IWP); the inverse Weibull uniform (IWU) and the inverse Weibull Burr XII (IWXII) are introduced.

Inverse Weibull Pareto Model
The second model is the inverse Weibull Pareto whose cdf is derived by substituting the following cdf () = 1 − ( ] ; , , ,  > 0,  <  < ∞. The probability density function of a random variable  having the IWP distribution, say ~(, , , ) is given by Furthermore, the reliability function and hrf are as follows and,

Inverse Weibull Uniform Model
Considering the baseline distribution is uniform on the interval (0, ),  > 0, the cdf of inverse Weibull uniform distribution is as follows The probability density function of a random variable  having the IWU distribution, say ~(, , ) is given by Furthermore, the reliability and hazard rate functions are as follows and,

Inverse Weibull Burr XII Model
Hence, the pdf of a random variable  has the inverse Weibull Burr XII distribution, say ~(, , , , ) is obtained from cdf (4) as follows The pdf of the IWBXII is given by The reliability and hazard rate functions are In Figure 1, we display some plots of the pdf of the IWW, IWP, IWU and IWBXII for selected parameter values.Figure1 reveals that the IWW, IWP, IWU and IWBXII densities generate various shapes such as symmetrical, left-skewed, reversed-J, unimodal and U shaped.Plots of the hrf of the IWW, IWP, IWU and IWBXII models are described in Figure 2 for some selected parameter values.From Figure 2, we observe that these models can produce hazard rate shapes such as constant, increasing, decreasing, and upside-down bathtub.This fact implies that the IW-G family can be very useful for fitting data sets with various shapes.

Mathematical Properties
In this section, we provide some main mathematical properties of the IW-G family.

Useful Expansions
Some mathematical properties of the IW-G family can be confirmed through an algebraic expansion which is more efficient than computing those directly by numerical integration of its density function.Here two important expansions are deduced for the IW-G pdf and cdf using mixture forms of exponentiated-G (Exp-G) distribution.
Since, the power series for the exponential function in pdf ( 5) can be written as follows: Inserting expansion (7) in pdf ( 5), then we have It is well-known that, if  > 0 is real non integer and || < 1, the generalized binomial theorem is written as follows Then, by applying the binomial theorem ( 9) in ( 8), the IW-G pdf, where  is real non integer becomes where,   (; ) = (; )((; )) −1 denotes the 'exp-G' pdf with power parameter .

Quantile Function
In this subsection, the quantile function of a random variable  has the IW-G distribution is derived.More specifically, the quantile function for the IWP model is obtained.Further, the skewness and kurtosis based on quantile function for IWP model are discussed.
where, (. ) denotes the quantile function.Figure 3 gives plots of the skewness and kurtosis for some choices of the parameter  as function of .These plots indicate that the skewness and kurtosis decrease when  increases for fixed .

Moments
Most of the necessary characteristics and features of a distribution can be studied through its moments.Here, the moments of a random variable  has the IW-G are derived.More specifically, the ℎ moment for the IWU model is obtained.
The ℎ moment of  about the origin is derived from (10) as follows Furthermore, the moment generating function of  is .
Example 2: Consider the IWU distribution discussed in subsection (3.3).The ℎ moment of IWU can be obtained from ( 13) with pdf and cdf as defined in subsection (3.3) as follows .
In particular, the mean and variance of the IWU distribution are obtained, respectively, as follows: ,

The probability Weighted Moments
The probability-weighted moments (PWMs) method of estimation has been proposed by Greenwood et al. (1979) for distribution expressible in inverse form.For a random variable  the PWMs, denoted by  , , can be calculated through the following relation The PWMs of IW-G family is obtained by inserting ( 10) and ( 12) into ( 14) as follows .

The Mean Deviation
In statistics, the mean deviation about the mean and mean deviation about the median measure the amount of scattering in a population.For random variable  with pdf (), cdf (), the mean deviation about the mean and mean deviation about the median, are defined by

Order Statistics
Suppose  1 ,  2 , … ,   be independent and identically distributed (i.i.d) random variables with their corresponding continuous distribution function () .Let  1: <  2: < … <  : the corresponding ordered random sample from a population of size .According to David (1981), the pdf of the ℎ order statistic, is defined by The pdf of the ℎ order statistic for IW-G family is derived by substituting ( 10) and ( 12) in ( 15 where,   1 * (; ) denotes the 'exp-G' pdf with power parameter  1 .In particular, the pdf of the smallest order statistics  1: is obtained from (16), by substituting  = 1, as follows

