In this chapter, we consider maximum likelihood estimation of the parameters of certain distributions, when there is the possibility that size-biased sampling has been used, rather than simple random sampling. If a statistical test is used to discriminate between these sampling procedures, and estimation proceeds on the basis of the outcome of that test, then we have a ‘preliminary-test estimation’ problem that has not previously been investigated. Using three common single-parameter distributions that are members of the generalized gamma family and various loss functions, we analyse the relative finite-sample biases and relative risks of such preliminary-test estimators, via Monte Carlo simulation. We also address the issue of selecting the optimal critical value for the test in question, based on the ‘mini-max regret’ criterion. We find that a constant critical value is optimal, regardless of the sample size, and that this value is very similar across the three distributions considered.