Null hypothesis H_0. New alternatives emerging from the previous hypothesis are alternative hypothesis H_1.
Testing to check whether H_0 should be rejected or failed to reject.
Distance between the sample data we have collected and the null hypothesis.
Set of all values of test statistic for which H_0 would be rejected. If the test statistic falls into RR then H_0 will be rejected.
When an erroneous conclusion is reached about the population parameter.
To reject a true null hypothesis
\alpha = P(\text{reject } H_0|H_0 \text{ is true})
To fail to reject a false null hypothesis
\beta = P(\text{accept } H_0|H_0 \text{ is false})
\text{power} = 1 - \beta = P(\text{reject } H_0|H_0 \text{ is false})
Probability of obtaining a test statistic at least as extreme as the observed one (given H_0 is true)
For testing H_0: \theta = \theta_0 versus a simple alternative hypothesis H_1: \theta = \theta_1, let c be a positive fixed number. Then:
R^* = \{(x_1, \cdots, x_n) : \frac{f(x_1, \cdots, x_n; \theta_1)}{f(x_1, \cdots, x_n; \theta_0)} \ge c \}
is the rejection region.
\alpha^* = P((X_1, \cdots, X_n) \in R^* | H_0)
\beta^* = P((X_1, \cdots, X_n) \notin R^* | H_1)
Are type 1 and 2 errors respectively.
Let H_0: \theta \in \Omega_0 vs H_1: \theta \in \Omega_1. The power function is defined as
\pi(\theta') = P((X_1, \cdots, X_n) \in R|\theta = \theta')
UMP test is one for which \pi(\theta') is maximized for any \theta' \in \Omega_1 subject to \pi(\theta') \le \alpha for any \theta' \in \Omega_0
TODO
We wish to test H_0 be \tilde \mu = 0 (median or mean) without normality assumptions.
Let R_1, \cdots, R_n be ranks of |X_1|, \cdots, |X_n| (ascending order), then
W_+ = \sum_{X_i:sign(X_i)=1} R_i
Is the sum of all positive signed-ranks.
For H_0 \tilde \mu = \mu_0 then just subtract \mu_0 from each observation and consider the Wilcoxon test for the modified sample.
Two independent samples, we wish to test H_0 \mu_X = \mu_Y. We order the pooled sample and assign ranks R_1, \cdots, R_{m+n}. Set
W = \text{Sum of } R_i \text{ associated with } X-\text{sample}