Lernkarten

1. The sample mean is an unbiased estimator of the expected value
2. The sample variance  \(\widehat{Var}(x)=\frac{1}{N}\sum\limits^N_{i=1}(x_i-\bar{x})^2\) is an not unbiased, but asymptotically unbiased estimator
3. However, the sample variance \(\widehat{Var}(x)=\frac{1}{N-1}\sum\limits^N_{i=1}(x_i-\bar{x})^2\) is an unbiased estimator. 

Given an estimate \(\widehat{\Theta}\) of \(\Theta\). An interval \((\widehat{\Theta}_L,\widehat{\Theta}_U)\) around \(\widehat{\Theta}\) is named a \((1-\alpha)\) confidence interval if 

\(P(\Theta\in (\widehat{\Theta}_L,\widehat{\Theta}_U))=1-\alpha\)

A 95% confidence interval covers the true value in 95% of the cases. 

t-Distribution: 

\(f_X(x;\nu)=c(\nu)\left(1+\frac{x^2}{\nu}\right)^{-\frac{\nu+1}{2}}\)

with a constant \(c(\nu)\). \(\nu\in\mathbb{N}\) is called the degree of freedom. For \(\nu\rightarrow \infty\) it converges to the \(\mathcal{N}(0,1)\) distribution. 

The t-Distribution is used to estimate the mean of a normally distributed population when the sample size is small and population standard deviation is unknown. 

The empirical mean \(\bar{x}\) has a distribution \(\mathcal{N}(\mu, \frac{\sigma}{\sqrt{n}})\) while

\(Z=\sqrt{n}(\bar{x}-\mu)/\sigma\)

has distribution \(\mathcal{N}(0,1)\) .

The value z is such that \(P(-z\leq Z\leq z)=1-\alpha\)

This yields: \(P(\bar{x}-z\frac{\sigma}{\sqrt{n}}\leq\mu\leq\bar{x}+z\frac{\sigma}{\sqrt{n}})=1-\alpha\)

The confidence interval can be easily found now. 

Significance tests aim at verification of a hypothesis based on statistical data:

1. Parametric tests consider hypothesis regarding parameters of the distribution 
2. Non-parametric tests consider hypotheses not involving paramters (e.g. distributaions are the same or different)

1. Formulate Null hypothesis and an alternative hypothesis 
2. Choose significance level \(\alpha\)
3. Choose significance test and test statistic; clarify assumptions to be made 
4. Calculate Null distribution and critical value 
5. Calculate test statistic and/or p-value 
6. Decide whether Null hypothesis is rejected or not 

Significance level and critical value are defined such that:

\(P(|T|\geq t_{crit})\equiv\alpha\)

i.e., the probability that T falls in the rejection region (Q) although the \(H_0\) (null hypothesis) is true (small probability).

Then the Null hypothesis is rejected in case \(p_{obs}\leq\alpha \) or \(|t_{obs}|\geq t_{crit}\)

error of the first kind or \(\alpha\)-error: Rection of the Null hypothesis \(H_0\) although it is true

Probability of this error is \(P(H_0 rejected\; | \;H_0 true)=\alpha\)

error of the second kind or \(\beta \)-error: No rejection of the Null hypothesis although it is wrong 

Probability of this error is \(P(H_0\;not\;rejected\;|\;H_0\;false)=\beta \)

The reduction ofthe one error leads to increase of the other, unless we can increase sample size!