Standard Estimators

• Arithmetic Mean
• Variance Sample Variance
• ECDF

You can use these estimators to estimate the Expectation , Variance and CDF of an n-fold experiment:

Estimate Expectation: ${\bar{X}}_{(n)}:\mu =\mathbb{E}\left(X_1\right),{\sigma }_n^2=\mathbb{V}\left(X_1\right)/n,\text{ if }\mathbb{E}\left(X_1^2\right)<\infty$ Estimate Variance: $S_{(n)}^2:\mu =\mathbb{V}\left(X_1\right),n{\sigma }_n^2\to \mathbb{V}\left({\left(X_1-\mathbb{E}\left(X_1\right)\right)}^2\right)\text{ as }n\to \infty ,\text{ if }\mathbb{E}\left(X_1^4\right)<\infty$ Estimate CDF: $F_n\left(t;X_1,\dots ,X_n\right):\mu =F(t),{\sigma }_n^2=F(t)(1-F(t))/n$

Their Distribution

For very large $n$ they are approximately normally distributed with the above Expectation and Variance.

$n{F}_n\left(t;X_1,\dots ,X_n\right)$ has Binomial Distribution with $n=n$ and $p=F(t)$. So at a given point $t$ we can approximate a Binomial Distribution with the ECDF times $n$.

If all datapoints come from a Normal Distribution, then the Sample Variance has Chi-Squared Distribution: $\Gamma \left(\frac{n-1}{2},\frac{1}{2}\right)=:{\chi }_{n-1}^2$ with $n-1$ Degrees of Freedom.