Method of Moments Estimator and its bias

Question

Let $X_1,.....,X_n$ be a random sample from population with density

$$f(x) =\frac{\beta}{x^{\beta+1}} \qquad x \geq 1$$

Use the method of moments to find an estimator for the parameter beta. Upon solving I found that

$$\hat{\beta} = \frac{\overline{X}}{\overline{X}-1}$$

However I am not sure how to show that the estimator is biased. Can I get some help with that?

Answer 1

As suggested by Henry, we write $$\hat{\beta} = \frac{\overline{X}}{\overline{X}-1}$$ A quick computation yields that $E(X) = \frac{\beta}{\beta-1}$, which you already probably obtained in computing your method of moments estimator. Notice that the function $f(x) = \frac{x}{x-1}$ is strictly convex on $[1, \infty )$, so Jensen's inequality gives us that: $$E(\hat{\beta}) = E \left( \frac{\overline{X}}{\overline{X}-1} \right) > \frac{E(\overline{X})}{E(\overline{X}) -1 } = \frac{\beta/(\beta-1)}{\beta/(\beta-1)-1} = \beta$$

showing that the estimator is biased.