Hydrological frequency analysis is a statistical prediction method that consists of studying past events that are characteristic of a particular hydrological process in order to determine the probabilities of the occurrence of these events in the future [1
]. It is widely used for planning, design, and management of water resource systems. The probability distributions containing four or more parameters may exhibit some useful properties [3
]: (1) versatility and (2) ability to represent data from mixed populations. Among these distributions, some popular distributions are Wakeby, two-component lognormal, two-component extreme value distributions, and the four-parameter kappa distribution. Since the pioneering stream flow records frequency analysis of Herschel and Freeman during the period from 1880 to 1890, hydrological frequency analysis has undergone extensive further development. There are a multitude of methods for estimating parameters of hydrologic frequency distributions. Some of the popular methods include [3
]: (1) the method of moments; (2) the method of probability weighted moments; (3) the method of mixed moments; (4) L-moments; (5) the maximum likelihood estimation; (6) the least square method; and (7) the entropy-based parameter estimation method.
Among the above parameter estimation methods, entropy, which is a measure of uncertainty of random variables, has attracted much attention and has been used for a variety of applications in hydrology [5
]. For example, an entropy-based derivation of daily rainfall probability distribution [24
], the Burrr XII-Singh-Maddala (BSM) distribution function derived from the maximum entropy principle using the Boltzmann-Shannon entropy with some constraints [25
]. “Entropy-Based Parameter Estimation in Hydrology
” is the first book focusing on parameter estimation using entropy for a number of distributions frequently used in hydrology [3
], including the uniform distribution, exponential distribution, normal distribution, two-parameter lognormal distribution, three-parameter lognormal distribution, extreme value type I distribution, log-extreme value type I distribution, extreme value type III distribution, generalized extreme value distribution, Weibull distribution, gamma distribution, Pearson type III distribution, log-Pearson type III distribution, beta distribution, two-parameter log-logistic distribution, three-parameter log-logistic distribution, two-parameter Pareto distribution, two-parameter generalized Pareto distribution, three-parameter generalized Pareto distribution and two-component extreme value distribution. Recently, two entropy-based methods, called the ordinary entropy method (ENT) and the parameter space expansion method (PSEM), that are both based on the principle of maximum entropy (POME) have been applied for estimating the parameters of the extended Burr XII distribution and the four-parameter kappa distribution [5
]. The results of the estimation show that the entropy method enables these two distributions to fit the data better than the other estimation methods. In the above method of entropy-based parameter estimation of a distribution, the distribution parameters are expressed in terms of the given constraints, and then the method can provide a way to derive the distribution from the specified constraints. The general procedure for the ENT for a hydrologic frequency distribution involves the following steps [3
]: (1) define the given information in terms of the constraints; (2) maximize the entropy subject to the given information; and (3) relate the parameters to the given information. The PSEM employs an enlarged parameter space and maximizes the entropy subject to the parameters and the Lagrange multipliers [3
]. The parameters of the distribution can be estimated by the maximization of the entropy function.
Inspired in large part by the two-parameter gamma distribution, a four-parameter exponential gamma distribution has been developed to apply in many areas, such as wind and flood frequency in Yellow River basin, Yangtse River basin, Aumer Basin and Liaohe River basin of China [4
]. Depending on the parameter values, the four-parameter exponential gamma distribution can be turned into a Pearson type III distribution, Weibull distribution, Maxwell distribution, Kritsky and Menkel distribution, Chi-square distribution, Poisson distribution, half-normal distribution and half-Laplace distribution. The properties of the four-parameter exponential gamma and relations between this distribution and other distributions have been investigated [4
]. These investigations suggest that the four-parameter exponential gamma distribution may have a potential in hydrology. Despite the advances mentioned above, the entropy-based parameter estimation for the four-parameter exponential gamma distribution has received comparatively little attention from the hydrologic community.
The objective of this paper is to apply two entropy-based methods that both use the POME for the estimation of the parameters of the four-parameter exponential gamma distribution; compute the annual precipitation quantiles using this distribution for different return periods; and compare these parameters with those estimated when the methods of moments (MOM) and maximum likelihood estimation (MLE) were employed for parameter estimation.