Inverse Gaussian distribution.

12 Let X1,...,Xn be a sample from the inverse Gaussian distribution I(µ, τ ), both parameters unknown.

(i) There exists a UMP unbiased test of µ ≤ µ0 against µ>µ0, which rejects when X>C ¯ [

(Xi + 1/Xi)], and a corresponding UMP unbiased test of

µ = µ0 against µ0 = µ0.

[The conditional distribution needed to carry out this test is given by Chhikara and Folks (1976).]

(ii) There exist UMP unbiased tests of H : τ = τ0 against both one- and two-sided hypotheses based on the statistic V = (1/Xi − 1/X¯).

(iii) When τ = τ0, the distribution of τ0V is χ2 n−1.

[Tweedie (1957).]