Variance of the T-year event in the log Pearson type-3 distribution

doi:10.1016/0022-1694(85)90203-3

Journal of Hydrology

Volume 77, Issues 1–4, 25 April 1985, Pages 141-158

https://doi.org/10.1016/0022-1694(85)90203-3 Get rights and content

Abstract

The computation of the variance of the T-yr. event evaluated by means of the log Pearson type-3 distribution is described. This computation involves the variances and covariances of the parameter estimates. Explicit formulae for these variances and covariances are analytically derived in this study for the case where the parameters are estimated by the methods of direct and mixed moments.

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Asymptotic bias in large quantiles and moments for four parameter estimation methods, including the maximum likelihood method (MLM), method of moments (MOM), method of L-moments (LMM), and least squares method (LSM), is derived when a probability distribution function (PDF) is falsely assumed. The first three estimation methods are illustrated using the lognormal and gamma distributions forming an alternative set of PDFs. It is shown that for every method when either the gamma or lognormal distribution serves as the true distribution, the relative asymptotic bias (RB) of moments and quantiles corresponding to the upper tail is an increasing function of the true value of the coefficient of variation (c_v), except that RB of moments for MOM is zero. The value of RB is the smallest for MOM and largest for MLM. The bias of LMM occupies an intermediate position. The value of RB from MLM is larger for the lognormal distribution as a hypothetical distribution with the gamma distribution being assumed to be the true distribution than it would be in the opposite case. For c_v=1 and MLM, it equals 30, 600, 320% for mean, variance and 0.1% quantile, respectively, while for MOM, the moments are asymptotically unbiased and the bias for 0.1% quantile amounts to 35%. An analysis of 39 70-year long annual peak flow series of Polish rivers provides an empirical evidence for the necessity to include bias in evaluation of the efficiency of PDF estimation methods.
A comparison of moment-based methods of estimation for the log Pearson type 3 distribution
2000, Journal of Hydrology
The log Pearson type 3 distribution is a very important model in statistical hydrology, especially for modeling annual flood series. In this paper we compare the various methods based on moments for estimating quantiles of this distribution. Besides the methods of direct and mixed moments which were found most successful in previous studies and the well-known indirect method of moments, we develop generalized direct moments and generalized mixed moments methods and a new method of adaptive mixed moments. The last method chooses the orders of two moments for the original observations by utilizing information contained in the sample itself. The results of Monte Carlo experiments demonstrated the superiority of this method in estimating flood events of high return periods when a large sample is available and in estimating flood events of low return periods regardless of the sample size. In addition, a comparison of simulation and asymptotic results shows that the adaptive method may be used for the construction of meaningful confidence intervals for design events based on the asymptotic theory even with small samples. The simulation results also point to the specific members of the class of generalized moments estimates which maintain small values for bias and/or mean square error.
A FORTRAN routine for the computation of gamma percentiles
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Derivation of the Pearson type (PT) III distribution by using the principle of maximum entropy (POME) - Comment
1987, Journal of Hydrology
The generalized method of moments as applied to problems of flood frequency analysis: Some practical results for the log-Pearson type 3 distribution
1987, Journal of Hydrology
The log-Pearson type 3 distribution is widely used in North America and Australia for fitting annual flood series. Four different versions of the method of moments used in fitting this distribution are compared using Monte Carlo simulated samples which reflect some of the characteristics of annual flood series observed at some Canadian rivers. The bias, standard error, root mean square error, and skew, of the parameter estimates, and of estimates of events associated with different probabilities of occurrence are examined. Also examined are the correlation coefficients between probabilities of occurrence are examined. Also examined are the correlation coefficients between the parameter estimates and between the sample moments that are used in each of the four methods of estimation. It is observed that variances, covariances and correlation coefficients calculated using the usual first-order asymptotic approximation might have considerable error and therefore should be used with caution. On the basis of mean square error of events with return period above the range covered by the sample it is observed that a method proposed earlier^★ which uses moments of order 1, 2 and 3 in real space performs better than the other three methods although certain of the other methods follow the recommendation put forward by some investigators that higher order moments (moments of order 3 or more) should be avoided in flood frequency estimation. It is argued in the present study that the use of higher order moments should not be avoided simply because they have high variability because it is not only the variability of the moments which determines the degree of variability of the estimated design flood events but also the correlation that exists between these moments. Some recommendations are given at the end of the study aimed at achieving better efficiency in flood frequency research at a period where more and more distributions and methods of estimation are being proposed.
Sampling properties of the maximum entropy estimators for the extreme-value type-1 distribution
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The extreme-value type-1 (EV1) distribution can be viewed as the distribution that satisfies two specified expected values. These expected values give rise to a method of parameter estimation referred to as the method of maximum entropy (MME). The main purpose of this note is to provide a scheme to estimate the variances and covariance of the MME estimators. As a by-product of the simulation runs used, some useful sampling properties of the MME estimators are obtained. These clearly show that the MME is a good method for fitting the EV1 distribution, and the approximations obtained analytically for the variance of estimates of the T-year event are of sufficient accuracy.

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Research paperVariance of the T-year event in the log Pearson type-3 distribution

Abstract

J. Hydrol.

J. Hydrol.

The probable maximum rainfall in Hong Kong

R. Obs., Hong Kong, R.O. Tech. Mem.

The log Pearson type 3 distribution and its application in hydrology

Water Resour. Res.

The log Pearson type 3 distribution: the T-year event and its asymptotic standard error by maximum likelihood theory

Water Resour. Res.

Research paper
Variance of the T-year event in the log Pearson type-3 distribution