Averaging quantiles, variance shrinkage, and overconfidence

Averaging quantiles as a way of combining experts' judgments is studied both mathematically and empirically. Quantile averaging is equivalent to taking the harmonic mean of densities evaluated at quantile points. A variance shrinkage law is established between equal and harmonic weighting. Data from 49 post-2006 studies are extended to include harmonic weighting in addition to equal and performance-based weighting. It emerges that harmonic weighting has the highest average information and degraded statistical accuracy. The hypothesis that the quantile average is statistically accurate would be rejected at the 5% level in 28 studies and at the 0.1% level in 15 studies. For performance weighting, these numbers are 3 and 1, for equal weighting 2 and 1.