Notes: HKUST notes on hypothesis testing
A major part of statistical inference is hypothesis testing, a decision procedure that gives probabilistic conclusions on underlying population from limited sample.
A working assumption is a statement used for pragmatic necessity to construct theoritical arguments. It should be falsifiable and frequently examined.
You start from an assumption that you believe is at least partially true, and the results will tell you that you used a good assumption, or that your assumption needs to be modified, or that your assumption is flat wrong.
Working assumptions cannot be logically false, nor logically true.
In contrast, a core assumption, like a principle, is a proposition that is either logically true or accepted as fundamental for pragmatic use. Core assumptions are thus rarely examined.
Hypothesis is a statement about a population parameter. We referred to the hypothesis we test on as the Null hypothesis \( H_0 \). If the null gets rejected, we accept the alternative hypothesis \( H_1 \) instead.
Hypothesis test is a rule that specifies for every sample point whether to reject or accept the null. A hypothesis test divides the sample space into a rejection region and an acceptance region.
A typical hypothesis testing procedure:
Types of hypothesis tests:
passive acceptance
statistical significant difference rejection
Def: likelihood ratio statistic
Def: likelihood ratio test
Def: Nuisance parameter
Def: Type I Error, Type II Error
Def: power function
Def: size
Def: level
Def: unbiased
Def: uniformly most powerful (UMP) test
Thm: (Neyman-Pearson)
Def: monotone likelihood ratio (MLR)
Thm: (Karlin-Rubin)