【統計學】第七章

Evernote Export

參數估計的基本原理

一個整體參數的區間估計

兩個整體參數的區間估計

樣本容量的肯定

掌握目標
估計量與估計值的概念
點估計與區間估計的區別
評價估計量優良性的標準
一個整體參數的區間估計方法
兩個整體參數的區間估計方法
樣本容量的肯定方法

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估計量與估計值

1.估計量：用於估計整體參數的隨機變量，如樣本均值，樣本比例，樣本方差，例如：樣本均值就是整體均值中的一個估計量
2.參數用θ表示，估計量使用θ^表示
3.估計值：估計參數時計算出來的統計量的整體值，若是樣本均值爲x=80,則80就是估計值

點估計與區間估計

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點估計：用樣本的估計量直接做爲整體參數的估計值，用樣本均值直接做爲整體均值的估計
用兩個樣本均值之差直接做爲整體均值之差的估計
沒有給出估計值接近整體參數程度的信息

區間估計：在點估計的整體基礎上，給出整體參數估計的一個區間範圍，該區間由樣本統計量加減抽樣偏差而獲得的。
根據樣本統計量的抽樣分佈可以對樣本統計量與整體參數的接近程度給出一個機率統計量，好比某班級平均分數在75-85分之間， 置信水平是95%
置信水平：將構造置信區間的步驟重複不少次，置信區間包含整體參數真值的次數所佔的比例稱爲置信水平
經常使用的置信水平有99%，95%，90%
由樣本統計量所構造的整體參數的估計區間稱爲置信區間
統計學家在某種程度上確信這個區間會包含真正的整體參數，因此給它取名爲置信區間
用一個具體的樣本所構造的區間是一個特定的區間，沒法知道這個樣本所產生的區間是否包含整體參數的真值。只能是但願這個區間是大量包含整體參數真值的區間中的一個，可是它也多是少數幾個不包含參數真值的區間中的一個。

評價估計量的標準---無偏性

無偏性估計量抽樣分佈的數學指望等於被估計的整體參數
有效性對同一整體參數的兩個無偏點估計量，有更小標準差的估計量更有效
一致性隨着樣本容量的增大，估計量的值愈來愈接近被估計的整體參數

一個整體參數的區間估計

整體參數	符號表示	樣本統計量
均值	μ	x
比例	π	P
方差	σ2	s2

整體均值的區間估計

正態整體、σ2已知，或非正態整體、大樣本
假定條件：整體服從正態分佈，且方差(σ2)已知
若是不是正態分佈，可由正態分佈來近似(大樣本n>=30)
使用正態分佈統計量z
整體均值μ在1−α的置信水平下的公式爲
x±z2α​​
或x±zn​s​​(α未知)

---

正態整體、σ2未知、小樣本
1.假定條件

• 整體服從正態分佈，且方差(σ2)未知
• 小樣本(n<30)
  2.使用t分佈統計量
  t=n​s​x−μ​ t(n−1)
  3.整體均值μ在1−α置信水平下的置信區間
  x±t2α​​n​s​
  t分佈t分佈式相似正態分佈的一種對稱分佈，它一般要比正態分佈平坦和分散，一個特定的分佈依賴於稱之爲自由度的參數，隨着自由度的增大，分佈更趨向於正態分佈。

整體比例的區間估計

1.假定條件

• 整體服從二項分佈
• 能夠由正態分佈來近似
  2.使用正態分佈統計量 Z
  z=nπ(1−π)​​p−π​ N(0,1)
  3.整體比例π在1−α置信水平下的置信區間
  p±z2α​​nπ(1−π)​​或p±z2α​​np(1−p)​​,π未知時

整體方差的區間估計

1.估計一個整體的方差或標準差
2.假設整體服從正態分佈
3.整體方差的點估計量爲s2,且σ2(n−1)s2​ χ2(n−1)
4.整體方差在1−α置信水平下的置信區間爲χ2α​2​(n−1)(n−1)s2​<=σ2<=χ12​2α​(n−1)(n−1)s2​

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#d585v4c9ynf .section{
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兩個整體參數的區間估計

