[Probability Theory] Final Review Notes: Hypothesis Testing

1. The basic concept of hypothesis testing

Hypothesis testing : Propose a hypothesis about the population → test the hypothesis with a sample
Accept the hypothesis: think the hypothesis is correct
Reject the hypothesis: think the hypothesis is wrong

Hypothesis testing is divided into parametric assumptions , distribution assumptions

1. Basic principles of hypothesis testing

Practical inference principle : It is almost impossible for a small probability event to occur in an experiment.
The idea of ​​hypothesis testing: Construct a suitable test hypothesis $H_0$The statistic ( test statistic ), if the hypothesis $H_0$is established, the test statistic satisfies a condition (if $H_0$If it is established, the probability of meeting the condition is very high), now see if the sample satisfies this condition, if it is satisfied, accept $H_0$, if not satisfied, reject $H_0$(This indicates that a small probability event has occurred, and the null hypothesis is not established, similar to rejection reasoning).

2. Two types of errors

deny domain $W$ : When the observed value of the test statistic falls within$When\; W$ , reject$H_0$
Receptive field $\overline{W}$: When the observed value of the test statistic falls on $\overline{W}$, accept $H_0$
Threshold : the critical point of rejecting domain and accepting domain

Two types of errors:
Type I errors : $H_0$True but rejected
Type II error : $H_0$false but accepted

Significance level $\alpha$ : Probability of making Type I error,$P\{\text{reject}{H}_0|H_0\text{true}\}=\alpha$ , reflecting the rejection of$H_0$The persuasiveness
$\beta$ : the probability of committing a type II error,$P\{\text{accept}{H}_0|H_0\text{not true}\}=\beta$
at sample size$When\; n$ is certain,$\alpha$ decreases,$\beta$ will increase

Significance test : Control the probability of making a Type I error not to exceed a value (significant level), regardless of the Type II error. The
significance level is $\alpha$ test method: the probability of making a Type I error does not exceed$\alpha$ , that is,$P\{\text{reject}{H}_0|H_0\text{true}\}\le \alpha$ is α \alpha
at all significance levels$In\; the\; test\; method\; of\; \alpha$ , the probability of making a Type II error$The\; test\; with\; the\; smallest\; beta$ is the best test

3. General steps of hypothesis testing

(1) Fully consider and use the known background knowledge to put forward the null hypothesis $H_0$Alternative Hypothesis $H_1$。

$H_1$is $H_0$The opposite of is called the alternative hypothesis or alternative hypothesis. $H_0$Generally, it is "hypothesis to be protected" or "hypothesis to maintain the status quo", false rejection hypothesis $H_0$Than incorrectly reject the hypothesis $H_1$lead to more serious consequences. In practice, $H_0$It should not be easily denied, and if it is denied, there must be sufficient reasons.

(2) Determine the test statistic $Z$ , and at$H_0$Export ZZ under the premise of establishment$The\; probability\; distribution\; of\; Z$ , requiring$The\; distribution\; of\; Z$ does not depend on any unknown parameters.

ZZ here$Z$ is similar to the pivot amount in parameter estimation, choose$The\; reason\; for\; Z$ is to eliminate the interference of unknown parameters and use a certain distribution to find the rejection domain.

(3) Determine the deny domain. First determine the form of the rejection domain based on intuitive analysis, and then according to the given level $a$和$The\; distribution\; of\; Z$ , by$P\{\text{reject}{H}_0|H_0\text{true}\}=\alpha$ determines the critical value of the rejection domain, thereby determining the rejection domain.

Determining the rejection region is similar to the process of determining confidence intervals in parameter estimation.

(4) Make a specific sampling, according to the obtained sample value and the rejection domain pair $H_0$Make a decision to reject or accept.

If ZZObservations of$Z$$W$ , then reject the null hypothesis$H_0$, accepting the alternative hypothesis $H_1$; if it falls into the receptive field $\overline{W}$, then accept the null hypothesis $H_0$, rejecting the alternative hypothesis $H_1$。
$$\begin{gathered} Z\in \overline{W}⟶H_0\sout{H_1} \\ Z\in W⟶H_1\sout{H_0} \end{gathered}$$

4. $p$ -value

$p$ -value: The minimum significance level for rejecting the null hypothesis using the sample value is called$p$ -value.
$$\begin{gathered} a<p⟶H_0\sout{H_1}=Z\text{falls within the receptive field} \\ a\ge p⟶H_1\sout{H_0}=Z\text{falls within the rejection domain} \end{gathered}$$$The\; p$ -value is the minimum significance level for "rejection", that is, if$\alpha$ greater than or equal to$p$ , then reject it. Intuitively,$The\; smaller\; \alpha$ is, the less likely it is to reject the null hypothesis$H_0$, reject $H_0$The more convincing; $The\; smaller\; \alpha$ is, the "smaller" the rejection domain will be.

