Gamma type distribution: Maximum likelihood values of the T-year event and their asymptotic variance.

Abstract

Maximum likelihood and censored sample theory are applied for flood frequency analysis purposes to the Two Parameter Gamma, log Two Parameter Gamma, Pearson Type III, log Pearson Type III (LP3), and Generalized Gamma distributions. The logarithmic likelihood functions are given in terms of the fully specified floods, the historical information, and the parameters to be estimated. Solution of the appropriate transcendental equations yields maximum likelihood estimators of the parameters. T-year floods are expressed as a function of these parameters and the standard normal variate. The asymptotic standard error of estimate of the T-year flood is derived using the general equation for the variance of estimate of a function. The variances and covariances of the parameters are obtained through inversion of Fisher's information matrix. The method is illustrated by application of the LP3 distribution to two sites having historical information. Monte Carlo studies were conducted for the LP3 distribution to analytically verify the accuracy of the derived asymptotic expression for the 10-, 50-, 100-, and 500-year floods. Results indicated that the asymptotic expressions were accurate for both Type I and Type II censored samples, while the bias was less than 2.5%. Subsequently, the Type II censored data were subjected to a random, multiplicative error. Results indicated that historical information contributes greatly to the accuracy of the estimate of the 100-year flood even when the error of its measurement becomes excessive. It is demonstrated that historical information can significantly reduce the standard error of estimate of flood quantiles.