Approximate Interval Estimation/ Approximate confidence Interval/ Slutsky's theorem/

<Approximate Interval Estimation>

We have covered Interval Estimation of Normal distribution. 

[Statistics] - Interval Estimation for parameter of Normal distribution/ Pivotal Quantity/

And this post is for other distributions than Normal distribution. We need some properties of approximation for the Interval Estimation for those distributions.

The typical property needed is central limit theorem, and by using this we can approximately estimate Interval of parameters. 

There are two cases for the Interval estimation for other distributions than Normal distribution. 

The first case is when variance of the distribution is known and it might be quite familiar with you. 

And the second case is when variance of the distribution is not known we use Slutsky's theorem here. 

<Confidence Interval of μ when σ is known>  

As the variance of distribution is known, we can approximate the confidence Interval of mean of population by only using central limit theorem. 

<Confidence Interval of μ when σ is unknown >

For the case that the variance of distribution is known, we can approximate the confidence Interval of mean of population by combining some theorems such as "Central limit theorem", "Slutsky's Theorem", "Convergence of Sn to σ."

Firstly, I separately explain what those properties mean. 

In summary, we proved that Sample Variance converges to the Standard deviation of the population in probability and shows what is central limit theorem. Combining two properties, by Slutsky's theorem, we can approximate Interval Estimation for the parameter of distributions when the population variance is not known. 

<Confidence Interval of μ(X)-μ(Y) when σ(X), σ(Y) is unknown >

<Confidence Interval of μ(X)-μ(Y) when σ(X), σ(Y) is unknown>

< The appropriate Sample Size to meet Limit of error >

If the condition for the size of confidence Interval is given, we can find appropriate sample size to meet the condition. 

However, when we cannot know the sigma which is needed for lower bound of n, we can find the maximum n by assume σ as the possibly largest number. Then n computed by the method should be bigger than any possible n.