  Neutral evolution assumes that there are no selective forces distinguishing
different variants in a population. Despite this striking assumption, many
recent studies have sought to assess whether neutrality can provide a good
description of different episodes of cultural change. One approach has been to
test whether neutral predictions are consistent with observed progeny
distributions, recording the number of variants that have produced a given
number of new instances within a specified time interval: a classic example is
the distribution of baby names. Using an overlapping generations model we show
that these distributions consist of two phases: a power law phase with a
constant exponent of -3/2, followed by an exponential cut-off for variants with
very large numbers of progeny. Maximum likelihood estimations of the model
parameters provide a direct way to establish whether observed empirical
patterns are consistent with neutral evolution. We apply our approach to a
complete data set of baby names from Australia. Crucially we show that analyses
based on only the most popular variants, as is often the case in studies of
cultural evolution, can provide misleading evidence for underlying transmission
hypotheses. While neutrality provides a plausible description of progeny
distributions of abundant variants, rare variants deviate from neutrality.
Further, we develop a simulation framework that allows for the detection of
alternative cultural transmission processes. We show that anti-novelty bias is
able to replicate the complete progeny distribution of the Australian data set.
