Repeat sampling of extreme observations: regression to the mean revisited

The phenomenon of classical regression to the mean was described by Sir Francis Galton in a series of prestigious works in the 19th century. This phenomenon refers to the fact that, in the presence of measurement error, experimental units which give rise to extreme values upon first sampling typically produce less extreme values upon a second independent (or repeat) sampling. This shift from the tails toward the population mean occurs even though there has been no intervention or change in the underlying population or error distributions. The mathematical ideas used to explain this shift typically appeal to correlation arguments and the classical Gaussian model. In this paper we study repeat sampling effects in the tails of arbitrary distributions. Perhaps surprisingly, we are able to show that there are actually three distinct asymptotic repeat sampling effects, of which only one corresponds to Galton’s classical result. These three effects depend on the heaviness of the population tails. In particular, for population distributions with relatively heavy tails the maximum shift occurs in the interior of the distribution. In this case the classical regression effect of Galton actually disappears out in the tails.