Parameter estimation (3) Interval estimation

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The concept of interval estimation
The case of a normal population
- $\mu$
- $\sigma^2$
The case of two normal populations
- $\mu_1-\mu_2$ interval estimate of
- $\sigma_1^2/\sigma_2^2$ interval estimate of
references

The concept of interval estimation

For unknown parameters, in addition to caring about its point estimate, sometimes we also need to estimate a range of it and the degree of confidence that this range contains the true value of the parameter. This range is usually given in the form of an interval, called a confidence interval. This form of estimation is calledinterval estimation.

设总体 $X$ 's distribution function $F(x;\theta)$ ， $\theta$ Unknown number, $X_1,...,X_n$ is from the population $Sample of X$ . For a given value $\alpha(0<\alpha<1)$ , young existence measurement $\underline{\theta}(X_1,...,X_n)$ 和 $\bar{\theta}(X_1,...,X_n)$ ，使得 $P\{\underline{\theta}(X_1,...,X_n)<\theta<\bar{\theta}(X_1,...,X_n)\} \ge 1-\alpha$ nomenclature $(\underline{\theta},\bar{\theta}) < /span>$ Bosin number $\theta$ is $1-\alpha$ position, $\underline{\theta}, \bar{\theta}$ are called upper and lower confidence limits respectively.

Usually the unknown parameters can be found according to the following ideas $\theta$ :

Unknown number $\theta$ , construct a sample containing only samples $X_1,..., X_n$ Number of sums $\theta$ 's function $G=G(X_1,. ..,X_n;\theta)$ , inside $The probability distribution of G$ should be easily determined and contain no unknown parameters ( $G$ also namepivot amount);
Confidence for the given $1-\alpha$ ，由等式 $P\{c<G(X_1,...,X_n;\theta)<d\}=1-\alpha$ Appropriate determination constant $c, d$ ；
把不等式 $c<G(X_1,...,X_n;\theta)<d$ Equal inequality $\underline{\theta}(X_1,...,X_n)<\theta<\bar{\theta}(X_1,...,X_n)$ Number of arrivals at this time $\theta$ is $1-\alpha$ $(\underline{\theta},\bar{\theta}) < /span>$ 。

Above calculation reference number $\theta$ is calledpivot measure method.

The case of a normal population

Definition $\sim N(\mu,\sigma^2)$ ， $X_1,...,X_n$ 为总体 $X$ の样本， $\overset{ˉ}{X} 、 S^{* 2}$ are the sample mean and corrected sample variance respectively. Discuss below $\mu,\sigma^2$ .

$\mu$

(1) We difference $\sigma^2$ 已智时，因为 $\bar{X}$ 是 $\mu$ , thus constructing the pivot quantity $U=\frac{\sqrt{n}(\bar{X}-\mu)}{\sigma}\sim N(0,1)$ $\mu$ is $1-\alpha$ 的置信区间为 $\left(\bar{X}-\frac{\sigma}{\sqrt{n}}u_{\alpha/2},\bar{X}+\frac{\sigma}{\sqrt{n}}u_{\alpha/2}\right)$

(2) We difference $\sigma^2$ Unknown time，Be careful $S^{*2}$ 是 $\sigma^2$ as $S^*$ Exchange (1) Central exchange $\sigma$ , the pivot quantity can be obtained $\frac{\sqrt{n}(\bar{X}-\mu)}{S^*}\sim t(n-1)$ $\mu$ is $1-\alpha$ 的置信区间为 $\left(\bar{X}-\frac{S^*}{\sqrt{n}}t_{\alpha/2}(n-1),\bar{X}+\frac{S^*}{\sqrt{n}}t_{\alpha/2}(n-1)\right)$

$\sigma^2$

这りJust讨论 $\mu$ unknown time $\sigma^2$ .

because $S^{*2}$ 是 $\sigma^2$ , so construct the pivot quantity $\chi^2=\frac{1}{\sigma^2}\sum_{i=1}^n (X_i- \bar{X})^2=\frac{(n-1)S^{*2}}{\sigma^2} \sim \chi^2(n-1)$ $\sigma^2$ Placement confidence $1-\alpha$ Differentiation and Range $\left(\frac{(n-1)S^{*2}}{\chi^2_{\alpha/2}(n -1)},\frac{(n-1)S^{*2}}{\chi^2_{1-\alpha/2}(n-1)}\right)$

