Let X1,...,Xm; Y1,...,Yn be samples from exponential distributions with densities for 1e(x)/, for x , and

Let X1,...,Xm; Y1,...,Yn be samples from exponential distributions with densities for σ−1e−(x−ξ)/σ, for x ≥ ξ, and τ −1e−(y−n)/τ for y ≥ η.

(i) For testing τ /σ ≤ ∆ against τ /σ > ∆, there exists a UMP invariant test with respect to the group G : X

i = aXi +

b, Y

j = aYj +

c, a > 0, −∞ <

b, c < ∞, and its rejection region is

[yj − min(y1,...,yn)]

[xi − min(x1,...,xm)] > C.

(ii) This test is also UMP unbiased.

(iii) Extend these results to the case that only the r smallest X’s and the s smallest Y ’s are observed.

[(ii): See Problem 5.15.]