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Abstract:
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Mean (or expected value) is a measurement of central tendency. However, it does not exist for some heavy-tailed distributions such as the Cauchy. The undefined or infinite mean of these heavy-tailed distributions explains the unreasonably large variance and bias in the sample mean, which, in turn, make all related statistical inference methods unreliable. We define kth-order mean (different from kth moment), which is 'always' finite and proportional to the natural location parameter of the underlying distribution for an appropriate choice of k. Then we propose a new nonparametric location shift estimator namely 'Kernel Weighted Average" (KWA). It is shown that the KWA estimator has a smaller empirical MSE than other existing location shift estimators such as the mean, the median and the Hodges-Lehmann (HL) estimator for heavy-tailed distributions, such as the Cauchy distribution.
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