Standard Estimators

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

Estimate Expectation:

Xˉ(n):μ=E(X1),σn2=V(X1)/n, if E(X12)<∞\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:μ=V(X1),nσn2→V((X1−E(X1))2) as n→∞, if E(X14)<∞S_{(n)}^{2}: \mu=\mathbb{V}\left(X_{1}\right), n \sigma_{n}^{2} \rightarrow \mathbb{V}\left(\left(X_{1}-\mathbb{E}\left(X_{1}\right)\right)^{2}\right) \text { as } n \rightarrow \infty, \text { if } \mathbb{E}\left(X_{1}^{4}\right)<\infty

Estimate CDF:

Fn(t;X1,…,Xn):μ=F(t),σn2=F(t)(1−F(t))/nF_{n}\left(t ; X_{1}, \ldots, X_{n}\right): \mu=F(t), \quad \sigma_{n}^{2}=F(t)(1-F(t)) / n

Their Distribution

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

nFn(t;X1,…,Xn)n F_n\left(t ; X_1, \ldots, X_n\right) has Binomial Distribution with n=nn=n and p=F(t)p=F(t). So at a given point tt we can approximate a Binomial Distribution with the ECDF times nn.

If all datapoints come from a Normal Distribution, then the Sample Variance has Chi-Squared Distribution:

Γ(n−12,12)=:χn−12\Gamma\left(\frac{n-1}{2}, \frac{1}{2}\right)=: \chi_{n-1}^2

with n−1n-1 Degrees of Freedom.