What is the invariance principle?

Question

Let $X_1,X_2,...,X_n$ be a random sample , $0<p<1$

$p_X(x)=(1-p)^{x-1}p$, $x=1,2,...$

I have worked out the MLE and have shown with further working that it is a maximum, but the next part of the question asks Find the maximum likelihood estimator for $\mathbf{θ=\frac{1}{p}}$. I think the invariance principle is required for this part?

Could someone explain what the invariance principle is? Many Thanks

Answer 1

the invariance principle simply says that

"if $\hat{p}$ is the MLE for $p$ then $g(\hat{p})$ is MLE for $g(p)$"

thus

$$\hat{\theta}_{ML}=\overline{X}_n$$