Mathematical Statistical Hypothesis Testing Definition
Definition
- A proposition about a parameter is called a Hypothesis.
- The problem of accepting hypothesis $H_{0}$ as true based on a given sample, or rejecting hypothesis $H_{0}$ and adopting hypothesis $H_{1}$ is called a Hypothesis Test. In hypothesis testing, the complementary hypotheses $H_{0}$, $H_{1}$ are called the Null Hypothesis and the Alternative Hypothesis, respectively.
- The subset $R \subset \Omega$ of the sample space $\Omega$ that leads to the rejection of the null hypothesis $H_{0}$ is called the Rejection Region.
Explanation
Not only for students majoring in statistics, but one also encounters explanations about hypothesis testing even at an introductory level of statistics. Many may find it sufficient, but the definitions above talk about hypothesis testing in as mathematically rigorous, unambiguous, and exact a manner as possible.
The explanations below are written assuming the reader is somewhat familiar with the concept of hypothesis testing. Let’s grasp the concept mathematically.
Hypothesis
According to the definition, a hypothesis is not just any ‘word’ but a proposition. The mention that it is a proposition about a parameter is key. For instance, tests like ’normality tests,’ which appear to be about the distribution itself rather than parameters, are ultimately based on parameters upon closer inspection. For example, the Jarque-Bera test is a test of normality, which, in fact, conducts hypothesis testing through skewness and kurtosis.
Hypothesis Testing
The word ‘adopting’ is underlined, which is a word of caution. As you know, while almost all textbooks use both the expressions Reject and Accept, most professors caution against the use of the term ‘accept’. Accepting the alternative hypothesis means rather than truly accepting the alternative hypothesis as true, it’s more about rejecting the null hypothesis, and accepting the null hypothesis is not about actively ‘adopting’ it but rather about not being able to reject it.
When defining the null and alternative hypotheses, the term complementary is used, and this emphasizes that $H_{0}$ and $H_{1}$ are not necessarily logical negations of each other. In hypothesis testing, showing that the null hypothesis is not true doesn’t automatically mean that the alternative hypothesis is true. More pragmatically, an alternative hypothesis that cannot coexist with the null hypothesis is sufficient. For example, hypothesis testing like $$ H_{0} : \theta = 0 \\ H_{1} : \theta < 0 $$ is fine, but in cases like $$ H_{0} : \theta \in [-1,0] \\ H_{1} : \theta \in [0,+1] $$ where $\theta = 0$, there’s a problem because both the null and alternative hypotheses could be true.
Rejection Region
According to the definition, the rejection region is an event. If hypothesis testing is considered a single trial, then the probability of $H_{0}$ being rejected is the same as the probability of the rejection event occurring. If this probability is quite low, for example, lower than $\alpha = 0.05$, yet it occurs, this is not an ordinary event but something noteworthy. In such storytelling, concepts like the significance level (p-value) come to mind naturally.
See Also
- A simple definition of hypothesis testing: Introduces definitions that are moderately easy to accept rather than being strictly rigid.
- How to determine null and alternative hypotheses: Explains any problems with those definitions.
- A rigorous definition of hypothesis testing: Introduces relatively strict mathematical statistical definitions of hypothesis testing.
- A simple definition of the rejection region