A STUDY ON THE DEVELOPMENT OF A DEFECTIVE GOMPERTZ-G FAMILY OF DISTRIBUTIONS

This study develops an advanced lifetime modeling framework through the formulation of the defective Gompertz-G family of distributions. Classical survival and reliability models often assume that all experimental units will eventually experience the event of interest. This assumption is frequently violated in biomedical, demographic, and industrial reliability studies, where a proportion of individuals may remain event-free indefinitely, representing a cured or long-term survivor group. To address this limitation, the defective distribution concept is integrated into the flexible Gompertz-G family, thereby extending its applicability to lifetime data characterized by incomplete failure.

The methodology employed in developing the proposed model is based on the T-X transformation technique. In this approach, a baseline random variable X with a specified distribution is transformed through a generator variable T following the Gompertz distribution. The T-X framework provides a systematic mechanism for constructing generalized families of distributions by compounding the cumulative distribution function of the baseline model within the generator structure. To introduce defectiveness, a defect parameter is incorporated into the transformed distribution, modifying the cumulative distribution function so that the total probability mass is less than unity, thereby capturing cure-fraction behavior.

Closed-form expressions for the probability density function, cumulative distribution function, survival function, and hazard rate function are derived. Additional statistical properties, including quantile function formulation and reliability measures, are established. Parameter estimation is developed to support inferential analysis and future empirical applications.