整體參數	符號表示	樣本統計量
均值之差	μ1​−μ2​	x1​​−x2​​
比例之差	π1​−π2​	p1​−p2​
方差比	σ22​σ12​​	s22​s12​​

兩個整體均值之差的區間估計

1.獨立大樣本
條件：兩個整體都服從正態分佈，σ12​,σ22​已知
若不是正態分佈，能夠用正態分佈來近似(n1​≥30 和 n2​≥30)
兩個樣本是獨立的隨機樣本
2.使用正態分佈統計量 z
z=n1​σ12​​+n2​σ22​​​(x1​​−x2​​−(μ1​−μ2​))​ N(0,1)
σ12​,σ22​已知時，兩個整體均值之差μ1​−μ2​在1−α置信水平下的置信區間爲(x1​​−x2​​)±z2α​​n1​σ12​​+n2​σ22​​​
σ12​,σ22​已知時，兩個整體均值之差μ1​−μ2​在1−α置信水平下的置信區間爲(x1​​−x2​​)±z2α​​n1​s12​​+n2​s22​​​

獨立小樣本(σ12​=σ22​)
1.假定條件
兩個整體都服從正態分佈
兩個整體方差未知可是相等(\sigma_1^2=\sigma_2^2)
兩個獨立的小樣本(\n_1<30和n2​<30)
2.整體方差的合併估計量
sp2​=n1​+n2​−2(n1​−1)s12​+(n2​−1)s22​​
3.估計量x1​​−x2​​的抽樣標準差
n1​sp2​​+n2​sp2​​​=sp​n1​1​+n2​1​​
兩個樣本均值之差的標準化
t=sp​n1​1​+n2​1​​(x1​​−x2​​)−(μ1​−μ2​)​ t(n1​+n2​−2)
兩個整體均值之差(μ1​−μ2​)在1−α在置信水平下的置信區間爲
(x1​​−x2​​)±t2α​​(n1​+n2​−2)sp2​(n1​1​+n2​1​)​

獨立小樣本(σ12​≠σ22​)
1.假定條件
兩個整體都服從正態分佈
兩個整體方差未知且不相等，α12​≠α22​
兩個獨立的小樣本(n1​<30,n2​<30)
2.使用統計量
t=n1​s12​​+n2​s22​​​(x1​​−x2​​)​ t(v)

兩個整體均值之差(μ1​−μ2​)在1−α置信水平下的置信區間爲
(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​​)​
匹配大樣本
1.假定條件
兩個匹配的大樣本(n1​≥30和n2​≥30)
兩個整體各觀察值的配對差服從正態分佈
2.兩個整體均值之差(μd​=μ1​−μ2​在1−α)置信水平下的置信區間
d±z2α​​n​σd​​
匹配小樣本
1.假定條件
兩個匹配的小樣本(n1​<30和n2​<30)
兩個整體各觀察值的配對差服從正態分佈
兩個整體均值之差(μd​=μ1​−μ2​)在1−α置信水平下的置信區間爲d±t2α​​(n−1)n​sd​​

兩個整體比例之差的區間估計

1.假定條件
兩個整體服從二項分佈
能夠用正態分佈來近似
兩個樣本是獨立的
2.兩個整體比例之差π1​−π2​在1−α置信水平下的置信區間爲
(p1​−p2​)±z2α​​n1​p1​(1−p1​)​+n2​p2​(1−p2​)​​

兩個整體方差比的區間估計

1.比較兩個整體的方差比
2.用兩個樣本的方差比來判斷

• 若是S22​S12​​接近於1，說明兩個整體方差很接近
• 若是S22​S12​​遠離1，說明兩個整體方差之間存在差別
  3.整體方差比在1−α置信水平下的置信區間
  \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}}

樣本容量的肯定

估計整體均值時樣本容量的肯定

1.估計整體均值時樣本容量n爲
n=E2z2α​2​σ2​
2.樣本容量n與整體方差σ2,邊際偏差E和可靠性係數Z或t之間的關係爲

• 與整體方差成正比
• 與邊際偏差成反比
• 與可靠性係數成正比

估計整體比例時樣本容量的肯定

1.根據比例區間估計公式可得樣本容量
n=E2(zfracα2​)2 dotπ(1−π)​
其中 E=z2α​