Rѐnyi Entropy
Entropy is a measure of variation or uncertainty of a random variable  (Rѐnyi, 1961).The Rѐnyi entropy of a random variable is defined by By applying the binomial theory ( 9) and exponential expansion, then the pdf ()  can be expressed as follows Inserting ( 18) in ( 17), then the Rѐnyi entropy of the family is as follows where, ℑ(, , ) = ∫ ((; ))  ((; ))

Estimation of Parameters
This section concerns with the maximum likelihood estimates (MLEs) of the unknown parameters for the new family based on complete samples.Let  1 ,  2 , … ,   be a simple random sample from pdf (5) with set of parameters Θ ≡ (, , ) .The log-likelihood function, denoted by ln , based on the observed random sample of size  from density ( 5) is given by: ln  =  ln  +  ln  + ∑ ln (  , ) .
Setting   ,   and   equal to zero and solving the equations simultaneously the MLE, say Θ ̂≡ ( ̂,  ̂,  ̂) of Θ ≡ (, , )  are obtained.Further the resulting equations cannot be solved analytically, so some software's can be used to solve them numerically.
For interval estimation of the model parameters, we obtain the 3 × 3 observed information matrix  = (Θ) (for ,  = , , ), whose elements are listed in Appendix.
Here,   2 ⁄ is the upper  2 ⁄ ℎ percentile of the standard normal distribution and (.)᾿s denote the diagonal elements of  −1 (Θ ̂) corresponding to the model parameters.

Applications To Real Data
The flexibility of IWW as especial model from IW-G family is examined using two real data sets.The superiority of IWW is clarified as compared with some main four models; additive Weibull (AW; Almalki and Yuan ( 2013 For both data sets, the unknown parameters of each distribution are estimated by the maximum-likelihood method.The model selection is carried out using Kolmogorov-Smirnov (K-S) statistic and corresponding P-value, -2 log-likelihood function (−2 ln ), Akaike information criterion (AIC), the correct Akaike information criterion (CAIC), Bayesian information criterion (BIC) and Hannan-Quinn information criterion (HQIC).However, the better distribution corresponds to the smaller values of AIC, CAIC, BIC, HQIC, K-S criteria and largest values of P-value.Furthermore, we plot the histogram for each data set and the estimated pdf of the four models.Moreover, plots of empirical cdf of the data sets and estimated cdf of four models are displayed.

Example 6.1:
The first data set is provided in Murthy et al. (2004)  In Table (1), we list the values of AIC, CAIC, BIC, HQIC, K-S and the P-value statistics.We observe that the IWW model has the smallest AIC, CAIC, BIC, HQIC, K-S values, and has the largest P-value as compared with those values of the other models.So, the IWW model seems to be a very competitive model to this data.More information is provided by a visual comparison of the histogram and estimated cumulative of the data with the fitted models as shown in Figure 4.It is clear from Figure 4 that the IWW distribution provides a better fit than the other competitive models.2009) and recorded as follows 5.1, 1.2, 1.3, 0.6, 0.5, 2.4, 0.5, 1.1, 8.0, 0.8, 0.4, 0.6, 0.9, 0.4, 2.0, 0.5, 5.3, 3.2, 2.7, 2.9, 2.5, 2.3, 1.0, 0.2, 0.1, 0.1, 1.8, 0.9, 2.0, 4.0, 6.8, 1.2, 0.4,0.2.
Results in Table 2, indicate that the IWW model is more suitable than the other competitive models for this data set based on the selected criteria.Further, it is clear from Figure 5 that the IWW distribution provides a better fit and therefore be one of the best models for this data set.

Concluding Remarks
In this paper, we introduce a new family of univariate distribution based on the inverse Weibull distribution as a new generator.Many new sub-models are obtained and four special models are provided.The IW-G density function can be expressed as a mixture of exponentiated-G distribution functions.Mathematical properties of the IW-G family are derived.We give explicit closed form expressions for the moments, probability weighted, entropy and distribution of order statistics.The maximum likelihood method of estimation is employed to derive the model parameters and the observed Fisher information matrix is obtained.We fit IWW distribution as special sub-model to two real data sets in order to explain the flexibility of this new family.We hope this new generation may attract wider applications in many areas.

Figure 3 :
Figure 3: Plots of skewness and kurtosis for IWP distribution based on quantile.

Figure 4 :Example 6 . 2 :
Figure 4: Estimated densities and estimated distributions of models for the first data set

Figure 5 :
Figure 5: Estimated densities and estimated distributions of models for the second data set.

Table ( 1
): Model Selection Criteria for the First Data Set

Table ( 2
): Model Selection Criteria for the Second Data Set