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

CC BY-SA 4.0
2019-11-15T21:06:03.567Z
Use Jensen's inequality
Amit
CC BY-SA 4.0
2019-11-15T21:25:36.243Z
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
<econometrics><statistics>
How can I prove that the following estimator is biased?
Nolan
# How can I prove that the following estimator is biased?

(Asked by: username_0 on 2019-11-15)

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?

**Tags:** econometrics, statistics

> **Comments:**
> **username_1**: Use Jensen's inequality  
> **username_1**: For convex function $g$, $\mathbb{E}(g(X)) \geq g(\mathbb{E}(X))$. If $g$ is strictly convex then inequality is strict: $\mathbb{E}(g(X)) > g(\mathbb{E}(X))$.  

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## 0 answers

