抽样调查

Sampling Design Notes

Jack H

June 5, 2025

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Introduction

Coefficient of Variation

$$\mathrm{cv} = \frac{\sqrt{\mathrm{Var}(\hat\theta)}}{E(\hat\theta)}$$

It measures the density of given data.

Mean Square Error

$$\mathrm{MSE}(\hat{\theta}) = \mathrm{Var}(\hat{\theta}) + (\mathrm{bias}(\hat{\theta}))^2$$

Point Estimation Error Control: MSE(V,C)
When $\hat\theta$ is unbiased, $\mathrm{MSE}(\hat\theta) = \mathrm{Var}(\hat\theta)$ , for a given upper bound of variance $V$, or a given upper bound of $\mathrm{cv}: C$, such that $V \geq \mathrm{var}(\hat\theta) $ or $C \geq \mathrm{cv}(\hat\theta)$

Margin of Error:
If $\hat \theta$ is a point estimator of $\theta$, for a given $\alpha \in (0,1)$, if $$P(|\hat\theta - \theta| \leq d) = 1-\alpha$$ we call $d$ the margin of error of $\hat\theta$ at confidence level $\alpha$.
Relative Margin of Error:
If $$P(\frac{|\hat\theta - \theta|}{\theta} \leq r) = 1-\alpha$$ we call $r$ the relative margin of error of $\hat\theta$ at confidence level $\alpha$.

Error Limit (d,r) Estimation Control:
For a given $\alpha \in (0,1)$, for given absolute margin of error $d$ or relative margin of error $r$, such that $P(|\hat\theta - \theta| \leq d) = 1-\alpha$ or $P(\frac{|\hat\theta - \theta|}{\theta} \leq r) = 1-\alpha$.

Assumptions in this course:

1. Consistent Estimation: $\hat \theta_n \xrightarrow{P}\theta,n\to \infty$
   Definition: If $\hat\theta$ is a consistent estimator of $\theta$, then for any $\epsilon > 0$,

$$P(|\hat\theta - \theta| > \epsilon) \to 0 \text{ as } n \to \infty$$

2. Asymptotically Normal Distribution CAN: $\frac{\hat\theta_n - E(\hat\theta)}{\sqrt{var(\hat\theta_n)}} \xrightarrow{d} N(0,1)$
   Denote $\sqrt{var(\hat\theta_n)}$ as $\mathrm{sd}(\hat\theta_n)$

You need to prove UE or AUE in this course
Theorem: When $\hat\theta$ is consistent and asymptotically normal, if $\hat \theta$ is an unbiased estimator(UE) or asymptotically unbiased estimator(AUE), then the distribution of $\hat\theta$ is approximately normal.

$$\frac{\hat\theta - \theta}{\sqrt{var(\hat\theta)}} = \frac{\hat\theta - \theta}{\mathrm{sd}(\hat\theta)} \xrightarrow{d} N(0,1)\quad \text{as } n \to \infty$$

Therefore $$P\left(\frac{\hat\theta - \theta}{\mathrm{sd}(\hat\theta)} \leq z_{\alpha/2}\right) = 1-\alpha$$
$d = z_{\alpha/2} \mathrm{sd}(\hat\theta), r = z_{\alpha/2} \frac{\mathrm{sd}(\hat\theta)}{\theta}$

$\hat\theta_L = \hat\theta - z_{\alpha/2} \mathrm{sd}(\hat\theta),\hat\theta_R = \hat\theta + z_{\alpha/2} \mathrm{sd}(\hat\theta)$
If $P(\hat\theta_L \leq \theta \leq \hat\theta_R) = 1-\alpha$, then $\hat\theta_L$ and $\hat\theta_R$ are the endpoints of a $(1-\alpha)$ confidence interval for $\theta$.
When $n$ is large, $$P(\theta \in [\hat\theta \pm z_{\alpha/2} \mathrm{sd}(\hat\theta)]) \approx 1-\alpha$$

In ci.r

conf.interval=function(para.hat, SD.hat, alpha)

