# UMP test for exponential family when sufficient statistics $T$ is a vector

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 $$H_0: \mu \leq \mu_0, H_1: \mu > \mu_0$$?