Kernel density estimation via diffusion
TL;DR: A new adaptive kernel density estimator based on linear diffusion processes that builds on existing ideas for adaptive smoothing by incorporating information from a pilot density estimate and a new plug-in bandwidth selection method that is free from the arbitrary normal reference rules used by existing methods.
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Abstract: We present a new adaptive kernel density estimator based on linear diffusion processes. The proposed estimator builds on existing ideas for adaptive smoothing by incorporating information from a pilot density estimate. In addition, we propose a new plug-in bandwidth selection method that is free from the arbitrary normal reference rules used by existing methods. We present simulation examples in which the proposed approach outperforms existing methods in terms of accuracy and reliability.
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Figures
Fig. 1. Boundary bias in the neighborhood of x= 0. 
Table 2 Results over 10 independent simulation experiments. In all cases the domain was assumed to be R 
Fig. 4. Right panel: plug-in rule with normal reference rule; left panel: the Improved Sheather–Jones method; the normal reference rule causes significant over-smoothing. 
Fig. 5. A two-dimensional example with 600 points generated uniformly within an ellipse. 
Fig. 3. The Improved Sheather–Jones bandwidth selection rule in Algorithm 1 leads to improved performance compared to the original plug-in rule that uses the normal reference rule. ![Table 3 Practical performance of the boundary bias correction of the diffusion estimator for the test cases: (1) exponential distribution with mean equal to unity; (2) test cases 1 through 8, truncated to the interval (−∞,0]](/figures/table3-1-1p5ziqv1n1a5.png)
Table 3 Practical performance of the boundary bias correction of the diffusion estimator for the test cases: (1) exponential distribution with mean equal to unity; (2) test cases 1 through 8, truncated to the interval (−∞,0]
![Table 3 Practical performance of the boundary bias correction of the diffusion estimator for the test cases: (1) exponential distribution with mean equal to unity; (2) test cases 1 through 8, truncated to the interval (−∞,0]](/figures/table3-1-1p5ziqv1n1a5.webp)
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