Extreme rainfall events have profound implications across various sectors, necessitating accurate modeling to assess risks and devise effective adaptation strategies. The common practice of employing three-parameter probability distributions, such as the Generalized Extreme Value (GEV) and Pearson Type III distributions, in rainfall frequency analysis often encounters limitations in capturing rare, heavy-tailed events with a lack of consensus as to which distribution is the most applicable. In this study, we explore the applicability of the four-parameter Kappa distribution (K4D) for modeling extreme daily rainfalls using annual maxima from the Global Historical Climatology Network-Daily database. Quality checks and thresholds were used to remove erroneous and poor-quality data, retaining 20,500 stations with 50 or more years of data. The variation in the second shape parameter (h $h$) was examined across regime characteristics, geospatial regions, and climate regional groupings to identify where the K4D is best able to model extreme rainfalls. Consistent with theoretical expectations, h $h$ converges toward zero (i.e., toward the limiting GEV distribution) as the average number of rainfall events per year increases (here approximated by rain days). However, in arid regions with a limited number of annual storm events, we observe average values of h $h$ greater than zero, with a strong regional and climatic coherence in h $h$. Our results suggest that there is merit in using the K4D for modeling heavy tail behavior, particularly in regions with a small number of events per year. These findings will contribute to advancing statistical modeling techniques for extreme rainfall, benefiting hydrological modeling and risk assessments.