seminars:stat:04162015
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| + | I will discuss a paper of Olga Y. Savchuk and Anton Schick. | ||
| + | Consider a random sample $X_1, | ||
| + | the density $g$ of $t(X_1) = |X_1|^\alpha sign(X_1)$ can be estimated in two ways: by a | ||
| + | kernel estimator based on the transformed data $t(X_1), | ||
| + | in estimator transformed from a kernel estimator based on the original data. | ||
| + | In this paper, they compare the performance of these two estimators using | ||
| + | MSE and MISE. For MSE, the plug-in estimator is better in the case $\alpha > 1$ | ||
| + | when $f$ is symmetric and unimodal, and in the case $\alpha \ge 2.5$ when $f$ is right- | ||
| + | skewed and/or bimodal. For $\alpha < 1$, the plug-in estimator performs better | ||
| + | around the modes of $g$, while the transformed data estimator is better in the | ||
| + | tails of $g$. For global comparison MISE, the plug-in estimator has a faster | ||
| + | rate of convergence for $0.4 \le \alpha < 1$ and $1 < \alpha < 2$. For $\alpha < 0.4$, the plug-in | ||
| + | estimator is preferable for a symmetric density $f$ with exponentially decaying | ||
| + | tails, while the transformed data estimator has a better performance when | ||
| + | $f$ is right-skewed or heavy-tailed. Applications to real and simulated data | ||
| + | illustrated these theoretical findings. | ||
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