On gamma inverse Weibull distribution

Salman Abbas1*, Mahjabeen Hameed2, Selen Cakmakyapan3 and Sarfraz Nawaz Malik4 1 Department of Statistics, COMSATS University Islamabad, Lahore Campus, Pakistan. 2 Department of Statistics and Computer Science, University of Veterinary and Animal Sciences, Pakistan. 3 Department of Statistics, Hacettepe University, Ankara, Turkey. 4 Department of Mathematics, COMSATS University Islamabad, Wah Campus, Pakistan.


INTRODUCTION
The inverse Weibull (IW) distribution is widely used in reliability analysis. Keller and Kanath (1982) first introduced IW distribution to investigate the decay of mechanical components. Nelson (1982) used the IW distribution for modelling of survival data to determine the time to breakdown insulating fluid and subject to the action of constant tension. Calabria and Pulcini (1990) discussed the estimation of parameter for IW distribution using method of maximum likelihood and least square. Mahmoud et al. (2003) described the recurrence relations for the single and product moments of order statistics for both non-truncated and truncated IW distribution. Aleem and Pasha (2003) defined the single, product, and ratio moments of order statistics of IW distribution. de Gusmão et al. (2011) introduced the generalised IW distribution and Hanook et al. (2009) developed the beta IW distribution. Using distribution function (cdf) of Kumaraswamy family of distribution, Shahbaz et al. (2012) computed the Kumaraswamy IW distribution. For lifetime modelling, IW distribution was generalised by Shuaib et al. (2016). Abbas et al. (2017) developed the Topp Leone IW distribution.
Generalisation of classical models is a widely practiced method in statistical analysis. To handle real data most of the time existing classical distributions are modified by adding one or more parameters.
In this article, we propose a new generalisation of IW distribution using a new class of gamma distribution introduced by Brito et al. (2017). The proposed model is named as gamma inverse Weibull distribution (GIW). The new model is more adaptable and flexible for the modelling of survival datasets.

December 2019
Journal of the National Science Foundation of Sri Lanka 47 (4) The cdf of gamma class of distribution is given as follows; where is regular incomplete gamma function, is the incomplete gamma function, and is gamma function. By differentiating equation (1) , the probability density function (pdf) of the proposed class is given as follows; a,b > 0, ... (2) where

GAMMA INVERSE WEIBULL DISTRIBUTION
In this section, we derive the cdf and pdf of gamma inverse Weibull distribution (GIW). Let Y follow the IW distribution. Then the cdf of one parameter IW distribution is given as, and the corresponding pdf is as, , y > 0, δ > 0, ... (4) Inserting equation (3) in equation (1), we have the cdf of GIW distribution as follows; , The corresponding density of the GIW distribution is obtained by taking derivative of (5) and is given as, y > 0, a, b, δ > 0, ... (6) For δ = 1, the proposed model reduces to gamma inverse exponential distribution. The new model converts into gamma inverse Rayleigh distribution for δ = 2.
The cfd and the pdf of the GIW distribution is expressed in form of exponentiated G-distributions as, ...(7) and ...(8)  It is noted that with the increase in a value the distribution again remains positively skewed and its peak increases, but there is no expressing change in the location and shape of the distribution. Figure 3 displays the pdf of GIW distribution for δ = 2, a = 2 with different values of b. It is noted that as we increase the value of b the peak decreases. For b < 1 the function initially shows a rapid increase and then slowly decreases as we increase the y -value, but for b > 1 the function initially shows a slow increase and the curve becomes flattened as y increases. Figure 4 displays the plot of pdf of GIW distribution for various values of δ and b when a = 2. These plots clearly indicate that the GIW distribution is unimodel.

PROPERTIES OF GAMMA INVERSE WEIBULL DISTRIBUTION
In this section, we derive and discuss some useful mathematical characteristics of the GIW distribution.

Moments, moment generating and characteristics function
Moment is an interesting feature in distribution theory.
It is used to define several important characteristics of the probability model including tendency, dispersion, skewness and kurtosis. The general expression for raw moment is given as, Using equation (8) and performing some calculations, the p -th moment of the GIW distribution is obtained as, For the above expression, δ > p The moment generating function of the proposed model is also obtained using equation (8). The general expression if given as, where After some simplification, the formulation is derived and presented as, Similarly, the characteristic function of the GIW distribution is provided as,

Central moments and general coefficient
Here, we derive and discuss the moment about origin and general coefficient. The expression of moment about origin of the gamma class of distributions is given as,  distribution is obtained as, , Figure 5 reveals that the hazard rate is upward-down shape. The survival function of the proposed distribution is given as, . ...(9)

Mode
For the mode, the density of GIW distribution given in

Quantile and risk function
The u th quantile function of GIW distribution is derived and given as, where is the inverse function of regular incomplete gamma function.
The risk function of any distribution is obtained by using the following method; Using equations (5) and (6), the risk function of GIW equation (6)  where . Solving equation (11) iteratively, the maxima can be obtained.

ESTIMATION
In this section, the maximum likelihood method is adapted for estimation of the model parameters. The estimation is based on complete sample. Let be the random sample of size n drawn from the density [equation (6)] of GIW distribution given as, The likelihood function of the above density is given as, where . The log-likelihood function of density [equation (6)] is given as follows; ... (12) The partial derivatives of log-likelihood function in equation (12) w:r:t parameters a,b,δ are given as follows: ... (13) ... (14) December 2019 Journal of the National Science Foundation of Sri Lanka 47 (4) where ψ is the diagamma function. The maximum likelihood estimates and of the parameters a, b and δ are defined by equating (13)      choose two datasets from the field of biomedical science. We obtained maximum likelihood estimates (MLE's), AIC, CAIC, and BIC statistics of GIW model and well known existing distributions mentioned in Table 3.   First, we consider the tubercle bacili dataset. Bjerkedal (1960) described that the data is comprising values of survival times of 72 Guinea pigs injected with virulent tubercle bacilli. Table 1 gives observed values of the survival times of Guinea pigs in days. This dataset is already studied and fitted with different models by Lazhar et al. (2017), Shanker et al. (2015), and Mahdavi and Jabbari (2017). Then the comparison is carried out by considering a few other competitive models. The values of the dataset are provided in Table 2. Table 3 gives the fitted distributions and their abbreviations. The Table 4 reveals that the GIW distribution yields the lowest values of these statistics and then explains the suitable fit to the datasets. The results suggest that the GIW distribution performs better than the other fitted distributions. Visual comparison of the histogram of the data with GIW distribution is given in Figure 6. The plot of empirical  distribution function and GIW distribution function are also displayed in Figure 6.
Journal of the National Science Foundation of Sri Lanka 47(4) December 2019 The second dataset consists of the relief times of twenty patients receiving an analgesic. The dataset in Table 5 is from Gross and Clark (1975). This dataset is studied and fitted with different models by Fayomi (2019). The values of the data set is provided in Table 5. The fitted distributions and the abbreviations for this dataset is listed in Table 6. Table 7 presents the lowest values of the AIC, CAIC, and BIC for the proposed model, which clearly indicates that the GIW distribution provides better fits than other models. The visual presentation also provides that the GIW distribution is adequately explaining the dataset.

CONCLUSION
In this article, we derived a new generalisation of inverse Weibull distribution using gamma class of distributions. Numerous mathematical characteristics for the proposed model was derived and discussed.
Some special cases of the proposed distribution were presented. Estimation of the model parameters were made through method of maximum likelihood. The evaluation of the model parameters were justified through simulation study. Two real dataset examples from the field of biomedical sciences were studied to check the applicability of the proposed model in real life phenomena. At the end, some remarks were stated.