Review of Weibull Statistics for Estimation of Wind Speed Distributions

K. Conradsen Department of Mathematical Statistics and Operations Research, Technical University of Denmark, DK-2800 Lyngby, Denmark

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L. B. Nielsen Department of Mathematical Statistics and Operations Research, Technical University of Denmark, DK-2800 Lyngby, Denmark

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L. P. Prahm Danish Meteorological Institute, Lyngbyvej 100, Copenhagen, Denmark

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Abstract

A statistical distribution commonly used for describing measured wind speed data is the Weibull distribution. A review of the methods found in the statistical literature for the purpose of estimation of the parameters in Weibull distributions is given, with a special emphasis on the efficiency of the different methods. From this review, the most appropriate method for a given application can be chosen. The general conclusion is that maximum likelihood estimators should be used due to their large sample efficiency. However, they require an iterative minimization. The recommended closed form estimators when there are few observations (say, less than 25) are the least-squares estimators. If wind speed data with a good speed resolution are available, the closed form ten fractile estimators are very good. The simplest are the two fractile estimators. They, however, require around three times as many observations as the maximum likelihood estimators in order to be of the same accuracy. If mean and standard deviation of the logarithms of the wind speed are available, Menon's moment estimators can be used. They are of closed form, and are slightly more efficient than the two fractile types. If only median and quantiles are known, the two fractile estimators can be used. For Danish wind speed data, measured by two cup anemometers with relatively high thresholds (2 m s−1), it is shown that improved estimations are obtained when the methods for censored data are used. Finally, use of the “effective” number of observations in evaluation of the quality of the distributional fits in cases with autocorrelated data is suggested.

Abstract

A statistical distribution commonly used for describing measured wind speed data is the Weibull distribution. A review of the methods found in the statistical literature for the purpose of estimation of the parameters in Weibull distributions is given, with a special emphasis on the efficiency of the different methods. From this review, the most appropriate method for a given application can be chosen. The general conclusion is that maximum likelihood estimators should be used due to their large sample efficiency. However, they require an iterative minimization. The recommended closed form estimators when there are few observations (say, less than 25) are the least-squares estimators. If wind speed data with a good speed resolution are available, the closed form ten fractile estimators are very good. The simplest are the two fractile estimators. They, however, require around three times as many observations as the maximum likelihood estimators in order to be of the same accuracy. If mean and standard deviation of the logarithms of the wind speed are available, Menon's moment estimators can be used. They are of closed form, and are slightly more efficient than the two fractile types. If only median and quantiles are known, the two fractile estimators can be used. For Danish wind speed data, measured by two cup anemometers with relatively high thresholds (2 m s−1), it is shown that improved estimations are obtained when the methods for censored data are used. Finally, use of the “effective” number of observations in evaluation of the quality of the distributional fits in cases with autocorrelated data is suggested.

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