On the uniform random upper bound family of first significant digit distributions

The first significant digit patterns arising from a mixture of uniform distributions with a random upper bound are revisited. A closed-form formula for its first significant digit distribution (FSD) is obtained. The one-parameter model of Rodriguez is recovered for an extended truncated Pareto mixing distribution. Considering additionally the truncated Erlang, gamma and Burr mixing distributions, and the generalized Benford law, for which another probabilistic derivation is offered, we study the fitting capabilities of the FSD's for various Benford like data sets from scientific research. Based on the results, we propose the general use of a fine structure index for Benford's law in case the data is well fitted by the truncated Erlang member of the uniform random upper bound family of FSD's.