Simple Random Sampling

In this cource, we consider picking $n$ units out of a population of $N$ without replacement, each pick has probability $p = 1/C_N^n$

In srs sampling.r

## simple random sampling without replacement
mysrs=sample(1:N, n)  
print(mysrs)
## simple random sampling with replacement
mysrs=sample(1:N, n, replace = TRUE)
print(mysrs)

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Mean :

$$\bar Y = \frac 1 N \sum_{i = 1} ^N Y_i$$

Total:

$$Y_T = N \bar Y$$

Variance:

$$S^2 = \frac{1}{N-1} \sum_{i=1}^N (Y_i - \bar Y)^2$$

Estimation of Population Mean $\bar Y$

1. Point Estimation

$$\bar y = \frac 1 n \sum_{i=1}^n y_i$$

2. Unbiased estimator of $\bar Y$

$$\mathrm E (\bar y ) = \bar Y \quad (UE)$$

3. Variance of estimation:

$$Var(\bar y ) = \frac{1-f}{n}S^2$$

where $f = \frac{n}{N}$ and $S^2$ is the variance of population $Y$ (unknown)
4. Estimation of variance:

$$\hat{\mathrm{Var} } (\bar y ) = \frac{1-f}{n} s^2$$

where

$$s^2 = \frac{1}{n-1}\sum_{i=1}^n (y_i - \bar y )^2$$

5. Confidence Interval

$$\left[ \bar y \pm z_{\alpha /2 } \sqrt{\widehat {\mathrm{Var}} (\bar y )}\right]$$

$$d = \sqrt{\widehat {\mathrm{Var}} (\bar y )}$$

$$r = \frac{d}{\bar y}$$

In srs.r

srs.mean=function(N, mysample, alpha) 

Proof of $E(s^2) = S^2$

Step 1: Express $ S^2 $ and $ s^2 $

The population variance $ S^2 $ is defined as:

$$S^2 = \frac{1}{N-1} \sum_{i=1}^N (Y_i - \bar{Y})^2$$

The sample variance $ s^2 $ is:

$$s^2 = \frac{1}{n-1} \sum_{i=1}^n (y_i - \bar{y})^2$$

Step 2: Expand the Sum of Squares

First, note that:

$$\sum_{i=1}^n (y_i - \bar{y})^2 = \sum_{i=1}^n y_i^2 - n \bar{y}^2$$

Step 3: Take the Expectation of $ s^2 $

Compute $ E(s^2) $:

$$E(s^2) = E\left( \frac{1}{n-1} \left[ \sum_{i=1}^n y_i^2 - n \bar{y}^2 \right] \right) = \frac{1}{n-1} \left[ \sum_{i=1}^n E(y_i^2) - n E(\bar{y}^2) \right]$$

Step 4: Compute $ E(y_i^2) $ and $ E(\bar{y}^2) $

For any $ y_i $:

$$E(y_i^2) = \text{Var}(y_i) + [E(y_i)]^2 = S^2 \left(1 - \frac{1}{N}\right) + \bar{Y}^2$$

For $ \bar{y} $:

$$E(\bar{y}^2) = \text{Var}(\bar{y}) + [E(\bar{y})]^2 = \frac{1-f}{n} S^2 + \bar{Y}^2$$

Where $ f = \frac{n}{N} $.

Step 5: Substitute Back into $ E(s^2) $

$$E(s^2) = \frac{1}{n-1} \left[ n \left( S^2 \left(1 - \frac{1}{N}\right) + \bar{Y}^2 \right) - n \left( \frac{1-f}{n} S^2 + \bar{Y}^2 \right) \right]$$

