Sample size calculations for the mean in a two component nonstandard mixture distribution

We are concerned with calculating the sample size required for estimating the mean of the continuous distribution in the context of a two component nonstandard mixture distribution (i.e., a mixture of an identifiable point degenerate function F at a constant with probability P and a continuous distribution G with probability 1 - P). A common ad hoc procedure of escalating the naïve sample size n (calculated under the assumption of no point degenerate function F) by a factor of 1/(1 - P), has about 0.5 probability of achieving the pre-specified statistical power. Such an ad hoc approach may seriously underestimate the necessary sample size and jeopardize inferences in scientific investigations. We argue that sample size calculations in this context should have a pre-specified probability of power ≥ 1 - β set by the researcher at a level greater than 0.5. To that end, we propose an exact method and an approximate method to calculate sample size in this context so that the pre-specified probability of achieving a desired statistical power is determined by the researcher.