## Unbiased Estimator (15/365)

In the coin tossing setup, we saw that the fraction $T_n(\omega) = \frac{S_n(\omega)}{n}$ of observed heads in $n$ trails $\omega$ approaches $p$ as $n \rightarrow \infty$. The function $T_n$ is called an estimator that takes on a value in $[0,1]$. It’s a special kind of estimator called an unbiased estimator because it satisfies

$\displaystyle E_\theta T_n = \theta \text{ for all } \theta \in [0,1]$

It says that on average, this estimator deduces the correct answer. Consider a different estimator $T_n(\omega) = \frac{S_n(\omega)}{n+1}$ which essentially starts with an assumed ‘tails’. Then

$\displaystyle E_\theta T_n = \frac{n}{n+1} E_\theta \frac{S_n}{n} = \frac{n\theta}{n+1}$

which, on average, slightly underestimates the success probability.

A problem asks the following. Let it be known a priori that $\theta$ has a value in the set $\Omega_0 \subseteq [0,1]$. Construct an unbiased estimator for $\theta$, taking values only in $\Omega_0$.

Consider the case where $\Omega_0 = \{ r \}$ for $r \in [0,1]$. Then $T_n = r$ is an unbiased estimator because

$\displaystyle E_\theta T_n = r\theta + (1-\theta)r \text{ for } \theta \in \Omega_0$

which is unbiased because $\theta$ can only be $r$. Now, I can’t seem to proceed further than this. For example, what is the estimator when $\Omega_0 = \{ r_1, r_2 \}$? I’ll have to return with an answer another day.

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