We consider statistical estimation of the Rényi exponent [math] , which characterizes the scaling behaviour of a singular measure [math] defined on a subset of [math] . The Rényi exponent is defined to be [math] , assuming that this limit exists, where [math] and, for [math] , [math] are the cubes of a [math] -coordinate mesh that intersect the support of [math] . In particular, we demonstrate asymptotic normality of the least-squares estimator of [math] when the measure [math] is generated by a particular class of multiplicative random cascades, a result which allows construction of interval estimates and application of hypothesis tests for this scaling exponent. Simulation results illustrating this asymptotic normality are presented.