  The Hirsch index (commonly referred to as h-index) is a bibliometric
indicator which is widely recognized as effective for measuring the scientific
production of a scholar since it summarizes size and impact of the research
output. In a formal setting, the h-index is actually an empirical functional of
the distribution of the citation counts received by the scholar. Under this
approach, the asymptotic theory for the empirical h-index has been recently
exploited when the citation counts follow a continuous distribution and, in
particular, variance estimation has been considered for the Pareto-type and the
Weibull-type distribution families. However, in bibliometric applications,
citation counts display a distribution supported by the integers. Thus, we
provide general properties for the empirical h-index under the small- and
large-sample settings. In addition, we also introduce consistent nonparametric
variance estimation, which allows for the implemention of large-sample set
estimation for the theoretical h-index.
