- max笔记
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四要素:
基本过程:
Note 1:关于 和
This is because hypothesis tests are designed to avoid rejecting when it is true. Therefore when the test rejects , one can be quite sure that is false. 这里涉及到下面要说的“假设检验中的两类错误”。
Note 2:关于统计量
Note 3:关于拒绝域
Type I error 和 Type II error 的关系:
We can always reduce the type I error by making the rejection region smaller. This will typically at the expense of larger type II error.
In practice,we want to have powerful tests with a given type I error.
The P-value is the smallest for which the given observed data (once you have done the random experiment) suggests rejection of
Smaller P-value indicates rejection of the null hypothesis.
are independently and identically distributed, with and known. Then
Note: If the variance (总体方差) is unknown, you can replace it by (样本方差), since is large.
小样本情况下,上述CLT中的正态分布可以用 分布近似,即
, then
过程同2.1.1 大样本均值检验
(1) for , the RR is
(2) for , the RR is
(3) for , the RR is
or
, then we have
(1) The likelihood of is
(2) Suppose ,
where are some sets of possible parameter values and .
Define generalized likelihood ratio as
where is the dimension of parameter space and is the dimension of parameter space
Note: 计算 时,涉及到 Maximum Likelihood Estimator.
Suppose each individual"s category is a multinomial draw with probability .
Let be the number of observed individuals in each category. Then
Let be the simplex, i.e. .
The maximum likelihood estimator (MLE) over all is:
vs
Under and using MLE, we can get the expected number for each category as . Then
Note: While we could apply a likelihood ratio test here, Pearson"s test has a bit more power.
检验两个分类变量是否相互独立。
Suppose we have observed an contingency table.
row and column variables are independent.
row and column variables are dependent.
Under we have following contingency table:
The MLEs for are
Then we can get expected number of individuals for each category.