Hypothesis testing (1) Basic concepts of hypothesis testing

Hypothesis testing (Test of Hypothesis) means to first put forward a hypothesis about the population or the nature of the population, and then use the information provided by the sample to test the hypothesis. to determine whether the hypothesis is true. Hypothesis tests can be divided into the following two categories:

1. Propose a hypothesis about a certain parameter of the population distribution, and use samples from the population to test whether the hypothesis is true. This type is called parametric hypothesis testing.
2. Propose a hypothesis about the nature of the population, and use samples from the population to test whether the hypothesis is true. This type is called non-parametric hypothesis testing .

Hypotheses and testing rules

The first task of hypothesis testing is to appropriately formulate hypothesis propositions. Taking parametric hypothesis testing as an example, assume that the population $X$'s distribution function $F(x;\theta ),i\in \Theta$, number $\theta$ Unknown.针对 $\theta$ General form of submission “$i\in {Th}_0$”. $H_0$Display the name of the programoriginal programorzero program< a i=4>，Immediate
$$H_0:i\in {Th}_0$$If the test result rejects the null hypothesis $H_0$, that means accepting the relevant $\theta$ 的另一个假设“$i\in {Th}_1$”. $H_1$ represents this hypothesis and is called alternative hypothesis (Alternative hypothesis), that is,
$$H_1:i\in {Th}_1$$ On the surface, reject $H_0$ means acceptance $H_1$, opposition, acceptance $H_0$ means reject $H_1$, either/or, the null hypothesis and the alternative hypothesis seem to have equal status. In fact, this is not the case. In mathematical statistics, the null hypothesis must be protected, and strong evidence is required to reject it, while the alternative hypothesis does not receive such treatment.

Proposed the null hypothesis $H_0$ 和备择假设 $H_1$ After that, it is necessary to test the pair of hypotheses, that is, to formulate a rule, which should be achieved when a sample value is obtained from a sampling$(x_1,...,x_n{)}^T$ Time, fixed reception $H_0$ Still accept $H_1$。

two types of errors

Testing the hypothesis $H_0$When , people are based on the sample values ​​obtained after sampling$(x_1,...,x_n{)}^T$ Rejection or acceptance $H_0$decision. Due to the random nature of the sample, errors may occur where the decisions made are inconsistent with the real situation. Errors fall into two categories:

1. Chapter Ⅰ错误（弃正错误）：本原唇设$H_0:i\in {Th}_0$ is true, but the sample value falls in the rejection region $W$ (therefore rejecting the null hypothesis), the error probability is $P_{\theta }(W)$
2. Second category 错误（existent伪错误）：本原聇设$H_0:i\in {Th}_0$ is not true, but the sample value falls in the acceptance region ${IN}^c$ (therefore accepting the null hypothesis), the error probability is$P_{\theta }(W^c)$

General steps for hypothesis testing

1. Properly propose the null hypothesis $H_0$ 和备择假设 $H_1$；
2. Construct test statistic $Z$，Besides $H_0$ Determined under the premise that $The\; probability\; distribution\; of\; Z$ requires $The\; distribution\; of\; Z$ does not depend on any unknown parameters;
3. Determine the deny domain. at $H_0$When is established, it is not conducive to $H_0$ Set the form of the rejection region W = { Z ∈ A } W=\{Z \in A\} IN={ Z∈A}, re-rooted definite horizontal $\alpha$ 和 $Z$ Infinite, equation P H 0 { W } ≤ α P_{H_0}\{W\} \le \alpha PH0​​{ W}≤α Confirmation rejection area $W$；
4. Carry out a sampling, and based on the obtained sample value and the rejection region determined above, $H_0$Make a decision to reject or accept.

references

[1] "Applied Mathematical Statistics", Shi Yu, Xi'an Jiaotong University Press.