Gaussian Kernel Density Estimation with Bounded Support

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Gaussian Kernel Density Estimation with Bounded Support

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Gaussian Kernel Density Estimation (KDE) is a nonparametric method for estimating the probability density function of a random variable. Unlike parametric approaches that assume a specific distribution form, KDE uses observed data points to construct a smooth density estimate by placing a Gaussian (normal) kernel at each data point and summing their contributions.

However, standard KDE operates over an unbounded domain, which can be problematic when estimating densities for variables with natural bounds, such as positive-only measurements or probabilities constrained between 0 and 1. To address this, adaptations like reflection or transformation methods are applied to ensure density estimates respect the bounded support.

Reflection methods mirror data points across boundaries before applying KDE, effectively preventing leakage outside the valid range. Alternatively, variable transformations (e.g., log-transform for positive data) can map bounded domains to unbounded ones, allowing standard KDE before inversely transforming the result. These techniques maintain the flexibility of nonparametric estimation while producing valid densities within the desired bounds.

Gaussian KDE with bounded support is particularly useful in fields like finance (volatility modeling), biology (concentration measurements), and engineering (reliability analysis), where ignoring natural constraints leads to unrealistic estimates. The choice between reflection and transformation depends on computational efficiency and the behavior of the underlying data near boundaries.