# Sufficient Estimators and Generalized Likelihood Ratios

If you can make the assumption that a sufficient statistic exists for some parameter - let's call it $\theta$.

How would you show that the critical region of a likelihood ratio test will depend on the sufficient statistic?

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If $T(\mathbf{X})$ is a sufficient statistic for $\mathbf{X}$ then by the factorisation theorem $f(\mathbf{X}|\theta) = h(\mathbf{X}) \, g(\theta, T(\mathbf{X}))$ so the likelihood ratio is $\Lambda(\mathbf{X})= \dfrac{f(\mathbf{X}|\theta_0)}{f(\mathbf{X}|\theta_1)} =\dfrac{g(\theta_0, T(\mathbf{X}))}{g(\theta_1, T(\mathbf{X}))}$, and so the likelihood ratio and its critical region depends on $\mathbf{X}$ only through the sufficient statistic $T(\mathbf{X})$.