In this thesis, the focus is developing an inference for the Downton's bivariate Weibull (DBW) distribution, particularly in the context of dependent competing risks. The study begins by investigating the statistical properties of the DBW distribution. Next, the thesis delves into likelihood inference for the DBW distribution using complete bivariate data, specifically in the context of dependent competing risks. Additionally, moment-based estimates based on complete bivariate data are developed and examined. To assess the performance of the point and interval estimates, extensive Monte Carlo simulations are conducted. Furthermore, two real datasets are presented as illustrative examples of the application of the DBW distribution with the inferential method. These illustrations serve to demonstrate the practical relevance and usefulness of the proposed model and methods. The thesis concludes with a comprehensive evaluation of the estimation techniques and suggestions for further research in this area.