⑧ Statistics
§1. Sampling¶
Calibration¶
(Sample reweighting)
Methods
- Post-stratification
- Raking
- Entropy-balancing
§2. Statistics¶
Let \(X^n = (X_i)_{i=1}^n\) denote an i.i.d. random sample of size \(n\).
Sample mean (an unbiased estimator of true mean):
\[\begin{equation}
\overline{X^n} = \frac{1}{n}\sum_{i=1}^n X_i
\end{equation}\]
Sample variance (an unbiased estimator of true variance):
\[\begin{equation}
\widehat{Var}(X^n) = \frac{1}{n-1}\sum_{i=1}^n (X_i - \overline{X^n })^2
\end{equation}\]
Sample standard deviation: \begin{equation} \widehat{SD}(X^n) = \sqrt{\frac{1}{n-1}\sum_{i=1}^n (X_i - \overline{X^n })^2} \end{equation}
§3. Parameter estimation¶
§4. Hypothesis testing¶
- Define a hypothesis.
- Select a test statistic \(T\).
- Derive the distribution of the test statistic \(T\) under the null hypothesis (e.g., a t distribution with known degrees of freedom; a normal distribution with know mean and variance).
- Select a significance level \(\alpha\) (e.g., 5%).
- Compute from the observations the observed value \(t\) of the test statistic T.
- Decide to either reject the null hypothesis in favor of the alternative or not reject it.
Null and alternative hypotheses¶
\(H_0\) and \(H_1\)
Significance levels¶
TBD
P-value¶
Type I and type II errors¶
False positive and false negative.
Multiple hypothesis testing¶
§5. Order statistics¶
TBD
References¶
TBD