Abstract:
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Estimation of the continuous risk of disease contraction over some spatial region $W$ is a core objective in applications in geographical epidemiology. Such pursuits require knowledge of the probability densities of both the infected and uninfected (at-risk) populations ($f$ and $g$ respectively). Their ratio, $r=f/g$, indicates spatial areas of increased and decreased risk with peaks and troughs respectively. The complexities associated with the often highly heterogeneous nature of the requisite density functions renders nonparametric 2-dimensional kernel density estimation particularly appealing for estimation of $f$ and $g$, given samples of data assumed to have arisen therefrom. However, a number of familiar practical problems associated with kernel smoothing are present, even exacerbated, in this density-ratio estimation. In this work, we use applied examples to discuss the latest research into the so-called relative risk estimator. Our findings indicate that this density-ratio estimation is considerably aided by using a variable (adaptive) smoothing regimen that can be symmetrized to stabilize the resulting surfaces, without disturbing beneficial asymptotic properties.
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