Comparison of Three Parameter Weibull and Gamma Distribution with Application to Age at Fertile Women Data: A Survival Analysis

Abstract

Fertility has long been recognized as one of the most critical measures for understanding population growth. From a demographic standpoint, fertility drives population expansion, whereas infertility acts as a limiting factor, often regarded as a “failure” in reproduc
tive terms. A comprehensive understanding of both is therefore vital for public health planning, reproductive health policy, and demographic research. In this study, two parametric models the three-parameter Weibull distribution and the three-parameter Gamma distributions are applied to examine the relationship between age and fertility. Using real data from UttarPradesh collected during the fifth round of the National Family Health Survey (NFHS-5), both models were found to be appropriate, with the Gamma distribution providing a superior fit compared to the Weibull. Through these models, fertility, infertility, and fertility persistence beyond a given age and projection of age at menopause and projection of fertility beyond a given time are analyzed using survival and hazard functions. The findings highlight the Gamma distribution as a particularly effective tool for capturing fertility dynamics,  specially at later reproductive ages, thereby offering valuable insights for demographic and survival analyses