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# On gamma Inverse Weibull Distribution

#### Salman Abbas,

##### COMSATS University Islamabad, Lahore Campus, PK
Department of Statistics

#### Mahjabeen Hameed,

##### University of Veterinary and Animal Sciences, PK
Department of Statistics and Computer Science

#### Selen Cakmakyapan,

##### Hacettepe University, TR
Department of Statistics

#### Sarfaraz Nawaz Malik

##### COMSATS University Islamabad, Wah Campus, PK
Department of Mathematics

## Abstract

The inverse Weibull distribution which was first introduced to investigate the decay of mechanical components is widely used in the field of biology, especially in the areas of
survival and reliability analysis. In this paper, a generalisation of inverse Weibull distribution is presented using a new class of gamma distributions. For the proposed model, important mathematical properties including quantile function, noncentral
moment, central moment, moment generating function, characteristics function, and general coeffi cient are discussed. The shapes of the probability density function and risk function were obtained and the graphical demonstrations of them were provided. To estimate model parameters of the proposed method, maximum likelihood was used. An illustrative simulation study was conducted to observe how the maximum likelihood estimators behave for different sample sizes and for different values of parameters. Two real data sets from the field of biomedical sciences were used to demonstrate
the practical applications of the proposed model in real life phenomena. Results were obtained from the data sets, using the gamma inverse Weibull distribution and some known existing distributions and compared. The comparative analysis justified
that the proposed model significantly fits with the survival data compared to other distributions.

##### Keywords: Inverse Weibull distribution, Gamma family of distribution, Risk Function, General Coefficient, Maximum Likelihood
How to Cite: Abbas, S., Hameed, M., Cakmakyapan, S. and Malik, S.N., 2020. On gamma Inverse Weibull Distribution. Journal of the National Science Foundation of Sri Lanka, 47(4), pp.445–453. DOI: http://doi.org/10.4038/jnsfsr.v47i4.8520
Published on 25 Jan 2020.
Peer Reviewed