# Parameter Estimation for Some Probability Distributions Used in Hydrology
## 4 Result and Discussion  
The errors of the new approximations are compared with the existing ones, having a uniform relative error below \(10^{-3}\). Figures 3-5 present the relative errors of the estimated parameters for the analyzed probability distributions. The improvement in the approximations was achieved by adopting other types of functions compared to the existing ones or by using some conditional functions, for example the approximation for the PE3 distribution in which the definition domain was divided into three equal intervals for \(\tau_{3}\). The proposed approximations were made for the estimated parameters, sometimes being necessary to transform them by logarithmization. Thus, there are approximations with polynomial functions to the logarithmic parameters, it being necessary to write them later in the exponent. In the following, this type of approximation will be referred to as a polynomial with a logarithmic argument. For the GEV distribution, a polynomial approximation was tried for the estimation of MOM, following an existing model calibrated in the range \(0<C_{\rm s}<4\) applied for the extended range \(0<C_{\rm s}<8\). The errors obtained were \(10^{-2}\), so a new approximation with a rational function was tried, which gave relative errors below \(10^{-3}\). The same type of rational function was used for the L-moments estimation, with errors below \(10^{-4}\) over the entire domain of \(\tau_{3}\). For the W3 distribution, rational functions were used to approximate the parameter. In the case of MOM estimation, the existing approximation was of a rational type, valid for \(C_{\rm s}\leq 2\), and the new approximation extended the definition domain to \(4<C_{\rm s}<8\) with a relative error for MOM below \(10^{-2}\). For the estimation of L-moments, there were two rational-type approximations with logarithmic and polynomial arguments with significant errors for \(\tau_{3}>0.5\) The proposed rational type approximation is characterized by relative errors less than \(10^{-2}\) over the entire domain. The W2 distribution requires approximation only for MOM, not existing so far, thus the proposed approximation with polynomial function with logarithmic argument has relative errors below \(10^{-4}\). For the F3 distribution, to estimate the parameters with MOM and L-moments, the rational function type was adopted, with relative errors below \(10^{-3}\) and \(10^{-4}\), respectively. For the F2 distribution, the approximation used is polynomial with the logarithmic argument having relative errors below \(10^{-3}\). For the P3 distribution, in the case of MOM estimation, the analysis resulted in the necessity of dividing the skewness coefficient into two intervals. For \(C_{\rm s}\leq 2\) a polynomial function of the 7 th degree was adopted with relative errors below \(10^{-6}\), respectively; for the interval \(2<C_{\rm s}\leq 8\) a rational function was chosen with relative errors below \(10^{-4}\). Figure 3: The variation of errors depending on \(\tau_{3}\). Figure 4: The variation of errors depending on \(C_{\rm s}\). For the LL3 distribution, the two existing approximations for parameter estimation with MOM have relative errors up to 0.1. The proposed approximation uses the same ranges for C\({}_{\rm s}\), but the approximation functions differ. For C\({}_{\rm s}\leq 2.5\) the approximation function is of the polynomial type with a logarithmic argument with relative errors below \(10^{-4}\), and for C\({}_{\rm s}>2.5\) the approximation function is of the polynomial type with relative errors below \(10^{-3}\). For the K3-Generalized Gumbel distribution, a polynomial type approximation function was adopted for MOM estimation. For L-moments estimation, the approximation function is of the conditional type defined on two intervals using a polynomial and a rational function. The relative errors for both parameter estimation methods are below \(10^{-3}\).