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

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  • 1 Institute of Atmospheric Sciences, South Dakota School of Mines and Technology, Rapid City, South Dakota
  • | 2 Department of Mathematics and Computer Science, South Dakota School of Mines and Technology, Rapid City, South Dakota
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Abstract

This paper complements an earlier one that demonstrated the bias in the method-of-moments (MM) estimators frequently used to estimate parameters for drop size distribution (DSD) functions being “fitted” to observed raindrop size distributions. Here the authors consider both the bias and the errors in MM estimators applied to samples from known gamma DSDs (of which the exponential DSD treated in the earlier paper is a special case). The samples were generated using a similar Monte Carlo simulation procedure. The skewness in the sampling distributions of the DSD moments that causes this bias is less pronounced for narrower population DSDs, and therefore the bias problems (and also the errors) diminish as the gamma shape parameter increases. However, the bias still increases with the order of the moments used in the MM procedures; thus it is stronger when higher-order moments (such as the radar reflectivity) are used. The simulation results also show that the errors of the estimates of the DSD parameters are usually larger when higher-order moments are employed. As a consequence, only MM estimators using the lowest-order sample moments that are thought to be well determined should be used. The biases and the errors of most of the MM parameter estimates diminish as the sample size increases; with large samples the moment estimators may become sufficiently accurate for some purposes. Nevertheless, even with some fairly large samples, MM estimators involving high-order moments can yield parameter values that are physically implausible or are incompatible with the input observations. Correlations of the sample moments with the size of the largest drop in a sample (Dmax) are weaker than for the case of sampling from an exponential DSD, as are the correlations of the MM-estimated parameters with Dmax first noted in that case. However, correlations between the estimated parameters remain because functions of the same observations are correlated. These correlations generally strengthen as the sample size increases.

Corresponding author address: Paul L. Smith, Institute of Atmospheric Sciences, SDSM&T, 501 East Saint Joseph St., Rapid City, SD 57701. Email: paul.smith@sdsmt.edu

Abstract

This paper complements an earlier one that demonstrated the bias in the method-of-moments (MM) estimators frequently used to estimate parameters for drop size distribution (DSD) functions being “fitted” to observed raindrop size distributions. Here the authors consider both the bias and the errors in MM estimators applied to samples from known gamma DSDs (of which the exponential DSD treated in the earlier paper is a special case). The samples were generated using a similar Monte Carlo simulation procedure. The skewness in the sampling distributions of the DSD moments that causes this bias is less pronounced for narrower population DSDs, and therefore the bias problems (and also the errors) diminish as the gamma shape parameter increases. However, the bias still increases with the order of the moments used in the MM procedures; thus it is stronger when higher-order moments (such as the radar reflectivity) are used. The simulation results also show that the errors of the estimates of the DSD parameters are usually larger when higher-order moments are employed. As a consequence, only MM estimators using the lowest-order sample moments that are thought to be well determined should be used. The biases and the errors of most of the MM parameter estimates diminish as the sample size increases; with large samples the moment estimators may become sufficiently accurate for some purposes. Nevertheless, even with some fairly large samples, MM estimators involving high-order moments can yield parameter values that are physically implausible or are incompatible with the input observations. Correlations of the sample moments with the size of the largest drop in a sample (Dmax) are weaker than for the case of sampling from an exponential DSD, as are the correlations of the MM-estimated parameters with Dmax first noted in that case. However, correlations between the estimated parameters remain because functions of the same observations are correlated. These correlations generally strengthen as the sample size increases.

Corresponding author address: Paul L. Smith, Institute of Atmospheric Sciences, SDSM&T, 501 East Saint Joseph St., Rapid City, SD 57701. Email: paul.smith@sdsmt.edu

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