The case of two normal populations

Definition $\sim N(\mu_1,\sigma^2_1)$ ,unless $\sim N(\mu_2,\sigma^2_2)$ ， $X_1,...,X_{n_1})^T$ ， $Y_1,...,Y_{n_2})^T$ Separate body from self $X, Y$ samples, and it is assumed that the two samples are independent of each other. Note $\bar{X},\bar{Y}$ is the sample mean of each of the two samples, $S^{*2 }_{1n_1},S^{*2}_{2n_2}$ is the corrected variance of each of the two samples. For a given confidence level $1-\alpha$ ，要手机 $\mu_1-\mu_2,\ \sigma_1^ 2/\sigma_2^2$ interval estimate.

$\mu_1-\mu_2$ interval estimate of

(1) $\sigma_1^2$ 和 $\sigma_2^2$ All are known.

图像枢長量 $U=\frac{\bar{X}}\bar{Y}-(\mu_1-\mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2} {n_2}}} \sim N(0,1)$ 故 $\mu_1-\mu_2$ The confidence of is $1-\alpha$ Differentiation and expansion $\left((\bar{X}-\bar{Y}) \mp u_{\alpha/2} \sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2 }{n_2}}\right)$

(2) $\sigma_1^2=\sigma_2^2=\sigma^2$ ，但 $\sigma^2$ Unknown.

构造枢轴量 $T=\frac{\bar{X}-\bar{Y}-(\mu_1-\mu_2)}{S_w \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}} \sim t(n_1+n_2-2)$ inside $S_w=\sqrt{\frac{(n_1-1)S^{*2}_{1n_1}+(n_2-1)S ^{*2}_{2n_2}}{n_1+n_2-2}}$ 故 $\mu_1-\mu_2$ The confidence of is $1-\alpha$ 的置信区间为 $\left((\bar{X}-\bar{Y}) \mp t_{\alpha/2}(n_1+n_2-2)S_w \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}\right)$

(3) $\sigma_1^2$ 和 $\sigma_2^2$ Unknown, however $n_1=n_2=n$

令 $Z_i=X_i-Y_i \sim N(\mu_1-\mu_2,\sigma_1^2+\sigma_2^2),i=1,2,...,n$ ，故 $\mu_1-\mu_2$ The confidence of is $1-\alpha$ is $\left(\bar{ Z} \mp \frac{S_Z^*}{\sqrt{n}} t_{\alpha/2}(n-1)\right)$ 其中 $\bar{Z}=\bar{X}-\bar{Y},S_Z^*=\sqrt{\frac{1}{n-1}\sum_{i=1}^n (Z_i-\bar{Z})^2}$

$\sigma_1^2/\sigma_2^2$ interval estimate of

这りJust讨论 $\mu_1,\mu_2$ 为未知时， $\sigma_1^2/\sigma_2^2$ interval estimate.

F = σ 2 2 S 1 n 1 ∗ 2 σ 1 2 S 2 n 2 ∗ 2 ∼ F ( n 1 − 1 , n 2 − 1 ) F=\frac{\sigma_2^2 S_{1n_1}^{*2}}{\sigma_1^2 S_{2n_2}^{*2}} \sim F(n_1-1,n_2-1)
$F = \frac{p _{2}^{2} S _{1 n_{1}}^{* 2}}{p _{1}^{2} S _{2 n_{2}}^{* 2}} \sim F (n_{1} - 1, n_{2} - 1)$ 故 $\sigma_1^2/\sigma_2^2$ The confidence of is $1-\alpha$ 的置信区间为
$\left(\frac{S_{1n_1}^{*2}/S_{2n_2}^{*2}}{F_{\alpha/2}(n_1-1,n_2-1)}, \frac{S_{1n_1}^{*2}/S_{2n_2}^{*2}}{F_{1-\alpha/2}(n_1-1,n_2-1)}\right)$ 利用 $F$ The upper quantile property of the distribution can be $\sigma_1^2/\sigma_2^2$ The confidence of is $1-\alpha$ is rewritten as
$\left(F_{1-\alpha/2} (n_2-1,n_1-1)\frac{S_{1n_1}^{*2}}{S_{2n_2}^{*2}}, F_{\alpha/2}(n_2-1,n_1-1) \frac{S_{1n_1}^{*2}}{S_{2n_2}^{*2}}\right)$

references

[1] "Applied Mathematical Statistics", Shi Yu, Xi'an Jiaotong University Press.