【统计学】第七章

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参数估计的基本原理

一个总体参数的区间估计

两个总体参数的区间估计

样本容量的确定

掌握目标
估计量与估计值的概念
点估计与区间估计的区别
评价估计量优良性的标准
一个总体参数的区间估计方法
两个总体参数的区间估计方法
样本容量的确定方法

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估计量与估计值

1.估计量：用于估计总体参数的随机变量，如样本均值，样本比例，样本方差，例如：样本均值就是总体均值中的一个估计量
2.参数用θ表示，估计量使用θ^表示
3.估计值：估计参数时计算出来的统计量的总体值，如果样本均值为x=80,则80就是估计值

点估计与区间估计

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点估计：用样本的估计量直接作为总体参数的估计值，用样本均值直接作为总体均值的估计
用两个样本均值之差直接作为总体均值之差的估计
没有给出估计值接近总体参数程度的信息

区间估计：在点估计的总体基础上，给出总体参数估计的一个区间范围，该区间由样本统计量加减抽样误差而得到的。
根据样本统计量的抽样分布能够对样本统计量与总体参数的接近程度给出一个概率统计量，比如某班级平均分数在75-85分之间， 置信水平是95%
置信水平：将构造置信区间的步骤重复很多次，置信区间包含总体参数真值的次数所占的比例称为置信水平
常用的置信水平有99%，95%，90%
由样本统计量所构造的总体参数的估计区间称为置信区间
统计学家在某种程度上确信这个区间会包含真正的总体参数，所以给它取名为置信区间
用一个具体的样本所构造的区间是一个特定的区间，无法知道这个样本所产生的区间是否包含总体参数的真值。只能是希望这个区间是大量包含总体参数真值的区间中的一个，但是它也可能是少数几个不包含参数真值的区间中的一个。

评价估计量的标准---无偏性

无偏性估计量抽样分布的数学期望等于被估计的总体参数
有效性对同一总体参数的两个无偏点估计量，有更小标准差的估计量更有效
一致性随着样本容量的增大，估计量的值越来越接近被估计的总体参数

一个总体参数的区间估计

总体参数	符号表示	样本统计量
均值	μ	x
比例	π	P
方差	σ2	s2

总体均值的区间估计

正态总体、σ2已知，或非正态总体、大样本
假定条件：总体服从正态分布，且方差(σ2)已知
如果不是正态分布，可由正态分布来近似(大样本n>=30)
使用正态分布统计量z
总体均值μ在1−α的置信水平下的公式为
x±z2α​​
或x±zn​s​​(α未知)

---

正态总体、σ2未知、小样本
1.假定条件

• 总体服从正态分布，且方差(σ2)未知
• 小样本(n<30)
  2.使用t分布统计量
  t=n​s​x−μ​ t(n−1)
  3.总体均值μ在1−α置信水平下的置信区间
  x±t2α​​n​s​
  t分布t分布式类似正态分布的一种对称分布，它通常要比正态分布平坦和分散，一个特定的分布依赖于称之为自由度的参数，随着自由度的增大，分布更趋向于正态分布。

总体比例的区间估计

1.假定条件

• 总体服从二项分布
• 可以由正态分布来近似
  2.使用正态分布统计量 Z
  z=nπ(1−π)​​p−π​ N(0,1)
  3.总体比例π在1−α置信水平下的置信区间
  p±z2α​​nπ(1−π)​​或p±z2α​​np(1−p)​​,π未知时

总体方差的区间估计

1.估计一个总体的方差或标准差
2.假设总体服从正态分布
3.总体方差的点估计量为s2,且σ2(n−1)s2​ χ2(n−1)
4.总体方差在1−α置信水平下的置信区间为χ2α​2​(n−1)(n−1)s2​<=σ2<=χ12​2α​(n−1)(n−1)s2​

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两个总体参数的区间估计

总体参数	符号表示	样本统计量
均值之差	μ1​−μ2​	x1​​−x2​​
比例之差	π1​−π2​	p1​−p2​
方差比	σ22​σ12​​	s22​s12​​

两个总体均值之差的区间估计

1.独立大样本
条件：两个总体都服从正态分布，σ12​,σ22​已知
若不是正态分布，可以用正态分布来近似(n1​≥30 和 n2​≥30)
两个样本是独立的随机样本
2.使用正态分布统计量 z
z=n1​σ12​​+n2​σ22​​​(x1​​−x2​​−(μ1​−μ2​))​ N(0,1)
σ12​,σ22​已知时，两个总体均值之差μ1​−μ2​在1−α置信水平下的置信区间为(x1​​−x2​​)±z2α​​n1​σ12​​+n2​σ22​​​
σ12​,σ22​已知时，两个总体均值之差μ1​−μ2​在1−α置信水平下的置信区间为(x1​​−x2​​)±z2α​​n1​s12​​+n2​s22​​​

