Assume we have a random sample $X_1,\dots,X_n$ from a distribution of the form $f(x_i;\theta) = h(x)g(\theta)\exp(\eta(\theta) T(x))$ and we wish to test $H_0: \theta \leq \theta_0, H_1: \theta > \theta_0$. If $T: \mathbb{R}^n \to \mathbb{R}$ then one might apply the Karlin-Rubin Theorem, namely in the case that the distribution has a monotone non-decreasing likelihood ratio in $T(X)$. Is there a similar procedure if $T: \mathbb{R}^n \to \mathbb{R}^m$ for $m > 1$?

As an example, for the inverse Gaussian distribution, the joint pdf is $$f(x;\mu,\lambda) = \left( \frac{\lambda}{2\pi x^3} \right)^{1/2} \exp \left( \frac{\lambda n}{\mu} \right) \exp\left( -\frac{\lambda}{2\mu^2} \sum x_i - \frac{\lambda}{2} \sum \frac{1}{x_i} \right).$$ In this case, $T(x) = \left( \sum x_i, \sum \frac{1}{x_i} \right)$. What would be a UMP test here (if there is one) for $H_0: \mu \leq \mu_0, H_1: \mu > \mu_0$?


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