1. 假设检验基本概念
1. 假设检验
对总体的分布类型或分布中的某些未知参数作出假设, 然后抽取样本并选择一个合适的检验统计量,利用检验统计量的观察值和预先给定的误差 \(\alpha\) ,对所作假设成立与否作出定性判断, 这种统计推断称为假设检验. 若总体分布已知, 只对分布中未知参数提出假设并作检验, 这种检验称为参数检验
2. 假设检验基本思想的依据是小概率原理
小概率原理是指概率很小的事件在试验中发生的频率也很小, 因此小概率事件在一次试验中不可能发生
当对问题提出待检假设 \({H}_{0}\) ,并要检验它是否可信时,先假定 \({H}_{0}\) 正确. 在这个假定下,经过一次抽样,若小概率事件发生了,就作出拒绝 \({H}_{0}\) 的决定; 否则,若小概率事件未发生,则接受 \({H}_{0}\)
3. 假设检验基本概念
在显著性水平 \(\alpha\) 下,检验假设\[{H}_{0} : \mu = {\mu }_{0},\;{H}_{1} : \mu \neq {\mu }_{0}\]\({H}_{0}\) 称为原假设或零假设. \({H}_{1}\) 称为备择假设
当检验统计量取某个区域 \(C\) 中的值时,我们拒绝原假设 \({H}_{0}\) ,则称区域 \(C\) 为拒绝域 (或否定域)
4. 假设检验过程
(1)提出原假设和备择假设
(2)选取检验统计量
(3)确定拒绝原假设的域
(4)计算检验统计量的观察值并作出判断
5. 两类错误
人们作出判断的依据是一个样本,样本是随机的,因而人们进行假设检验判断 \({H}_{0}\) 可信与否时, 不免发生误判而犯两类错误
第一类错误: \({H}_{0}\) 为真,而检验结果将其否定,这称为"弃真"错误
第二类错误: \({H}_{0}\) 不真,而检验结果将其接受,这称为"取伪"错误
分别记犯第一、二类错误的概率为 \(\alpha ,\beta\) ,即 \(\alpha = P\left\{ \right.\) 拒绝 \(\left. {{H}_{0} \mid {H}_{0}\text{ 为真 }}\right\} ,\beta = P\left\{ \right.\) 接受 \({H}_{0} \mid {H}_{0}\) 不真}. 当样本容量 \(n\) 固定时, \(\alpha\) 越小, \(\beta\) 就越大. 一般采取的原则是: 固定 \(\alpha\) ,通过增加样本容量 \(n\) 降低 \(\beta\)
6. 假设检验与区间估计的联系
假设检验与区间估计是从不同角度来对同一问题的回答, 其解决问题的途径相通
下面以正态总体 \(N\left( {\mu ,{\sigma }_{0}^{2}}\right)\) ,其中 \({\sigma }_{0}^{2}\) 已知,关于 \(\mu\) 的假设检验和区间估计为例加以说明:
假设 \({H}_{0} : \mu = {\mu }_{0}\) ,当 \({H}_{0}\) 为真时,则 \(\displaystyle U = \frac{\bar{X} - {\mu }_{0}}{\frac{\sigma }{\sqrt{n}}} \sim N\left( {0,1}\right)\) ,对于给定的显著性水平 \(\displaystyle \alpha ,P\{ \left| U\right|\) \(\displaystyle \left. { \leq {u}_{\frac{\alpha }{2}}}\right\} = 1 - \alpha\) ,那么 \({H}_{0}\) 的接受域为 \(\displaystyle \left( {\bar{X} \pm {u}_{\frac{\alpha }{2}}\frac{{\sigma }_{0}}{\sqrt{n}}}\right)\) ,即认为以 \(1 - \alpha\) 的概率接受 \({H}_{0}\) ,事实上这个接受域也是 \(\mu\) 的置信度为 \(1 - \alpha\) 的置信区间. 这充分说明两者解决问题的途径相同,假设检验判断的是结论是否成立, 而参数估计解决的是范围问题
2. 正态总体参数的假设检验
1. 一个正态总体的假设检验
设 \(X \sim N\left( {\mu ,{\sigma }^{2}}\right) ,\left( {{X}_{1},{X}_{2},\cdots ,{X}_{n}}\right)\) 为其样本
(1) \({\sigma }^{2}\) 已知,检验假设 \({H}_{0} : \mu = {\mu }_{0};{H}_{1} : \mu \neq {\mu }_{0}\)
检验步骤为:
①提出待检假设 \({H}_{0} : \mu = {\mu }_{0}\left( {\mu }_{0}\right.\) 已知)
②选取样本 \(\left( {{X}_{1},{X}_{2},\cdots ,{X}_{n}}\right)\) 的统计量 \(\displaystyle U = \frac{\bar{X} - {\mu }_{0}}{\frac{{\sigma }_{0}}{\sqrt{n}}}\left( {{\sigma }_{0}\text{ 已知 }}\right)\) ,在 \({H}_{0}\) 成立时, \(U \sim N\left( {0,1}\right)\)