At fixed $In\; the\; case\; of\; \alpha$ ,$The\; larger\; p$ is, the easier it is to accept$H_0$(referred to as " $The\; larger\; the\; p$ , the better").

I feel that$\alpha$ is a pair rejecting$H_0$The "tolerance" of $H_0$"degree of doubt"; $The\; smaller\; \alpha$ is, for$H_0$The smaller the degree of suspicion, the easier it is to accept $H_0$。$The\; p-$ values ​​are for$H_0$The minimum "suspicious degree" of , if $a<p$ , then this$\alpha$ to$p$ to$H_0$"more confident", so accept $H_0$. To sum up, it is: $The\; smaller\; \alpha$ is, for$H_0$The more confident you are.

The method of judging by rejecting the domain is called the critical value method, using $The\; p-$ value is called$p$ -value method.

2. Hypothesis testing of normal population parameters

The test statistic follows a normal distribution $\to u$ test method
The test statistic obeys$h^2$ distribution$\to h^2$ test method
The test statistic obeys$t$ distribution$\to t$ test method
The test statistic obeys$F$ distribution$\to F$ test

For the case of a single population, we set $X\text{~}N(\mu ,p^2)$; for the case of two populations, we assume$X\text{~}N(m_1,p_1^2)$，$Y\text{~}N(m_2,p_2^2)$。$X$ has a sample size of$n$ , the sample variance is$S_X^2$；$The\; sample\; size\; of\; Y$ is$m$ , the sample variance is$S_Y^2$。

The following is the case of a single population ( $X\text{~}N(\mu ,p^2)$）。

$p^2\text{known, check}\mu \text{and}{\mu }_0\text{Relationship}$

Default settings : Note: $$\begin{array}{rlr}& H_0:m=m_0& \\ & H_1:m\ne m_0& \end{array}$$Recall that $\alpha$ is the probability of making the first type of error, that is, when$H_0$When established $P\{\text{reject}{H}_0\}=\alpha$ . Because only the hypothesis$H_0$Established we can continue to analyze, so we must make $H_0$Established first, that is, $m=m_0$Variable value $X\text{~}N(m_0,p^2)$, we givethe test statistic$U=\frac{\sqrt{n}\left(\overline{X}-m_0\right)}{p}\text{~}N(0,1)$。$H_0$When was it established? is the sample mean $\overline{X}$与$m_0$It is established when it is relatively close, and it is rejected when it is too outrageous. Now we have assumed $H_0$Established, as long as UUThe probability that the observed value of$U$$\alpha$ will do. This is similar to interval estimation, becauseP { U ≥ u α / 2 or U ≤ − u α / 2 } = α P\{U\ge u_{\alpha/2}\text{or}U\le-u_{ \alpha/2}\}=\alphaP{ U≥ua / 2or U≤−ua / 2}=α , so when rejecting it is$|U|\ge u_{a/}$when. For a certain sample, we calculate $U$ value, then judge$|U|$和$u_{a/}$relationship is fine. If $|U|\ge u_{a/}$, then say that at $H_0$Under the established condition, a probability is only $The\; event\; of\; \alpha$ happened, which means that$H_0$unlikely to hold, so we reject $H_0$. Attention, $U$ measures$\overline{X}$与$The\; difference\; of\; \mu$ , if the difference is too large, reject$H_0$；$U$ divided by$The\; purpose\; of\; \sigma$ is to remove the dimension and standardize.

Remove the default settings : $$\begin{array}{rlr}& H_0:m=m_0& \\ & H_1:m\ge m_0& \end{array}$$Because the mean $\mu$ will not be less than$m_0$，soU $U\le -u_{a/}$The rejection field of does not exist, only $U\ge \text{A reject field for a value}$ . What is this value? don't forget$The\; probability\; that\; U$ falls into the rejection domain is$\alpha$ , so obviously this value is$u_{}$, rejection is $U\ge u_{}$。把$H_0$Type $H_0:m\le m_0$, the deny domain is still the same.

