Bayesian selector of adaptive bandwidth for multivariate gamma kernel estimator on [0,∞ ) .

Area of Science:

• Statistics
• Probability Theory
• Computational Statistics

Background:

• Classical methods like cross-validation for kernel density estimation can be slow and suboptimal.
• Multivariate probability density function estimation is crucial in various scientific fields.
• Existing methods struggle with accurate bandwidth selection in complex density estimations.

Purpose of the Study:

• To introduce a novel Bayesian adaptive bandwidth selection for multivariate kernel density estimation.
• To utilize a multivariate gamma kernel suitable for estimating densities on positive support.
• To improve execution time and smoothing quality compared to conventional techniques.

Main Methods:

• Employed Bayesian adaptive estimation for the bandwidths vector under a quadratic loss function.
• Derived the exact posterior distribution and the optimal bandwidths vector.
• Utilized a multivariate gamma kernel for density estimation on positive support.

Main Results:

• The proposed Bayesian approach significantly outperforms global cross-validation bandwidth selection.
• Achieved superior smoothing quality and reduced execution time in simulation studies.
• Demonstrated excellent performance under integrated squared errors criterion.

Conclusions:

• Bayesian bandwidth selection provides a more efficient and accurate alternative for multivariate kernel density estimation.
• The developed method is effective for estimating densities with positive support.
• Successfully applied to real-world datasets, including geyser eruption data and drinking water quality.