Using a point estimator $\hat{\theta }$ to estimate $\theta$

Suppose a random sample $X_1,\dots ,X_n$ is collected from a population and it is assumed that $X_i\stackrel{iid}{\sim }f(x;\theta )$
Let the estimator of $\theta$ be $\hat{\theta }=\hat{\theta }(X_1,\dots ,X_n)$ be a function of $X_i^{\prime }$s meant to estimate $\theta$

Finding point estimators

Evaluation

Properties of a good estimator:

1. Unbiasedness: $E(\hat{\theta })=E(\hat{\theta }(X_1,\dots ,X_n))=\theta$
2. Efficiency: $var(\hat{\theta }(X_1,\dots ,X_n))$ is as small as possible (more efficient per sample size)
3. Consistency: $\hat{\theta }(X_1,\dots ,X_n)\to \theta$ as $n\to \mathrm{\infty }$
4. Sufficiency: Does $\hat{\theta }$ utilize all the information contained in the data to estimate $\theta$?
5. Likeliness: $\hat{\theta }(X_1,\dots ,X_n)$ is the most likely value of $\theta$ give the data ($\hat{\theta }$ is the choice with highest probability for the data)
6. Robustness: $\hat{\theta }(X_1,\dots ,X_n)$ has a sampling distribution that is not too adversely affected by violations of assumptions made in the model/analysis

An estimator $\hat{\theta }(X_1,\dots ,X_n)$ is asymptotically unbiased for $\theta$ if

$E(\hat{\theta })\to \theta$ as $n\to \mathrm{\infty }$

If $\hat{\theta }$ is unbiased for $\theta$, and $\underset{n\to \mathrm{\infty }}{lim}\frac{Var(\hat{\theta })}{CRLB}=1$, then $\hat{\theta }$ is asymptotically efficient