The problem of combining n independent tests as n→∞ for testing that variables are uniformly distributed over the interval (0, 1) compared to their having a conditional shifted exponential distribution with probability density function f (xθ ) = e−(x−γθ) , x ≥γθ , θ ∈[a,∞), a ≥ 0 was studied. This was examined for the case where θ1, θ2, … are distributed according to the distribution function (DF) F and when the DF is Gamma (1, 2). Six omnibus methods were compared via the Bahadur efficiency. It is shown that, as γ → 0 and γ → ∞ , the inverse normal method is the best among the methods studied.