  Statistical hypotheses are translations of scientific hypotheses into
statements about one or more distributions, often concerning their centre.
Tests that assess statistical hypotheses of centre implicitly assume a specific
centre, e.g., the mean or median. Yet, scientific hypotheses do not always
specify a particular centre. This ambiguity leaves the possibility for a gap
between scientific theory and statistical practice that can lead to rejection
of a true null. In the face of replicability crises in many scientific
disciplines, significant results of this kind are concerning. Rather than
testing a single centre, this paper proposes testing a family of plausible
centres, such as that induced by the Huber loss function (the Huber family).
Each centre in the family generates a testing problem, and the resulting family
of hypotheses constitutes a familial hypothesis. A Bayesian nonparametric
procedure is devised to test familial hypotheses, enabled by a novel pathwise
optimization routine to fit the Huber family. The favourable properties of the
new test are demonstrated theoretically and experimentally. Two examples from
psychology serve as real-world case studies.
