Log-transformed kernel density estimation for positive data
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Kernel density estimators (KDEs) are ubiquitous tools for nonpara- metric estimation of probability density functions (PDFs), when data are obtained from unknown data generating processes. The KDEs that are typically available in software packages are defined, and designed, to estimate real-valued data. When applied to positive data, these typical KDEs do not yield bona fide PDFs as outputs. A log-transformation can be applied to the kernel functions of the usual KDEs in order to produce a nonparametric estimator that is appropriate and yields proper PDFs over positive supports. We call the KDEs obtained via this transformation log-KDEs. We derive expressions for the pointwise biases, variances, and mean-squared errors of the log-KDEs that are obtained via various underlying kernel functions. Mean integrated squared error (MISE) and asymptotic MISE results are also provided and used to derive a plug-in rule for log-KDE bandwidths. The described log-KDEs are implemented through our R package logKDE, which we describe and demonstrate. A set of numerical simulation studies and real data case studies are provided to demonstrate the strengths of our log-KDE approach.
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