Simplify the expression:

$$E(s^2) = \frac{1}{n-1} \left[ n S^2 \left(1 - \frac{1}{N}\right) - (1 - f) S^2 \right]$$

$$= \frac{1}{n-1} \left[ n S^2 - \frac{n}{N} S^2 - S^2 + \frac{n}{N} S^2 \right]$$

$$= \frac{1}{n-1} \left[ (n - 1) S^2 \right] = S^2$$

Conclusion

Thus, we have shown that:

$$E(s^2) = S^2$$

Estimation of Population Total $Y_T = N \bar Y = \sum_{i=1}^N Y_i$

1. Point Estimation

$$\hat y_T = N \bar y$$

2. Unbiased Estimator

$$\mathrm E (\hat y_T) = N\cdot \mathrm E (\bar y) = N \bar Y = Y_T$$

3. Variance of Estimation$$Var(\hat y_T) = N^2 Var (\bar y ) = N^2 \frac{1-f}n S^2$$
4. Estimation of Variance $$\hat Var(\hat Y_T) = N^2 \frac{1-f}n s^2$$
5. Confidence Interval

$$\left[\hat y_T \pm z_{\alpha /2 } \sqrt{\widehat {\mathrm{Var}} (\hat y_T)}\right]$$

$$d = z_{\alpha /2 }\sqrt{\widehat {\mathrm{Var}} (\hat y_T)}$$

$$r = \frac{d}{\hat y_T}$$

In srs.r

srs.total=function(N, mysample, alpha)

Estimation of Population Proportion $P$

Define:

• Population Proportion $P = \frac{1}N \sum_{i=1}^N Y_i = \bar Y $
• Population Total $A = \sum Y_i = NP$
• Population Variance $$S^2 = \frac{N}{N-1} P(1-P) =\frac{N}{N-1} PQ \quad \text{where } Q = 1-P$$

Let the observed $y_1 ,\ldots, y_n$ have property with count $a$

1. $\hat p = \bar y = \frac a n$
2. UE
3. Variance of Estimation $$Var(\hat p) = \frac{1-f}n (\frac{N}{N-1}PQ)$$
4. Estimation of Variance $$\hat Var(\hat p ) = \frac{1-f}{n-1}\hat p \hat q$$

In srs.r

srs.prop=function(N=NULL, n, event.num, alpha)

Estimation of Population total $A$

1. $\hat A = N\bar y = N \hat p$
2. UE

3. $$Var(\hat A) = N^2 \frac{1-f}{n} \frac {N}{N-1} PQ$$

4. $$\hat Var(\hat A) = N^2 \frac{1-f}{n} \frac {n}{n-1} \hat p \hat q$$

In srs.r

srs.num=function(N=NULL, n, event.num, alpha)

Determining the Sample size

The sample size is determined by the accuracy needed

$$(V,C,d,r) \implies n_{\min}$$

V: Variance upper bound
C: CV upper bound
d: Error upper bound
r: Relative error upper bound

Sample Size $n_{\min}$ for Estimating Population Mean $\bar Y$

Step 1 Calculate $n_0$
Here $S^2$ and $\bar Y$ are given from historical data.

$$n_0 = \frac{S^2}{V} = \begin{cases} \frac{S^2}V & V=V\\ \frac{S^2}{C^2\bar Y ^2 } & C = \sqrt V/\bar Y \\ \frac{z_{\alpha/2}^2 S^2}{d^2} & d = z_{\alpha /2}\sqrt V\\ \frac{z_{\alpha / 2 } ^2 S^2}{r^2 \bar Y ^2 } & r = z_{\alpha /2} \sqrt V / \bar Y \end{cases}$$

Step 2

$$n_{\min} = \begin{cases}\frac{n_0}{1+ \frac{N_0}{N} } & \text{given reasonable } N \\ n_0 & \text{when $N$ is very big}\end{cases}$$

In srs size.r

size.mean=function(N=NULL, Mean.his=NULL, Var.his, method, bound, alpha)

Sample Size for Estimating Proportion $P$

Here $P$ and $Q = 1 - P$ are given from historical data.

$$n_0 = \frac{PQ}{V} = \begin{cases} \frac{PQ}V \\ \frac{Q}{C^2P} \\\frac{z_{\alpha /2}^2 PQ }{d^2}\\ \frac{z_{\alpha /2}^2 Q }{r^2 P } \end{cases}$$

$$n_{\min} = \begin{cases} \frac{n_0}{1 + \frac{n_0- 1}{N} } & \text{Given } N\\ n_0 & N >> n_0 \end{cases}$$

