  The beta distribution is a two-parameter family of probability distributions
whose distribution function is the (regularised) incomplete beta function. In
this paper, the inverse incomplete beta function is studied analytically as
univariate function of the first parameter. Monotonicity, limit results and
convexity properties are provided. In particular, logarithmic concavity of the
inverse incomplete beta function is established. In addition, we provide
monotonicity results on inverses of a larger class of parametrised
distributions that may be of independent interest.
