[Paper Review] Unified treatment of the asymptotics of asymmetric kernel density estimators

Till Hoffmann, Nick S. Jones|arXiv (Cornell University)|Dec 10, 2015

Gaussian Processes and Bayesian Inference5 references3 citations

TL;DR

This paper introduces a unified framework for analyzing asymptotic properties of asymmetric kernel density estimators by extending balloon and sample-smoothing estimators with a shift parameter. It derives general expressions for bias, variance, and mean integrated squared error, enabling systematic study of gamma, log-normal, Birnbaum-Saunders, inverse Gaussian, and reciprocal inverse Gaussian kernels, and proposes two new properly normalized estimators for positive random variables with plugin bandwidth rules.

ABSTRACT

We extend balloon and sample-smoothing estimators, two types of variable-bandwidth kernel density estimators, by a shift parameter and derive their asymptotic properties. Our approach facilitates the unified study of a wide range of density estimators which are subsumed under these two general classes of kernel density estimators. We demonstrate our method by deriving the asymptotic bias, variance, and mean (integrated) squared error of density estimators with gamma, log-normal, Birnbaum-Saunders, inverse Gaussian and reciprocal inverse Gaussian kernels. We propose two new density estimators for positive random variables that yield properly-normalised density estimates. Plugin expressions for bandwidth estimation are provided to facilitate easy exploratory data analysis.

Motivation & Objective

• To unify the asymptotic analysis of variable-bandwidth kernel density estimators, including balloon and sample-smoothing types, by introducing a shift parameter.
• To address the limitations of fixed-bandwidth KDEs in balancing bias and variance across high- and low-density regions.
• To extend the applicability of asymmetric kernels—such as gamma, log-normal, and inverse Gaussian—to improve boundary bias and ensure proper normalization.
• To derive plugin bandwidth selection rules for practical exploratory data analysis.
• To propose two new density estimators for positive random variables that yield properly normalized density estimates.

Proposed method

• Introduces shifted balloon and sample-smoothing estimators by incorporating a shift parameter into the kernel bandwidth function to control bias and variance.
• Derives general asymptotic expressions for bias, variance, and mean (integrated) squared error using complex analysis and residue theory, particularly via contour integration and Laurent series expansion.
• Applies Lemma 1 and Lemma 2 to express the first and second moments of the estimator in terms of derivatives of the kernel and bandwidth functions.
• Uses the residue theorem to evaluate contour integrals arising from the moment-generating function approach, enabling systematic expansion in powers of the bandwidth.
• Derives plugin bandwidth rules by minimizing the asymptotic MISE, based on the derived bias and variance expressions.
• Validates the framework by applying it to five specific asymmetric kernels: gamma, log-normal, Birnbaum-Saunders, inverse Gaussian, and reciprocal inverse Gaussian.

Experimental results

Research questions

• RQ1 How can the asymptotic bias, variance, and mean integrated squared error be uniformly derived across a broad class of asymmetric kernel density estimators?
• RQ2 What is the impact of introducing a shift parameter on the bias and normalization properties of balloon and sample-smoothing estimators?
• RQ3 Can a single framework unify the analysis of both balloon and sample-smoothing estimators with asymmetric kernels?
• RQ4 How can plugin bandwidth rules be derived consistently across different asymmetric kernels to minimize MISE?
• RQ5 What new properly normalized density estimators can be constructed for positive random variables using this unified framework?

Key findings

• The unified framework enables consistent derivation of asymptotic bias, variance, and MISE for a wide class of asymmetric kernel estimators, including gamma, log-normal, Birnbaum-Saunders, inverse Gaussian, and reciprocal inverse Gaussian kernels.
• The asymptotic variance of the shifted sample-smoothing estimator is $ \frac{f(x)\kappa}{nh(x)} + o(n^{-1}h^{-1}(x)) $, where $ \kappa $ is the integral of the squared kernel.
• The first moment of the shifted sample-smoothing estimator is $ \left\langle \hat{f}(x) \right\rangle = B_0(x) + B_p(x) + o(h^p(x)) $, with $ B_k(x) $ defined via higher-order derivatives of the density and bandwidth functions.
• The framework yields two new density estimators for positive random variables that are properly normalized, ensuring $ \int \hat{f}(x) \, dx = 1 $.
• Plugin bandwidth rules are derived for each kernel type, facilitating practical implementation and exploratory data analysis.
• The method confirms that asymmetric kernels such as gamma and inverse Gaussian eliminate boundary bias on the positive half-line, while maintaining favorable asymptotic properties under the unified framework.