③对给定的显著性水平 \(\alpha\) ,查表确定临界值 \(\displaystyle {u}_{\frac{\alpha }{2}}\) ,使得 \(\displaystyle P\left\{ {\left| U\right| > {u}_{\frac{\alpha }{2}}}\right\} = \alpha\) ,计算检验统计量 \(U\) 的观察值并与临界值 \(\displaystyle {u}_{\frac{\alpha }{2}}\) 比较
④作出判断:若 \(\displaystyle \left| U\right| > {u}_{\frac{\alpha }{2}}\) ,则拒绝 \({H}_{0}\) ；若 \(\displaystyle \left| U\right| < {u}_{\frac{\alpha }{2}}\) ,则接受 \({H}_{0}\)
(2)\(\sigma ^2\)未知,检验假设 \({H}_{0} : \mu = {\mu }_{0};{H}_{1} : \mu \neq {\mu }_{0}\)
选取统计量 \(\displaystyle T = \frac{\bar{X} - {\mu }_{0}}{\frac{S}{\sqrt{n}}}\) ,其中 \(\displaystyle {S}^{2} = \frac{1}{n - 1}\mathop{\sum }\limits_{{i = 1}}^{n}{\left( {X}_{i} - \bar{X}\right) }^{2}\) ,当 \({H}_{0}\) 为真时, \(T \sim t\left( {n - 1}\right)\)
拒绝域为 \(\displaystyle \left| T\right| > {t}_{\frac{\alpha }{2}}\left( {n - 1}\right)\)
(3)\(\mu\) 已知,检验假设 \({H}_{0} : {\sigma }^{2} = {\sigma }_{0}^{2};{H}_{1} : {\sigma }^{2} \neq {\sigma }_{0}^{2}\)
选取统计量 \(\displaystyle {\chi }^{2} = \frac{\mathop{\sum }\limits_{{i = 1}}^{n}{\left( {X}_{i} - {\mu }_{0}\right) }^{2}}{{\sigma }_{0}^{2}} \sim {\chi }^{2}\left( n\right)\)
拒绝域为 \(\displaystyle {\chi }^{2} > {\chi }_{\frac{\alpha }{2}}^{2}\left( n\right)\) 或 \(\displaystyle {\chi }^{2} < {\chi }_{1 - \frac{\alpha }{2}}^{2}\left( n\right)\)
(4) \(\mu\) 未知,检验假设 \({H}_{0} : {\sigma }^{2} = {\sigma }_{0}^{2};{H}_{1} : {\sigma }^{2} \neq {\sigma }_{0}^{2}\)
选取统计量 \(\displaystyle {\chi }^{2} = \frac{\left( {n - 1}\right) {S}^{2}}{{\sigma }_{0}^{2}}\) . 当 \({H}_{0}\) 为真时, \({\chi }^{2} \sim {\chi }^{2}\left( {n - 1}\right)\)
拒绝域为 \(\displaystyle {\chi }^{2} > {\chi }_{\frac{\alpha }{2}}^{2}\left( {n - 1}\right)\) 或 \(\displaystyle {\chi }^{2} < {\chi }_{1 - \frac{\alpha }{2}}^{2}\left( {n - 1}\right)\)
2. 两个正态总体的假设检验
设 \(X \sim N\left( {{\mu }_{1},{\sigma }_{1}^{2}}\right) ,Y \sim N\left( {{\mu }_{2},{\sigma }_{2}^{2}}\right) ,\left( {{X}_{1},{X}_{2},\cdots ,{X}_{{n}_{1}}}\right)\) 和 \(\left( {{Y}_{1},{Y}_{2},\cdots ,{Y}_{{n}_{2}}}\right)\) 分别是来自总体 \(X\) 和 \(Y\) 的样本, \(\bar{X}\text{ 、 }{S}_{1}^{2}\) 和 \(\bar{Y}\text{ 、 }{S}_{2}^{2}\) 是相应的样本的均值和方差
(1) \({\sigma }_{1}^{2},{\sigma }_{2}^{2}\) 已知,检验假设 \({H}_{0} : {\mu }_{1} = {\mu }_{2};{H}_{1} : {\mu }_{1} \neq {\mu }_{2}\)
选取统计量 \(\displaystyle U = \frac{\bar{X} - \bar{Y}}{\sqrt{\frac{{\sigma }_{1}^{2}}{{n}_{1}} + \frac{{\sigma }_{2}^{2}}{{n}_{2}}}} \sim N\left( {0,1}\right)\)
