HT-Adjusted Kernel Density Plug-In

• The Horvitz-Thompson-adjusted kernel density plug-in is a nonparametric estimator that reweights data to correct for unequal inclusion probabilities in survey samples.
• It integrates kernel smoothing with plug-in divergence minimization and minimum Hellinger distance estimation to achieve robust and design-consistent parameter estimates.
• The method offers theoretical guarantees such as L1-consistency, large-deviation bounds, and asymptotic normality, validated through empirical applications.

The Horvitz-Thompson-Adjusted Kernel Density Plug-In is a nonparametric estimation procedure for complex survey designs that corrects for unequal inclusion probabilities and leverages the Horvitz–Thompson framework within kernel density estimation. This methodology enables robust, consistent density estimation and subsequent statistical inference in finite populations where conventional approaches are sensitive to sampling weights, outliers, and design-based leverage. It integrates survey sampling principles, kernel smoothing, and plug-in divergence minimization, underpinning modern nonparametric estimation for superpopulation models and empirical functionals.

1. Horvitz–Thompson Adjustment in Kernel Density Estimation

The Horvitz–Thompson (HT) principle assigns each unit a weight inverse to its inclusion probability, $\pi_{\gamma i}$, ensuring unbiased finite-population inference under unequal probability sampling. For kernel density estimation, the classic estimator is reweighted:

$$\hat f_\gamma(y) = \frac{T_\gamma(y)}{S_\gamma}, \quad T_\gamma(y) = \frac{1}{N_\gamma}\sum_{i\in U}\frac{\delta_{\gamma i}}{\pi_{\gamma i}} K_{h_\gamma}(y - Y_{\gamma i}), \quad S_\gamma = \frac{1}{N_\gamma} \sum_{i\in U}\frac{\delta_{\gamma i}}{\pi_{\gamma i}},$$

where $\delta_{\gamma i}$ flags sample membership, $N_\gamma$ is the finite population size, and $K_{h_\gamma}$ is a kernel with scale parameter $h_\gamma$. The normalization $S_\gamma$ ensures that $\hat f_\gamma(y)$ integrates to one despite design weights. This approach generalizes directly under post-stratification and calibration, preserving unbiasedness in expectation for the population density.

2. Integration in Divergence-Based Plug-In Estimation

The plug-in estimator leverages the weighted kernel density $\hat f_\gamma$ as a proxy for the unknown true superpopulation density $g$. In minimum Hellinger distance estimation (MHDE)—a robust parametric procedure—the objective is:

$$\hat\theta_\gamma = \arg\min_{\theta\in\Theta} H^2(f_\theta, \hat f_\gamma) = \arg\max_{\theta\in\Theta}\; \int \sqrt{f_\theta(y)\,\hat f_\gamma(y)}\,dy,$$

where $f_\theta$ is the parametric model and $H^2$ the squared Hellinger distance. The HT–adjusted kernel density plug-in supplies a design-consistent, self-normalizing empirical density for MHDE, yielding parameter estimates less sensitive to sampling anomalies than weighted MLE.

3. Theoretical Guarantees: Consistency, Large-Deviation Bounds, Asymptotic Normality

The HT-adjusted KDE attains $L^1$-consistency and explicit large-deviation tail bounds. As established:

$\|\hat f_\gamma - g\|_{L^1} \xrightarrow{P} 0,$

$$P\left(\|\hat f_\gamma - g\|_{L^1} > \tau \right) \le C_1 \exp\{-C_2\,h_\gamma^d\,\min(\tau^2,\tau)\} + C_1 \exp\{-C_3\,N_\gamma\, h_\gamma^d\,\tau^2\} + o(1),$$

where $C_1, C_2, C_3$ are fixed constants, $d$ is the dimension, $h_\gamma$ the bandwidth, and $N_\gamma h_\gamma^d \to \infty$ ensures exponential decay of tail probabilities.

For downstream inference, the MHDE achieves asymptotic normality:

$$\sqrt{N_\gamma h_\gamma^d} (\hat\theta_\gamma - \theta_0) \;\Rightarrow\; \mathcal{N}_p(0,\,A^{-1}\,\bm{\Sigma}\,A^{-T}),$$

with $A$ the Hessian of the population Hellinger affinity at $\theta_0$ and $\bm{\Sigma}$ a design-adjusted sandwich covariance. Finite-population correction factors (e.g., $1 - \alpha_\gamma$ for sampling without replacement) further adjust the variance as necessary.

4. Robustness and Influence Functions

Robustness arises from the MHDE’s bounded influence function and $\alpha$-influence curves in the Hellinger topology. Specifically, for contaminating mass at $z$:

$$\text{IF}(z; T, G) = -Q^{-1}\phi_g(z), \quad \phi_g(z) = \frac{1}{4} u_{\theta_0}(z)\sqrt{\frac{f_{\theta_0}(z)}{g(z)}},$$

where $$u_{\theta_0}(y) = \nabla_\theta \log f_\theta(y)\big|_{\theta_0}$$ and $Q$ is the gradient matrix of the affinity’s score. An $\alpha$-influence curve characterizes first-order behavior under contamination:

$$T(G_\varepsilon) = \theta_0 - \varepsilon Q^{-1}\phi_g(z) + O(\varepsilon^2),$$

demonstrating controlled sensitivity to outliers, essential for robust survey inference when extreme design weights or responses are present.

5. Implementation and Computational Considerations

Computation of the HT-adjusted KDE is straightforward, especially when evaluated on a pre-defined grid via quadrature (e.g., Gauss–Kronrod for numerical integration). The methodology handles large samples efficiently due to its additive structure over the sample units. No iterative or high-dimensional optimization is required for density estimation; MHDE parameter optimization proceeds over a single or limited-dimensional parametric space. Simulations confirm tractable execution and scalability.

6. Empirical Performance and Applications

Simulations under Gamma and lognormal superpopulation models document the superiority of the MHDE (using the HT–adjusted KDE plug-in) over weighted MLE in efficiency-robustness trade-offs, especially under contamination by extreme observations or high-leverage units. Applications to dietary survey data (NHANES 2021–2023 total water consumption) show stable estimation despite outliers, in direct contrast to weighted likelihood estimates that exhibit considerable bias.

7. Relationship to Kernel Density Plug-In and Extensions

The plug-in principle is mirrored in related nonparametric functionals: the HT–adjusted KDE, when inserted into empirical Bayes, compound decision, or population total estimation, achieves bias reduction analogous to kernel plug-in methods in classical statistics. Deconvolution methods, as discussed for Horvitz–Thompson estimator modification (Greenshtein et al., 2013), use similar inversion machinery to estimate unknown mixing distributions and functionals thereof—suggesting extensibility of the HT-adjusted kernel plug-in framework to broader empirical Bayes and multiple testing contexts.

In sum, the Horvitz–Thompson–adjusted kernel density plug-in is established as a robust, design-consistent estimator for density and functional estimation under complex survey designs. It yields favorable large-sample properties, controls outlier impact, and supports efficient computation for population-level inference, aligning with the evolving requirements of survey methodology and robust statistics (Keepplinger et al., 15 Oct 2025).