独立小样本(σ12​=σ22​)
1.假定条件
两个总体都服从正态分布
两个总体方差未知但是相等(\sigma_1^2=\sigma_2^2)
两个独立的小样本(\n_1<30和n2​<30)
2.总体方差的合并估计量
sp2​=n1​+n2​−2(n1​−1)s12​+(n2​−1)s22​​
3.估计量x1​​−x2​​的抽样标准差
n1​sp2​​+n2​sp2​​​=sp​n1​1​+n2​1​​
两个样本均值之差的标准化
t=sp​n1​1​+n2​1​​(x1​​−x2​​)−(μ1​−μ2​)​ t(n1​+n2​−2)
两个总体均值之差(μ1​−μ2​)在1−α在置信水平下的置信区间为
(x1​​−x2​​)±t2α​​(n1​+n2​−2)sp2​(n1​1​+n2​1​)​

独立小样本(σ12​≠σ22​)
1.假定条件
两个总体都服从正态分布
两个总体方差未知且不相等，α12​≠α22​
两个独立的小样本(n1​<30,n2​<30)
2.使用统计量
t=n1​s12​​+n2​s22​​​(x1​​−x2​​)​ t(v)

两个总体均值之差(μ1​−μ2​)在1−α置信水平下的置信区间为
(x1​​−x2​​)±t2α​​(v)n1​s12​​+n2​s22​​​
其中v是指自由度,v=n1​−1(n1​s12​​)​+n2​−1(n1​s22​​)​(n1s12​​+n2​s22​​)​
匹配大样本
1.假定条件
两个匹配的大样本(n1​≥30和n2​≥30)
两个总体各观察值的配对差服从正态分布
2.两个总体均值之差(μd​=μ1​−μ2​在1−α)置信水平下的置信区间
d±z2α​​n​σd​​
匹配小样本
1.假定条件
两个匹配的小样本(n1​<30和n2​<30)
两个总体各观察值的配对差服从正态分布
两个总体均值之差(μd​=μ1​−μ2​)在1−α置信水平下的置信区间为d±t2α​​(n−1)n​sd​​

两个总体比例之差的区间估计

1.假定条件
两个总体服从二项分布
可以用正态分布来近似
两个样本是独立的
2.两个总体比例之差π1​−π2​在1−α置信水平下的置信区间为
(p1​−p2​)±z2α​​n1​p1​(1−p1​)​+n2​p2​(1−p2​)​​

两个总体方差比的区间估计

1.比较两个总体的方差比
2.用两个样本的方差比来判断

• 如果S22​S12​​接近于1，说明两个总体方差很接近
• 如果S22​S12​​远离1，说明两个总体方差之间存在差异
  3.总体方差比在1−α置信水平下的置信区间
  \frac{\frac{s_1^2}{s_2^2}}{F_{\frac{\alpha}{2}}} \leq \frac{\sigma_1^2}{\sigma_2^2} \leq \frac{\frac{s_1^2}{s_2^2}}{F_{1-\frac{\alpha}{2}}

样本容量的确定

估计总体均值时样本容量的确定

1.估计总体均值时样本容量n为
n=E2z2α​2​σ2​
2.样本容量n与总体方差σ2,边际误差E和可靠性系数Z或t之间的关系为

• 与总体方差成正比
• 与边际误差成反比
• 与可靠性系数成正比

估计总体比例时样本容量的确定

1.根据比例区间估计公式可得样本容量
n=E2(zfracα2​)2 dotπ(1−π)​
其中 E=z2α​nπ(1−π)​​​
2.E的取值一般小于0.1
3.π未知时，可取最大值0.5

估计两个总体均值之差时样本容量

1.设n1​和n2​来自两个总体的样本，并假定n1​=n2​
2.根据均值之差的区间估计公式可得两个样本的容量n为
n1​=n2​=n=E2(z2α​​)(σ12​+σ22​)​
其中E=z2α​n​σ1​+σ2​​​

估计两个总体比例之差时样本容量

1.设n1​和n2​来自两个总体的样本，并假定n1​=n2​
2.根据比例之差的区间估计公式可得两个样本的容量n为
n1​=n2​=n=E2(zfracα2​)2[π1​(1−π1​)+π2​(1−π2​)]​
其中E=z2α​​n​π1​(1−π1​)+π2​(1−π2​)​

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来源：oschina

链接：https://my.oschina.net/u/4312544/blog/3503577