拒绝域为 \(\displaystyle \left| U\right| > {u}_{\frac{\alpha }{2}}\)
(2)\({\sigma }_{1}^{2},{\sigma }_{2}^{2}\) 未知,检验假设 \({H}_{0} : {\mu }_{1} = {\mu }_{2};{H}_{1} : {\mu }_{1} \neq {\mu }_{2}\) . 常见的三种特殊情形:
①当 \({n}_{1},{n}_{2}\) 较大时:
选取统计量 \(\displaystyle U = \frac{\bar{X} - \bar{Y}}{\sqrt{\frac{{S}_{1}^{2}}{{n}_{1}} + \frac{{S}_{2}^{2}}{{n}_{2}}}}\xrightarrow[]{\text{ 近似 }}N\left( {0,1}\right)\)
拒绝域为 \(\displaystyle \left| U\right| > {u}_{\frac{\alpha }{2}}\)
②\({\sigma }_{1}^{2} = {\sigma }_{2}^{2}\) 时:
选取检验统计量 \(\displaystyle T = \frac{\bar{X} - \bar{Y}}{\sqrt{\frac{\left( {{n}_{1} - 1}\right) {S}_{1}^{2} + \left( {{n}_{2} - 1}\right) {S}_{2}^{2}}{{n}_{1} + {n}_{2} - 2}}\sqrt{\frac{1}{{n}_{1}} + \frac{1}{{n}_{2}}}}\) ,当 \({H}_{0}\) 为真时, \(T \sim t\left( {{n}_{1} + {n}_{2} - 2}\right)\)
显著性水平为 \(\alpha\) 的拒绝域为 \(\displaystyle \left| T\right| > {t}_{\frac{\alpha }{2}}\left( {{n}_{1} + {n}_{2} - 2}\right)\)
③\({\sigma }_{1}^{2} \neq {\sigma }_{2}^{2}\) ,但 \({n}_{1} = {n}_{2}\) (配对问题):
令 \({D}_{i} = {X}_{i} - {Y}_{i}\left( {i = 1,2,\cdots ,n}\right)\) ,则 \({D}_{i} \sim N\left( {{\mu }_{D},{\sigma }_{D}^{2}}\right)\) ,其中 \({\mu }_{D} = {\mu }_{1} - {\mu }_{2},{\sigma }_{D}^{2} = {\sigma }_{1}^{2} + {\sigma }_{2}^{2}\) (未知)
此时检验假设等价于 \({H}_{0} : {\mu }_{D} = 0;{H}_{1} : {\mu }_{D} \neq 0\)
选取统计量 \(\displaystyle T = \frac{\bar{D} - {\mu }_{D}}{\frac{{S}_{D}}{\sqrt{n}}} \sim t\left( {n - 1}\right)\)
拒绝域为 \(\displaystyle \left| T\right| > {t}_{\frac{\alpha }{2}}\left( {n - 1}\right)\)
(3)\({\mu }_{1},{\mu }_{2}\) 已知,检验假设 \({H}_{0} : {\sigma }_{1}^{2} = {\sigma }_{2}^{2};{H}_{1} : {\sigma }_{1}^{2} \neq {\sigma }_{2}^{2}\)
选取统计量 \(\displaystyle F = \frac{\frac{\mathop{\sum }\limits_{{i = 1}}^{{n}_{1}}{\left( {X}_{i} - {\mu }_{1}\right) }^{2}}{{n}_{1}}}{\frac{\mathop{\sum }\limits_{{j = 1}}^{{n}_{2}}{\left( {Y}_{j} - {\mu }_{2}\right) }^{2}}{{n}_{2}}} \sim F\left( {{n}_{1},{n}_{2}}\right)\)
拒绝域为 \(\displaystyle F > {F}_{\frac{\alpha }{2}}\left( {{n}_{1},{n}_{2}}\right)\) 或 \(\displaystyle F < {F}_{1 - \frac{\alpha }{2}}\left( {{n}_{1},{n}_{2}}\right)\)
(4)\({\mu }_{1},{\mu }_{2}\) 未知,检验假设 \({H}_{0} : {\sigma }_{1}^{2} = {\sigma }_{2}^{2};{H}_{1} : {\sigma }_{1}^{2} \neq {\sigma }_{2}^{2}\)
