Estimators

Learning Outcomes

• Estimators

• Unbiased Estimators

• Bias

• Mean Square Error

Estimators

An estimator is an operation computing the value of an estimate, that targets the parameter, using measurements from a sample.

Unbiased Estimator

An unbiased estimator \(\hat\theta\) is an estimator that satisfies the following condition:

\[ E(\hat\theta) = \theta \]

Bias

The bias of an estimator is defined as

\[ B(\hat\theta) = E(\hat\theta)-\theta \]

Mean Square Error

The mean square of an estimator is \(\hat\theta\) is given as

\[ \begin{eqnarray} MSE(\hat\theta) & = & E\{(\hat\theta-\theta)^2\} \\ & = & Var(\hat\theta) + B(\hat\theta)^2 \end{eqnarray} \]

Is \(\bar X\) an unbiased estimator of \(\mu\)?

Let \(X_1,\ldots,X_n\overset{iid}{\sim}N(\mu,\sigma^2)\), find the bias of \(\bar X\).

Why is \(S²\) divided by \(n-1\) instead of \(n\)?

Let \(X_1,\ldots,X_n\overset{iid}{\sim}N(\mu,\sigma^2)\), find the bias of \(S²\).

Problem

Let \(X_1,X_2,X_3\) follow and exponential distribution with mean and variance \(\lambda\) and \(\lambda²\), respectively. Using the following estimators:

• \(\hat\theta_1 = X_1\)

• \(\hat\theta_2 = \frac{X_1+X_2}{2}\)

• \(\hat\theta_3 = \frac{X_1+2X_2}{3}\)

• \(\hat\theta_4 = \frac{X_1+X_2+X_3}{3}\)

Identify which estimator

1. Is unbiased?
2. Has the lowest variance?