Suppose $X_1,\ldots,X_n$ is a sample from a population with parameter $\theta$. Prove that if $T$ is a sufficient statistic for $\theta$, and $\theta=h(\eta)$ where $h$ is differentiable, then $T$ is a sufficient statistic for $\eta$. Use the chain rule to construct your proof. You may assume that $f_{X_1,\ldots,X_n\mid T}(x_1,\ldots,x_n\mid t;\theta)$ is a differentiable function of $\theta$.

I was wondering if it would be better to approach this problem using the definition of sufficiency, or if I should try to implement the Neyman Factorization Theorem somewhere. A hint or idea about how to start this proof would be appreciated.


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