选取检验统计量 \(\displaystyle F = \frac{{S}_{1}^{2}}{{S}_{2}^{2}}\) ,当 \({H}_{0}\) 为真时 \(F \sim F\left( {{n}_{1} - 1,{n}_{2} - 1}\right)\)
显著性水平为 \(\alpha\) 的拒绝域为 \(\displaystyle F > {F}_{\frac{\alpha }{2}}\left( {{n}_{1} - 1,{n}_{2} - 1}\right)\) 或 \(\displaystyle F < {F}_{1 - \frac{\alpha }{2}}\left( {{n}_{1} - 1,{n}_{2} - 1}\right)\)
3. 单侧检验
在假设检验中, 如果只关心总体参数是否偏大或偏小, 此时可将拒绝域确定在某一侧, 这种检验称为单侧检验. 单侧检验可由双侧检验修改转化而得到. 常用基本类型举例:
(1)\({\sigma }^{2}\) 已知,检验假设 \({H}_{0} : \mu \leq {\mu }_{0};{H}_{1} : \mu > {\mu }_{0}\) (有时也写成 \({H}_{0} : \mu = {\mu }_{0};{H}_{1} : \mu > {\mu }_{0}\) )
选取 \(\displaystyle U = \frac{\bar{X} - {\mu }_{0}}{\frac{\sigma }{\sqrt{n}}}\) ,拒绝域为 \(U > {u}_{a}\)
(2) \({\sigma }^{2}\) 已知,检验假设 \({H}_{0} : \mu \geq {\mu }_{0};{H}_{1} : \mu < {\mu }_{0}\)
选取 \(\displaystyle U = \frac{\bar{X} - {\mu }_{0}}{\frac{\sigma }{\sqrt{n}}}\) ,拒绝域为 \(U < - {u}_{a}\)
(3)\({\sigma }^{2}\) 未知,检验假设 \({H}_{0} : \mu \leq {\mu }_{0};{H}_{1} : \mu > {\mu }_{0}\)
选取 \(\displaystyle T = \frac{\bar{X} - {\mu }_{0}}{\frac{S}{\sqrt{n}}}\) ,拒绝域为 \(T > {t}_{a}\left( {n - 1}\right)\)
(4)\({\sigma }^{2}\) 未知,检验假设 \({H}_{0} : \mu \geq {\mu }_{0};{H}_{1} : \mu < {\mu }_{0}\)
选取 \(\displaystyle T = \frac{\bar{X} - {\mu }_{0}}{\frac{S}{\sqrt{n}}}\) ,拒绝域为 \(T < - {t}_{\alpha }\left( {n - 1}\right)\)
(5)\(\mu\) 未知,检验假设 \({H}_{0} : {\sigma }^{2} \leq {\sigma }_{0}^{2};{H}_{1} : {\sigma }^{2} > {\sigma }_{0}^{2}\)
选取 \(\displaystyle {\chi }^{2} = \frac{\left( {n - 1}\right) {S}^{2}}{{\sigma }_{0}^{2}}\) ,拒绝域为 \({\chi }^{2} > {\chi }_{\alpha }^{2}\left( {n - 1}\right)\)
(6)\(\mu\) 未知,检验假设 \({H}_{0} : {\sigma }^{2} \geq {\sigma }_{0}^{2};{H}_{1} : {\sigma }^{2} < {\sigma }_{0}^{2}\)
选取 \(\displaystyle {\chi }^{2} = \frac{\left( {n - 1}\right) {S}^{2}}{{\sigma }_{0}^{2}}\) ,拒绝域为 \({\chi }^{2} < {\chi }_{1 - \alpha }^{2}\left( {n - 1}\right)\)
(7)\({\mu }_{1},{\mu }_{2}\) 未知,检验假设 \({H}_{0} : {\sigma }_{1}^{2} \leq {\sigma }_{2}^{2};{H}_{1} : {\sigma }_{1}^{2} > {\sigma }_{2}^{2}\)
选取 \(\displaystyle F = \frac{{S}_{1}^{2}}{{S}_{2}^{2}}\) ,拒绝域为 \(F > {F}_{\alpha }\left( {{n}_{1} - 1,{n}_{2} - 1}\right)\)
(8)\({\mu }_{1},{\mu }_{2}\) 未知,检验假设 \({H}_{0} : {\sigma }_{1}^{2} \geq {\sigma }_{2}^{2};{H}_{1} : {\sigma }_{1}^{2} < {\sigma }_{2}^{2}\)
选取 \(\displaystyle F = \frac{{S}_{1}^{2}}{{S}_{2}^{2}}\) ,拒绝域为 \(F < {F}_{1 - \alpha }\left( {{n}_{1} - 1,{n}_{2} - 1}\right)\)
其他类型可仿照上述类型得到解决