Parameter Estimation and Data Analysis Using a Novel Extension of Weibull Distribution

Recently, statistical modeling has become essential for analyzing real-world datasets across various fields. However, classical statistical distributions often lack the flexibility to capture the complexity of observed phenomena accurately. For example, in reliability engineering and biomedical analysis, some distributions perform well for monotonic failure rates but are inadequate for non-monotonic patterns. Similarly, in insurance analytics, distributions that model small losses effectively may fail to accurately represent extensive claim data. To overcome these limitations, several approaches have been proposed to enhance classical models. This study introduces the inverse Weibull-Weibull (IWW) distribution, a novel four-parameter extension of the Weibull distribution, derived using the inverse Weibull-G family. The IWW distribution offers greater flexibility for modeling lifetime data and can be expressed as an infinite mixture of exponentiated Weibull distributions. Various parameter combinations reveal the wide range of shapes its density and hazard rate functions can assume. We examine the statistical properties of the IWW distribution and consider three parameter estimation methods: maximum likelihood, least squares, and the Cramér–von Mises statistic. Their effectiveness is evaluated through a Monte Carlo simulation, which shows that maximum likelihood estimation generally yields superior results. To demonstrate the practical relevance of the IWW distribution, we apply it to two real-world datasets. In both cases, the IWW model provides better fits than several well-known distributions, underscoring its potential for accurately modeling complex data.