# Basic Properties of Confidence Intervals

Posted by Beetle B. on Sun 16 July 2017

$$\newcommand{\Cov}{\mathrm{Cov}}$$ $$\newcommand{\Corr}{\mathrm{Corr}}$$ $$\newcommand{\Sample}{X_{1},\dots,X_{n}}$$

Assume you have a normal distribution with unknown $$\mu$$ but known $$\sigma$$ (highly implausible). A sample is collected. The 95% confidence interval for $$\mu$$ is $$\left(\bar{x}-1.96\frac{\sigma}{\sqrt{n}},\bar{x}+1.96\frac{\sigma}{\sqrt{n}}\right)$$

This is the classical confidence interval (CI) (i.e. the frequentist).

The $$100(1-\alpha)$$ % CI for the normal is $$\left(\bar{x}-z_{\alpha/2}\frac{\sigma}{\sqrt{n}},\bar{x}+z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\right)$$

## Deriving a Confidence Interval

Let $$\Sample$$ be the sample, and $$\theta$$ the quantity to be estimated. If you can find a random variable $$Y$$ that satisfies:

1. The random variable depends on $$\Sample$$ and $$\theta$$.
2. The pdf of the random variable does not depend on $$\theta$$ or any unknown parameters

Then you can find $$a,b$$ such that $$P(a<Y<b)=1-\alpha$$ (for any $$\alpha$$). Manipulate the expression to get $$P(A<\theta<B)=1-\alpha$$. Then the CI for $$100(1-\alpha)$$ is $$(A,B)$$.

### Normal

Use $$Z=\frac{X-\mu}{\sigma/\sqrt{n}}$$

### Exponential

Use $$Y=2\lambda\sum X_{i}$$ (you get the chi-squared distribution).

## Bootstrap Confidence Intervals

You can use bootstrapping to estimate the confidence interval.

As an example, let $$B=1000$$. Calculate $$\hat{\theta_{i}^{\ast}}$$ for all of them, as well as the mean of all of them. Note the differences $$\hat{\theta^{\ast}}-\hat{\theta_{i}^{\ast}}$$. Then look at the 25th largest and smallest values and you have your interval.