# Parameter Estimation for Some Probability Distributions Used in Hydrology
## 3 Methods
### Method for Parameter Approximation  
The approach method for three-parameter distributions consisted of determining the domain of definition of the \(\alpha\) variable as a function of \(\mathrm{C_{s}}\) for MOM and \(\tau_{3}\) for L-moments. The limits of the definition domain of the parameter \(\alpha\) were established according to the extreme values of \(\mathrm{C_{s}}\) and \(\tau_{3}\), thus \(\alpha\) was calculated for \(0<\mathrm{C_{s}}\leq 8\), respectively, \(0<\tau_{3}<1\). The \(\mathrm{C_{s}}(\alpha)\) and \(\tau_{3}(\alpha)\) functions were plotted for both methods.  
The variation graph of skewness (\(\mathrm{C_{s}}\)) for the first vertical axis and L-skewness (\(\tau_{3}\)) and for the vertical secondary axis, depending on the parameter \(\alpha\) are presented in Figure 1.  
Figure 1: _Cont_.  
For two-parameter distributions, the definition range of the \(\alpha\) variable was determined as a function of \(\mathrm{C_{v}}\) for MOM and \(\tau_{2}\) for L-moments. The limits of the definition domain of the parameter \(\alpha\) were established according to the extreme values of \(\mathrm{C_{v}}\) and \(\tau_{2}\), thus \(\tau_{2}\) was calculated for \(0<\mathrm{C_{v}}\leq 2\), respectively, for \(0<\tau_{2}<1\). The functions \(\mathrm{C_{v}}(\alpha)\) and \(\tau_{2}(\alpha)\) were plotted for both methods.  
The variation graph of coefficient of variation for the first vertical axis (\(\mathrm{C_{v}}\)) and L-coefficient of variation (\(\tau_{2}\)) for the vertical secondary axis, depending on the parameter \(\alpha\) are presented in Figure 2.  
Graphical examination indicated the type of approximation functions. The first-order derivatives were calculated to determine the slope variation, so for uniform variations over certain intervals a type of function is defined. If the variation is uniform in logarithmic  
Figure 1: The variation of parameter \(\alpha\) for three parameters distributions.  
Figure 2: The variation of parameter \(\alpha\) for two parameters distributions.  
scale, then the argument of the function is chosen logarithmically for the approximation with a polynomial, otherwise the rational function is chosen without logarithmization. The rational function is the ratio of two polynomial functions. There were situations in which the rational function was reduced only to a polynomial function, because \"the denominator\" represents a constant equal to one. For some probability distributions, the same type of functions, rational or polynomial, were used, and in some cases both functions were used on some distinct defining intervals.  
The attempts that were made for approximation functions took into account existing approximations for some distributions [6; 11]. The approximation functions considered optimal in terms of relative errors and number of terms were chosen, ensuring a balance between the relative error allowed and the complexity of the relationship.  
The calibration of the approximation functions was carried out with the linear least squares method for the polynomial functions and nonlinear for the rational functions.