Kernel flood frequency estimators: Bandwidth selection and kernel choice

Abstract

Kernel density estimation methods have recently been introduced as viable and flexible alternativesto parametric methods for flood frequency estimation. Key properties of such estimators are reviewed in this paper. Attention is focused on the selection of the kernel function and the bandwidth. These are the parameters of the method. Existing techniques for kernel and bandwidth selection are applied to three situations: Gaussian data, skewed data (three-parameter gamma), and mixture data. The intent was to investigate issues relevant to parameter estimation as well as to the likely performance of these methods with the small sample sizes typical in hydrology. Bandwidths chosen by minimizing a performance criterion related to the distribution function lead to much smaller mean square errors of tail probabilities than those chosen by cross-validation methods designed for density estimation. However, this can lead to estimates that degenerate to the empirical distribution function, and hence to an unusable flood frequency curve. Variable bandwidths with heavy tailed kernels appear to do best. Kernel estimators are increasingly more competitive in terms of mean square error of estimate as the underlying distribution gets more complex. INTRODUCTIONEstimating exceedance frequencies of annual maximum flood events at a gaged site is a classical problem of hydrology. A finite data set (usually 20-100 points) is used to extrapolate flood magnitudes corresponding to recurrence intervals of up to 1000 years. A "curve-fitting" or parametric approach is traditionally used for the purpose. An a priori choice of a probability distribution function (e.g., log Pearson type iii) is made, and its parameters are estimated using one of several methods (e.g., moments, entropy, or likelihood maximization). Despite intensive research and legislation, no particular curve has emerged as a clear "winner" across different sites. Indeed, for the typical sample sizes given above, methods (e.g., the Kolmogorov-Smirnov test or the chi-square test) for selecting between probability distributions at a site cannot discriminate among candidate families of distributions and among members of a family [e.g., Kite, 1977]. A comparison between a variable kernel estimate (VK-C-AC) of the cumulative distribution function tCDF) for the St. Mary's River data used by Kite [1977] and Kite's results for a variety of parametric distributions is presented in Figure 1. The kernel estimate is close to the empirical CDF, and none of the parametric alternatives appear reasonable. The empirical CDF and the kernel estimator suggest a bimodal probability density for the data. A hydrologist forced to choose between the parametric alternatives would find from Kite that he cannot discriminate among two-parameter lognormal, three-parameter lognormal, type I extremal, Pearson type III, and log Pearson type III on the basis of standard tests. However, none of these distributions provides a fit consistent with the empirical distribution function. There are often a number of cau...

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Kernel bandwidth selection for a first order nonparametric streamflow simulation model

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