$p^2\text{Unknown, check}\mu \text{and}{\mu }_0\text{Relationship}$

Test statistic $T=\frac{\sqrt{n}\left(\overline{X}-m_0\right)}{S}\text{~}t(n-1)$
Rejection field and$p^2$ is similar when known, that is,$u_{}$Replaced by $t_{}(n-1)$ That's all.

$\mu \text{is known, check}{\sigma }^2\text{与}{p}_0^2\text{Relationship}$

Infinite value $h^2=\frac{\sum _{i=1}^n{\left(X_i-m\right)}^2}{p_0^2}\text{~}{h}^2(n)$

$\mu \text{unknown, test}{\sigma }^2\text{and}{\mu }_0\text{Relationship}$

2 = ∑ i = 1 n ( X i − X ‾ ) 2 σ 0 2 = ( n − 1 ) S 2 σ 0 2 ~ χ 2 ( n − 1 ) \newcommand{\td}{\ , $h^2=\frac{\sum _{i=1}^n{\left(X_i-\overline{X}\right)}^2}{p_0^2}=\frac{(n-1)S^2}{p_0^2}\text{~}{h}^2(n-1)$

For example $H_0:p^2=p_0^2,H_1:p^2\ne m_0^2$Inequality $\left\{x^2\le h_{1-a/2}^2(n-1)\right\}\bigcup \left\{x^2\ge h_{a/2}^2(n-1)\right\}$ .

One-sided hypothesis test$H_0:p^2=p_0^2,H_1:p^2>p_0^2$The rejection region of is $h^2\ge h_a^2(n-1)$ .
One-sided hypothesis test$H_0:p^2=p_0^2,H_1:p^2<p_0^2$The rejection region of is $h^2\le h_{1-a}^2(n-1)$。

$p_1^2,p_2^2\text{Known, check}{\mu }_1-m_2\text{Relationship}\text{with}\Delta \mu$

设计管理量$U=\frac{\left(\overline{X}-\overline{Y}\right)-D}{\sqrt{\frac{p_1^2}{n}+\frac{p_2^2}{m}}}\text{~}N(0,1)$

$p_1^2=p_2^2\text{unknown, test}{\mu }_1-m_2\text{Relationship}\text{with}\Delta \mu$

检验统计量$T=\frac{\left(\overline{X}-\overline{Y}\right)-D}{S_W\sqrt{\frac{1}{n}+\frac{1}{m}}}\text{~}t(n+m-2)$，其中$S_W=\sqrt{\frac{(n-1)S_X^2+(m-1)S_Y^2}{n+m-2}}$

$m_1,m_2\text{known, tested}\frac{p_1^2}{p_2^2}\text{relationship}\text{with}c$

检验统计量$F=\frac{\sum _{i=1}^n\frac{{(X_i-m_1)}^2}{n}}{\sum _{j=1}^m\frac{{(Y_j-m_2)}^2}{m}}/c=\frac{1}{c}\frac{m\sum _{i=1}^n{(X_i-m_1)}^2}{n\sum _{j=1}^m{(Y_j-m_2)}^2}\text{~}F(n,m)$

$m_1,m_2\text{unknown, test}\frac{p_1^2}{p_2^2}\text{relationship}\text{with}c$

Test statistic $F=\frac{1}{c}\frac{S_X^2}{S_Y^2}\text{~}F(n-1,m-1)$

注意$F_{1-}(n,m)=\frac{1}{F_{}(m,n)}$(There are three things to do: ① take the inverse, ② $1-\alpha$ change$\alpha$ , ③ Exchange$n,$order of$m$ ) . The rejection domain and$h^2$ tests are similar.

$\text{Memorization of Test Statistics}$

1. Single variable test mean : In order to standardize the distribution, we need to exclude the influence of three factors: mean ( $\mu$ ), sample size ($n$ ), standard deviation ($\sigma$ or$S$ ), where excluding the standard deviation also makes the data dimensionless. Therefore, corresponding to the known and unknown standard deviations respectively, we have the statistic$U=\frac{\sqrt{n}(\overline{X}-m}{p}$和$T=\frac{\sqrt{n}(\overline{X}-m}{S}$, the former obeys $N(0,1)$ , the latter follows$t(n-1)$。

