[Paper Review] Gamma kernel estimation of multivariate density and its derivative on the nonnegative semi-axis by dependent data

L. A. Markovich|arXiv (Cornell University)|Oct 9, 2014

Statistical Methods and Inference3 references3 citations

TL;DR

This paper proposes gamma kernel-based nonparametric estimators for multivariate probability density functions and their partial derivatives on the nonnegative semi-axis using dependent data under strong mixing conditions. It derives asymptotic bias, variance, and covariance, and establishes optimal bandwidths that achieve a mean integrated squared error (MISE) rate of $ n^{-4/7} $, matching the optimal rate for symmetric kernels under independent data, but adapted for asymmetric, boundary-robust gamma kernels on $[0,\infty)^d$. The method effectively handles boundary bias and dependent observations, with theoretical guarantees on convergence and bandwidth selection.

ABSTRACT

In this paper, we consider the nonparametric estimation of the multivariate probability density function and its partial derivative with a support on $[0,\infty)$. To this end we use the class of kernel estimators with asymmetric gamma kernel functions. The gamma kernels are nonnegative. They change their shape depending on the position on the semi-axis and are robust to the boundary bias problem. We investigate the mean integrated squared error (MISE) assuming dependent data with strong mixing and find the optimal bandwidth of the kernel as a minimum of the MISE. We derive the bias, the variance and the covariance of the density and of its partial derivative.

Motivation & Objective

• To develop nonparametric density and derivative estimators for multivariate data supported on the nonnegative semi-axis, where standard symmetric kernels suffer from boundary bias.
• To extend gamma kernel estimation—previously used for univariate or i.i.d. cases—to multivariate, dependent data under strong mixing.
• To derive the asymptotic bias, variance, and covariance of the gamma kernel density and derivative estimators under weak dependence.
• To determine optimal bandwidths that minimize the mean integrated squared error (MISE) for both density and derivative estimation under dependence.
• To establish the optimal rate of convergence for MISE of the derivative estimator, matching known optimal rates under independent data.

Proposed method

• Uses a product gamma kernel estimator for multivariate density on $[0,\infty)^d$, where each marginal kernel is a gamma density with shape parameter depending on the evaluation point.
• Applies the gamma kernel's inherent nonnegativity and shape adaptivity to reduce boundary bias at $x=0$ without reflection or transformation.
• Derives the bias, variance, and covariance of the density and its partial derivative estimators under strong mixing dependence.
• Uses moment bounds and mixing coefficients $\alpha(k)$ to control the dependence structure and derive the asymptotic behavior of the MISE.
• Applies $L^p$-norm inequalities with conjugate exponents $p,q$ and $r = p/(p-1)$ to bound the covariance of kernel functions at different time lags.
• Minimizes the MISE by balancing bias and variance terms, leading to an optimal bandwidth of order $n^{-2/7}$ for the derivative estimator.

Experimental results

Research questions

• RQ1 Can gamma kernel estimators effectively reduce boundary bias in multivariate density estimation on $[0,\infty)^d$ under dependent data?
• RQ2 What are the asymptotic bias, variance, and covariance of the gamma kernel estimator for the multivariate density and its partial derivative under strong mixing?
• RQ3 What is the optimal bandwidth selection rule that minimizes the MISE for the derivative estimator under dependent data?
• RQ4 Does the MISE of the derivative estimator achieve the same optimal rate $n^{-4/7}$ as in the i.i.d. case, despite dependence?
• RQ5 How does the choice of $p,q,r$ conjugate exponents affect the convergence rate and the MISE bound under dependence?

Key findings

• The gamma kernel estimator achieves zero boundary bias when the underlying density has a shoulder at zero, i.e., $f''(0) = 0$, which holds for exponential and gamma families.
• The optimal bandwidth for the derivative estimator is of order $n^{-2/7}$, leading to an optimal MISE rate of $n^{-4/7}$, matching the known optimal rate for symmetric kernels under i.i.d. data.
• The variance of the derivative estimator scales as $b^{-d/2}/n$, where $b$ is the bandwidth, and the bias is of order $b$ and $b^2$.
• The covariance term decays slower than the variance but is shown to be negligible in the MISE minimization due to its lower order of magnitude.
• The MISE bound incorporates the strong mixing coefficient $\alpha(\tau)$, and the integral $\int_1^\infty \alpha(\tau)^\upsilon d\tau$ must be finite for the asymptotic results to hold.
• The final MISE bound is derived by minimizing the sum of squared bias and variance terms, with the optimal bandwidth satisfying $b \sim n^{-2/7}$.