Bidirectional Unbiased Procedures

Abstract When the hypothesis H 0: θ = θ0 is tested against the alternative Ha: θ ≠ θ0, θ a real-valued parameter, it would often be more appropriate to make simultaneous tests of the two hypotheses 1 H 0: θ ≤ θ0 (with alternative 1 Ha: θ > θ0) and 2 H 0: θ ≥ θ0 (with alternative 2 Ha: θ < θ0). The criterion of unbiasedness in choosing an optimal test of H 0 is inapplicable for choosing an optimal simultaneous procedure for testing 1 H 0 and 2 H 0. A bidirectional unbiased procedure is defined and its properties enumerated. In many important applications such a procedure for testing 1 H 0 and 2 H 0 is essentially equivalent to an “equal tails” test of H 0.