Interval Estimation for parameter of Normal distribution/ Pivotal Quantity/

<Weakness of Point Estimation> 

Some topics that I have written so far are the ways of point estimation.

To summarize, Unbiased estimator, Minimum variance unbiased estimator, Maximum likelihood method and Method of moments are the ways of point estimation.  

However, we cannot use those methods for 'accuracy'. That is, we cannot estimate how far or close the estimator is to the parameter. 

To complement the weakness of point estimation, we can define accuracy of estimator by using confidence Interval(Random Interval) of parameters and compute the probability that the parameter in the confidence interval. 

<Interval Estimation > 

When θ is the parameter of a distribution, we can estimate the parameter and also contains the information of statistical accuracy by using Interval Estimation. There are some core vocabularies such as "confidence Interval", "confidence coefficient", "upper limit of random interval" and "lower limit of confidence Interval" when defining confidence interval.  

When L(x1,x2,x3,...,xn) and U(x1,x2,x3,...,xn) are fixed, there are only two cases either the Interval contains parameter θ or not, and that is the weakness of point estimation I mentioned. 

However, when using confidence interval of plenteous random samples, the probability that the confidence interval contains the parameter θ is 100(1-a)%. That is, among plenteous confidence Interval which is computed by sample method with plenteous random samples, 100(1-a)% of those confidence Interval contains the parameter and 100*a%  of the confidence Interval does not contain the parameter. It's like Binomial distributions with p=1-a, success=the parameter being contained in confidence Interval. 

<The way to find confidence Interval - Pivotal Estimation>

Pivotal Estimation is a way of finding confidence Interval especially when the distribution is continuous. We can use Pivotal Estimation only if T(X1,X2,X3,...,Xn;θ), function of random samples and the parameter of continuous distribution, does not depend on the parameter θ. However, we cannot sure the pivotal estimation exists or not and unique or not. 

<Confidence Interval of parameters of N(μ,σ^2) using Pivotal Quantity>

 As Normal distribution has lots of related distributions, It is easier to find Pivotal Quantity of the distribution. So for this post, I will introduce Interval Estimation of Normal distribution's parameters using Pivotal Quantity.