2. Single variable test variance : Our statistics should obey the chi-square distribution, pay attention to $h^{The\; expectation\; of\; 2}\left(n\right)$is$n$ , so we require that the expectation of our statistic should be$n$ , n − 1 n-1when the mean is unknown$n-1$。注意$E\left[\sum _{i=1}^n{\left(X_i-m\right)}^2\right]=andE\left[{\left(X_1-m\right)}^2\right]$，而$m=E\left(X_1\right)$ , so according to the definition of variance$E\left[{\left(X_1-m\right)}^2\right]=p^2$，因此$E\left[\sum _{i=1}^n{\left(X_i-m\right)}^2\right]=n{p}^2$ . At the same time we know,$E\left(S^2\right)=p^2$，即$E\left[\sum _{i=1}^n{\left(X_i-\overline{X}\right)}^2\right]=(n-1)p^2$ . Therefore, the two test statistics we use are$\frac{\sum _{i=1}^n{\left(X_i-m\right)}^2}{p^2}$和$\frac{\sum _{i=1}^n{\left(X_i-m\right)}^2}{p^2}$, subject to $n$ and$n-Chi-square\; distribution\; of\; 1$ . It can be understood as: in the test statistics we use, the dimension is square and the expectation is$n$ is subject to$h^2(n)$。

3. Two variables test mean difference : note that $\overline{X}-\overline{Y}\text{~}N\left(m_1-m_2,\frac{p_1^2}{n}+\frac{p_2^2}{m}\right)$ . Imitate the situation when a single variable is tested for the mean, and the situation where the variance is known will not be repeated here; for the situation where the variance is unknown, it is required that$p_1^2=p_2^2$, for convenience, we denote it as $p^2$。此时$D\left(\overline{X}-\overline{Y}\right)=p^2\left(\frac{1}{n}+\frac{1}{m}\right)$ , the test statistic used should be$\frac{\left(\overline{X}-\overline{Y}\right)-(m_1-m_2)}{p\sqrt{\frac{1}{n}+\frac{1}{m}}}$, but $\sigma$ is unknown, so we have to put$\sigma$ is replaced by a S_X with$S_X$和$S_Y$The estimated value indicated. How to estimate it? Let us estimate $p^{The\; quantity\; of\; 2}$ is$S_W^2$. First, this $S_W^2$The expectation must be $p^2$ ; secondly, it is divided by$p^{After\; 2}$ must obey$h^2(n+m-2)$ (requires degrees of freedom to be full). Let's think about it,$\sum _{i=1}^n{\left(X_i-\overline{X}\right)}^2+\sum _{j=1}^m{\left(Y_j-\overline{Y}\right)}^2$ what is it? It is actually$(n-1)S_X^2+(m-1)S_Y^2$,Let $\frac{(n-1)S_X^2+(m-1)S_Y^2}{p^2}\text{~}{h}^2(n+m-2)$，$E\left(\frac{(n-1)S_X^2+(m-1)S_Y^2}{n+m-2}\right)=p^2$ . So, we use$S_W=\sqrt{\frac{(n-1)S_X^2+(m-1)S_Y^2}{n+m-2}}$To replace the σ \sigma in the test statistic$\sigma$ , get$T=\frac{\left(\overline{X}-\overline{Y}\right)-(m_1-m_2)}{S_W\sqrt{\frac{1}{n}+\frac{1}{m}}}$。

4. Two variables test variance ratio : in fact, the test statistic is a $\frac{\text{Measured Variance}}{\text{required variance ratio}}$form. There is another way of understanding: we know that $h_X^2=\frac{(n-1)S_X^2}{p_1^2}\text{~}{h}^2(n-1)$ ,$h_Y^2=\frac{(m-1)S_Y^2}{p_2^2}\text{~}{h}^2(m-1)$ , according toFFWe know the properties of the$F$$\frac{h_X^2/(n-1)}{h_Y^2/(m-1)}\text{~}F(n-1,m-1)$ ,let$\frac{S_X^2/p_1^2}{S_Y^2/p_2^2}\text{~}F(n-1,m-1)$。

$\text{Hypothesis Testing on Paired Data}$

Sometimes, $X_1,X_2,\cdots ,X_n$和$Y_1,Y_2,\cdots ,Y_n$are not necessarily independent of each other, $X_i$Give $Y_i$may have a strong correlation, but different $(X_i,Y_i)$ are not related. How do we examine the mean difference in this case?

with $n$ pairs of mutually independent samples$(X_1,Y_1),(X_2,Y_2),\cdots ,(X_n,Y_n)$，令$Z_i=Y_i-X_i(i=1,2,\cdots ,n)$，显然$Z_1,Z_2,\cdots ,Z_n$Independent. Let $Z_1,Z_2,\cdots ,Z_n$is from the population $N(\mu ,p^2)$, then$\mu$ for hypothesis testing. In general$p^2$ is unknown, so the test statistic we choose is generally$T=\frac{\sqrt{n}\left(\overline{X}-m_0\right)}{S}$。

