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?

  • $\begingroup$ Use Jensen's inequality $\endgroup$ – Amit Nov 15 '19 at 21:06
  • 2
    $\begingroup$ For convex function $g$, $\mathbb{E}(g(X)) \geq g(\mathbb{E}(X))$. If $g$ is strictly convex then inequality is strict: $\mathbb{E}(g(X)) > g(\mathbb{E}(X))$. $\endgroup$ – Amit Nov 15 '19 at 21:25

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