# Parameter Estimation for Some Probability Distributions Used in Hydrology
## 5 Conclusions  
The probability distributions presented here are frequently used in hydrology to calculate maximum rainfall, maximum and minimum flows, and volumes of synthetic floods.  
The need to approximate the parameters is given by obtaining some initial values for the numerical calculation of the parameters, reducing the number of iterations and thus the calculation time.  
The approximate values calculated with the formulas presented here can be used directly to estimate the parameters of the probability distributions due to very small errors.  
The relative errors of the parameter estimates are generally well below \(10^{-3}\), which has a much smaller implication on the relative errors of the inverse function values and are independent of the length of the observed data. The first-order derivatives of the parameters determined from the approximations show negligible errors, especially for \"better approximations\". The choice of the approximation functions using mathematical analysis tools, especially the analysis of the parameter variation graph, allowed the improvement of the approximation functions and the choice of new approximation functions, especially for the distributions for which there were no approximation formulas. The functions used in the approximation relations are of the rational and polynomial type, sometimes with a logarithmic argument.  
The comparative presentation of the variation in the estimated parameter for the two methods of estimating the parameters of the probability distributions is useful in choosing the skewness coefficient in the case of MOM, considering that the linear moments are closer to reality, an important aspect in Romania where the skewness is chosen according to the genesis of the flows, a legacy from the USSR normative standards [23].  
In general, in Romania, tabular calculation is used for a small number of distributions (Pearson III and Kritsky-Menkel) using linear interpolation, so the approximations presented in this article prove to be extremely useful, facilitating the use of a larger number of distributions, a necessity regarding the updating of Romanian normative standards at the international level.  
It is worth mentioning that this article can be used as a guideline for the simpler implementation of these probability distributions in various software.  
All research was carried out by authors in the Faculty of Hydrotechnics with hydrological data provided by the National Institute of Hydrology and Water Management and National Administration \"Romanian Waters\".  
The presentation of some approximate forms of the parameters, especially for the estimation with the L-moments method, represents a step forward in the implementation stage of a transition from MOM to a regionalization based on L-moments in Romania.  
Conceptualization, C.I. and C.G.A.; methodology, C.I. and C.G.A.; software, C.I. and C.G.A.; validation, C.I. and C.G.A.; formal analysis, C.I. and C.G.A.; investigation, C.I. and C.G.A.; resources, C.I. and C.G.A.; data curation, C.I. and C.G.A.; writing--original draft preparation, C.I. and C.G.A.; writing--review and editing, C.I. and C.G.A.; visualization, C.I. and C.G.A.; supervision, C.I. and C.G.-A.; project administration, C.I. and C.G.A.; funding acquisition, C.I. and C.G.A. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.