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

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−η)/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 = a Xi +

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.]