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By Michel Hervé

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In particular, it is more stable than the cross-validation approach described above. The two techniques take different approaches to the same problem of minimising ISE. Cross-validation estimates the ISE function and locates the minimum. The plug-in approach minimises the function theoretically and then estimates this minimising value directly. The good performance of this estimation process produces a method which is subject to less variability. 016. 018 and so in this particular example there is little difference between the two approaches.

The combined effect of these properties is that, in order to produce an estimator which converges to the true density function /, it is necessary that both h and 1/nh decrease as the sample size increases. A suitable version of the central limit theorem can also be used to show that the distribution of the estimator is asymptotically normal. A similar analysis enables approximate expressions to be derived for the mean and variance of a density estimate in the multivariate case. In p dimensions, with a kernel function defined as the product of univariate components w, and with smoothing parameters (h 1 , .

The left panel shows the cross-validation function based on the tephra data, on a logit scale. The MISE function for a normal density has been superimposed, on a shifted vertical scale, for comparison. The right panel shows density estimates from the tephra data using the cross-validatory smoothing parameter (full line), and the normal optimal smoothing parameter (dashed line). Cross-validation has therefore successfully chosen a suitable amount of smoothing to apply to the data. In the left panel, the MISE function for a normal distribution is also displayed.

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