Let X1,..., Xn be a sample from the inverse Gaussian distribution I(, ) with density

Let X1,..., Xn be a sample from the inverse Gaussian distribution I(μ, τ ) with density

τ

2πx 3 exp

− τ

2xμ2 (x − μ)

2

, x > 0, τ , μ > 0.

Show that there exists a UMP test for testing

(i) H : μ ≤ μ0 against μ > μ0 when τ is known;

(ii) H : τ ≤ τ0 against τ > τ0 when μ is known. In each case give the form of the rejection region.

(iii) The distribution of V = r(Xi − μ)2/Xiμ2 is χ2 1 and hence that of τ

[(Xi −

μ)2/Xiμ2] is χ2 n.

[Let Y = min(Xi, μ2/Xi), Z = τ (Y − μ)2/μ2Y . Then Z = V and Z is χ2 1 [Shuster (1968)].] Note. The UMP test for (ii) is discussed in Chhikara and Folks (1976).