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The Bias in Moment Estimators for Parameters of Drop Size Distribution Functions: Sampling from Exponential Distributions

Paul L. SmithInstitute of Atmospheric Sciences, South Dakota School of Mines and Technology, Rapid City, South Dakota

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Donna V. KlicheInstitute of Atmospheric Sciences, South Dakota School of Mines and Technology, Rapid City, South Dakota

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Abstract

The moment estimators frequently used to estimate parameters for drop size distribution (DSD) functions being “fitted” to observed raindrop size distributions are biased. Consequently, the fitted functions often do not represent well either the raindrop samples or the underlying populations from which the samples were taken. Monte Carlo simulations of the process of sampling from a known exponential DSD, followed by the application of a variety of moment estimators, demonstrate this bias. Skewness in the sampling distributions of the DSD moments is the root cause of this bias, and this skewness increases with the order of the moment. As a result, the bias is stronger when higher-order moments are used in the procedures. Correlations of the sample moments with the size of the largest drop in a sample (Dmax) lead to correlations of the estimated parameters with Dmax, and, in turn, to spurious correlations between the parameters. These things can lead to erroneous inferences about characteristics of the raindrop populations that are being sampled. The bias, and the correlations, diminish as the sample size increases, so that with large samples the moment estimators may become sufficiently accurate for many purposes.

Corresponding author address: Dr. Paul L. Smith, IAS, South Dakota School of Mines and Technology, Rapid City, SD 57701. paul.smith@sdsmt.edu

Abstract

The moment estimators frequently used to estimate parameters for drop size distribution (DSD) functions being “fitted” to observed raindrop size distributions are biased. Consequently, the fitted functions often do not represent well either the raindrop samples or the underlying populations from which the samples were taken. Monte Carlo simulations of the process of sampling from a known exponential DSD, followed by the application of a variety of moment estimators, demonstrate this bias. Skewness in the sampling distributions of the DSD moments is the root cause of this bias, and this skewness increases with the order of the moment. As a result, the bias is stronger when higher-order moments are used in the procedures. Correlations of the sample moments with the size of the largest drop in a sample (Dmax) lead to correlations of the estimated parameters with Dmax, and, in turn, to spurious correlations between the parameters. These things can lead to erroneous inferences about characteristics of the raindrop populations that are being sampled. The bias, and the correlations, diminish as the sample size increases, so that with large samples the moment estimators may become sufficiently accurate for many purposes.

Corresponding author address: Dr. Paul L. Smith, IAS, South Dakota School of Mines and Technology, Rapid City, SD 57701. paul.smith@sdsmt.edu

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