# How can I prove that the following estimator is biased?

I'm trying to prove that $$e^{x\hat{\beta}}-1$$ is a biased estimator for $$e^{x \beta}-1$$. I know that this involves taking the expected value of the estimator and showing that it is not equal to $$e^{x \beta}-1$$. Unfortunately, I'm stuck on the math. Any hints?

• Use Jensen's inequality – Amit Nov 15 '19 at 21:06
• For convex function $g$, $\mathbb{E}(g(X)) \geq g(\mathbb{E}(X))$. If $g$ is strictly convex then inequality is strict: $\mathbb{E}(g(X)) > g(\mathbb{E}(X))$. – Amit Nov 15 '19 at 21:25