3. Distribution hypothesis testing

1. The concept of distribution fitting test

Distribution fitting test : Let $(X_1,X_2,\cdots ,X_n)$ is from the overall$The\; sample\; of\; X$ , according to the sample test hypothesis$$\begin{array}{rlr}& H_0:The\; distribution\; function\; of\; X\text{is}F\left(x\right)& \\ & H_1:\text{The distribution function of}X\; is\; notF(x& \end{array}$$Here $F\left(x\right)$ isa theoretical distribution function, which can be known, or a function whose form is known but contains unknown parameters. The distribution fitting test is to investigate the use of$F\left(x\right)$ fitXXHow good is the fit for the distribution of$X\; ?$

2. Pearson's Theorem

Pearson's theorem Let a random experiment $r$ results$A_1,A_2,\cdots ,A_r$Constitute a mutually exclusive complete event group, and their occurrence probabilities in one trial are $p_1,p_2,\cdots ,p_r$，其中$p_i>0(i=1,2,\cdots ,r)$，且$\sum _{i=1}^r{p}_i=1$。以$m_i$expressed in nnA i A_i$in\; n$ independent repeated experiments$A_i$The number of occurrences, then when $n\to \infty$ random variable$h^2=\sum _{i=1}^r\frac{{(m_i-n{p}_i)}^2}{n{p}_i}$The distribution of converges to degrees of freedom $r-$χ 2 \chi^2of$1$$h^2$ distribution.

3. $h^2$ Fitting test method

(1) Theoretical distribution $The\; case\; where\; F\left(x\right)$ is completely known

设$X_1,X_2,\cdots ,X_n$is from overall $A\; sample\; of\; X$ ,$F\left(x\right)$ is a fully known distribution function, at the significance level$Under\; \alpha$ , test the hypothesis$$\begin{array}{rlr}& H_0:The\; distribution\; function\; of\; X\text{is}F\left(x\right)& \\ & H_1:\text{The distribution function of}X\; is\; notF(x& \end{array}$$Methods as below:

1. will overall $The\; value\; range\; of\; X$ is divided into$r$ mutually disjoint subsets (intervals)$A_1,A_2,\cdots ,A_r$, then it constitutes a mutually exclusive complete event group ( $r$ is not large,$n$ is large). Note that the sample size is large, generally$n\ge 50$ ; when grouping, the number of samples contained in each interval$m_i\ge 5$。
2. at $H_0$Under the premise that holds, calculate the event $A_i$The frequency of $P(A_i)=p_i$, then event $A_i$The frequency of occurrence is $n{p}_i$。
3. Count the sample values $x_1,x_2,\cdots ,x_n$middle, incident $A_i$The actual frequency of occurrence $m_i$。
4. At the significance level $Under\; \alpha$ , consider the statistic$h^2=\sum _{i=1}^r\frac{{(m_i-n{p}_i)}^2}{n{p}_i}$. If the data does obey F ( x ) F(x)The distribution specified by$F$$($$x$$)$$h^2$ should be as small as possible, so its rejection domain should be$h^2$ is greater than or equal to a certain number. Therefore, the rejection region is$h^2\ge h_a^2(r-1)$。

(2) The theoretical distribution contains unknown parameters

Functional function $F(x;i_1,i_2,\cdots ,i_l)$ ,includeθ$i_1,i_2,\cdots ,i_l$is an unknown parameter. We'll do two things before continuing with the process when the theoretical distribution is known:

1. According to samples $X_1,X_2,\cdots ,X_n$Rangeθ $i_1,i_2,\cdots ,i_l$The maximum likelihood estimates of ${\hat{i}}_1,{\hat{i}}_2,\cdots ,{\hat{i}}_l$,the function $F(x;{\hat{i}}_1,{\hat{i}}_2,\cdots ,{\hat{i}}_l)$ is treated as a fully known distribution function;
2. Don't forget to put $h^2$ degrees of freedom is changed to$r-l-1$ ! ! ! This is a modification of Pearson's theorem. The rejection region is$h^2\ge h_a^2(r-l-1)$ . It can be understood that we need to start fromnnDraw ll$out\; of\; n$ samples$l$ to make up for the unknown parameters, which caused the loss of degrees of freedom, so the final degrees of freedom become$r-l-1$。