Estimating conditional means with heavy tails

When a conditional distribution has an infinite variance, commonly employed kernel smoothing methods such as local polynomial estimators for the conditional mean admit non-normal limiting distributions (Hall et al., 2002). This complicates the related inference as the conventional tests and confidence intervals based on asymptotic normality are no longer applicable, and the standard bootstrap method often fails. By utilizing the middle part of data nonparametrically and the tail parts parametrically based on extreme value theory, this paper proposes a new estimation method for conditional means, resulting in asymptotically normal estimators even when the conditional distribution has infinite variance. Consequently the standard bootstrap method could be employed to construct, for example, confidence intervals regardless of the tail heaviness. The same idea can be applied to estimating the difference between a conditional mean and a conditional median, which is a useful measure in data exploratory analysis.