# Parameter Estimation for Some Probability Distributions Used in Hydrology
## 1 Introduction  
Probability distributions are used in hydrology to determine the probability of occurrence of extreme events: maximum flows, maximum volumes and maximum precipitation. A criticism of the probabilistic estimation of extreme events is that it does not take climate changes into account, this being generally solved with hydrological modeling using learning machines [1] of the neural network type and others. However, statistical modeling is complementary and necessary to other types of black-box and gray-box modeling, because the input and output data from these models must be presented probabilistically, especially for extreme events with low probability, necessary for the design of hydrotechnical constructions. In the case of determining the maximum flows, three or more parameter distributions are recommended, while for the determination of precipitation and maximum volumes, two parameter distributions are recommended [2; 3; 4]. In general, finding approximate relations to express the parameters is very important because many of these probability distributions, for exact determination, require solving with the Levenberg-Marquardt algorithm [5] for nonlinear least squares curve-fitting problems, which can sometimes represent a difficulty requiring the use of specific calculation programs. Another important aspect is the expression of the quantile function (inverse of the distribution function) of these distributions, both of which facilitate the use of these distributions in the determination of the analyzed extreme values. This article presents approximate forms of the expressions of some parameters of some frequently used probability distributions in hydrology, such as: Pearson III (PE3), Generalized extreme value (GEV), Weibull (W3), Weibull (W2), Frechet (F3), Frechet (F2), Generalized Pareto (P3), Log-Logistics (LG3), Kappa (K3-Generalized Gumbel), Kappa (K3-Park), Pearson V (PV3), Pearson V (PV2) and Generalized Exponential (EG2). The approximate forms are for determining the parameters using the method of ordinary moments (MOM) and the method of linear moments (L-moments). For the method of ordinary moments, the parameters of the three-parameter distributions are determined from relationships that use mean, variance and skewness, the latter representing the ratio between the centered moment of order 3 and variance to the 1.5 power. For the method of linear moments, the parameters of the distribution are determined from relations that use the arithmetic mean (L\({}_{1}\)), L\({}_{2}\) and \(\tau_{3}\), the latter representing the ratio between the moment L\({}_{3}\) and L\({}_{2}\). These methods are some of the most frequently used methods for estimating parameters in hydrology [2; 3; 4; 6; 7; 8]. The approximations use rational and polynomial functions in which the coefficients are obtained by least squares nonlinear regression on the domain of definition considered. In some cases, the approach is similar by using the same type of function but with more terms, and in other cases the approach is different by using other types of functions. For the method of ordinary moments, the approximation range most often used for skewness is 0-4, but in certain regions of the world they meet up to a skewness equal to 9 [3], or the skewness is adopted by multiplying the coefficient of variation with a chosen positive coefficient [10], so the skewness is always positive, even if that of the given sample may have negative skewness values. The definition range of approximations 0-8 was chosen in the case of MOM, and the definition range for the method of linear moments is the default 0-1. Considering that the approximation relations must be as accurate as possible, the relative error graphs are also presented. The relative error is the ratio between the difference between the exact and approximate value of the skewness, and the exact value. In this article, the new approximation relations are noted as \"a better approximation\".