Five-parameter log-normal distribution and its modification

The introduction of the LN5 and mLN5 distributions extends the commonly used three-parameter log-normal distribution (LN3) by enhancing tail modeling, which is critical for accurate representation of extreme values in hydrology and climatology. This paper details two methods for parameter estimation: the established local maximum likelihood method and the newly developed triangular method, an adaptation of the relative least squares approach. The effectiveness of these distributions is demonstrated through their application to datasets from the Czech Hydrometeorological Institute, encompassing average daily flow, precipitation, atmospheric pressure, and air temperature. Results show significant improvements in modeling extreme events with LN5 and mLN5 over LN3, as well as over other compared distributions such as generalized Gamma and generalized Weibull, particularly in tail behavior, underscoring their potential for advancing environmental studies. Appendices include comprehensive derivations of the functional characteristics of LN5 and mLN5 and introduce an alternative parametrization for LN5.