Two-sided t-test. (i) Let X1,...,Xn be a sample from N(, 2). For testing = 0 against

Two-sided t-test.

(i) Let X1,...,Xn be a sample from N(ξ, σ2). For testing ξ = 0 against ξ = 0, there exists a UMP invariant test with respect to the group X

i = cXi, c = 0, given by the two-sided t-test (5.17).

(ii) Let X1,...,Xm, and Y1,...,Yn be samples from N(ξ, σ2) and N(η, σ2)

respectively. For testing η = ξ against η = ξ there exists a UMP invariant test with respect to the group X

i = aXi +

b, Y

j = aYj +

b, a = 0, given by the two-sided t-test (5.30).

[(i): Sufficiency and invariance reduce the problem to |t|, which in the notation of Section 4 has the probability density pδ(t) + pδ(−t) for t > 0. The ratio of this density for δ = δ1 to its value for δ = 0 is proportional to ∞
0 (eδ1v +
e−δ1v)gt2 (v) dv, which is an increasing function of t 2 and hence of |t|.]