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
- Is unbiased?
- Has the lowest variance?
