Definition: An Estimator or a function of statistic \small T_{n}(x_{1},x_{2},x_{3},......x_{n}) is called unbiased estimator for a function of parameter \small \theta , say \small \gamma(\theta) if;
\small E(T_{n})=\small \gamma(\theta), where \small \theta belongs to the parameter space \small \Theta
Biased: let, \small b_{\theta} is the biased then
\small b_{\theta}=E(T_{n})-\small \gamma(\theta)
Remarks:
1. If, \small E(T_{n} > \small \theta then it is called positively biased.
2. If, \small E(T_{n} < \small \theta then it is called negatively biased.
1.\small (x_{1},x_{2},x_{3},......x_{n}) is a random sample from a normal population, \small N(\mu,1), show that \small t=\frac{\sum_{i=1}^{n} x_{i}^2}{n} is an unbiased estimator of \small \mu ^2 +1.
Ans: According to the question,
\small E(x_{i})=\small \mu,
\small V(x_{i})=\small 1 for all \small i=1,2,3,.........n
\small E(x_{i}^2)=\small V(x_{i}) +\small [E(x_{i})]^2 = \small 1+\mu^2
\small E(t)=\small E(\frac{\sum_{i=1}^{n} x_{i}^2}{n}) = \small 1+\mu^2
hence t is an unbiased estimator for \small 1+\mu^2 .
2. If T is unbiased estimator \small \theta then show that \small T^2 is biased estimator for \small \theta ^2.
Ans: According to the question,
\small E(T)= \theta and,
\small V(T)= E(T^2) - [E(T)]^2
or, \small E(T^2)= \theta ^2+ V(T)
or, \small E(T^2) \neq \small \theta ^2
\small T^2 is a biased estimator for \small \theta ^2
3. Show that \frac{\sum x_{i}(\sum x_{i}-1)}{n(n-1)} is an unbiased estimator for \small \theta ^2 for the sample \small x_{1}, x_{2},..... x_{n} drawn on X which takes the values 0 or 1 with respective probabilities \small \theta and \small (1-\theta).
Ans: T=\small \sum_{i=1}^{n}x_{i}\simB(n,\theta)
\small E(T)=\small \theta
\small V(T)=\small n \theta (1 -\theta)
So,
E[\frac{\sum x_{i}(\sum x_{i}-1)}{n(n-1)}]=E\small \frac{T(T-1)}{n(n-1)}
= \small \frac{ E(T^2)-E(T)}{n(n-1)}
=\small \frac{ Var(t)+{E(T)}^2-E(T)}{n(n-1)}
=\small \frac{n\theta (1-\theta)+n^2\theta62-n\theta}{n(n-1)}
=\small \theta ^2
So,[\frac{\sum x_{i}(\sum x_{i}-1)}{n(n-1)}]is an unbiased estimator for \theta^2
4.Suppose \small X and \small Y are independent random variable with the same unknown mean \small \mu. Both \small X and \small Y have same variances. let \small T= aX+bY be an estimator of \small \mu.
i. Show that \small T is an unbiased estimator of \small \mu if a+b=1.
[Hint: E(T)=\small a\mu+b\mu=\small \mu]
ii. Find the var(T)=?(when \small a=\frac{1}{3},b=\frac{2}{3})
[Hint: Var(T)=\small a^2 Var(X)+ b^2 Var(Y), Var(X)=36=Var(Y)]
5.Examine the unbiasednes of following estimates.
i. \small s_{1}^2=\frac{\sum_{i=1}^{n} (x_{i} - \bar x)}{n}
ii. \small s_{2}^2=\frac{\sum_{i=1}^{n} (x_{i} - \mu)}{n}
[Hint(i)]
\small E(s_{1}^2)=\small \frac{n-1}{n} \sigma ^2
[Hint(ii)]