In srs size.r

size.prop=function(N=NULL, Prop.his, method, bound, alpha)

Sample Size for Estimating Population Total $Y_T$

Use size.mean and adjust inputs
Apply the Sample Number $n_{\min}$ for Estimating Population Mean $\bar Y$ Methods
Bounding Total is the same as bounding $\bar Y$ with different bounds:

$$Var(\hat y_T) \leq V \iff Var(\bar y ) \leq \frac{V}{N^2}$$

$$CV(\hat y_T) \leq C \iff CV(\bar y ) \leq C$$

$$\text{Error}(\hat y_T)\leq d\iff\text{Error} (\bar y)\leq \frac d N$$

$$\text{Absolute Error}(\hat y_T) \leq r \iff\text{Absolute Error}(\bar y) \leq r$$

Stratified Random Sampling

Stratified Random Sampling Formulas

Concept	Population ($Y_{h1}, \ldots, Y_{hN_h}$)	Sample ($y_{h1}, \ldots, y_{hn_h}$)
Size (Size)	$N_h$ ($\sum_{h=1}^L N_h = N$)	$n_h$ ($\sum_{h=1}^L n_h = n$)
Mean	$\overline{Y}_h = \frac{1}{N_h} \sum_{i=1}^{N_h} Y_{hi}$	$\overline{y}_h = \frac{1}{n_h} \sum_{i=1}^{n_h} y_{hi}$
Variance	$S_h^2 = \frac{1}{N_h - 1} \sum_{i=1}^{N_h} (Y_{hi} - \overline{Y}_h)^2$	$s_h^2 = \frac{1}{n_h - 1} \sum_{i=1}^{n_h} (y_{hi} - \overline{y}_h)^2$
Stratum weight	$W_h = \frac{N_h}{N}$	$f_h = \frac{n_h}{N_h}$

Esimation of Population Mean $\bar Y$

1. $$\bar y_{st} = \sum_{h = 1}^ L W_h \bar y_h$$

2. $$E(\bar y_{st}) = \bar Y$$

3. $$Var(\bar y_{st}) = \sum_{h=1}^L W_h^2 \frac{1- f_h }{n_h} S_h^2$$

4. $$\hat Var (\bar y_{st}) =\sum_{h=1}^L W_h^2 \frac{1- f_h }{n_h} s_h^2$$

See stratified mean.r

stra.srs.mean1=function(Nh, nh, yh, s2h, alpha)
stra.srs.mean2=function(Nh, mysample, stra.index, alpha)

Estimation of Population Total $Y_T$

1. Total Estimate:

$$\hat{y}_{st} = N \cdot \overline{y}_{st} = N \left( \sum_{h=1}^{L} W_h \overline{y}_h \right)$$

2. Expected Value of the Estimator:

$$E(\hat{y}_{st}) = \hat{Y}$$

3. Variance of the Estimator:

$$\operatorname{Var}(\hat{y}_{st}) = N^2 \left( \sum_{h=1}^{L} W_h^2 \cdot \frac{1 - f_h}{n_h} \cdot S_h^2 \right)$$

4. Estimated Variance:

$$\widehat{\operatorname{Var}}(\hat{y}_{st}) = N^2 \left( \sum_{h=1}^{L} W_h^2 \cdot \frac{1 - f_h}{n_h} \cdot s_h^2 \right)$$

See stratified mean.r

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Estimation of Proportion

Symbol	Population ($Y_{h1}, \ldots, Y_{hN_h}$)	Sample ($y_{h1}, \ldots, y_{hn_h}$)
Size	$N_h$ ($N = \sum_{h=1}^L N_h$)	$n_h$ ($n = \sum_{h=1}^L n_h$)
Proportion with attribute	$A_h$	$a_h$
Proportion	$P_h = \frac{A_h}{N_h}$	$\hat{p}_h = \frac{a_h}{n_h}$
Variance	$S_h^2 = \frac{N_h}{N_h - 1} P_h Q_h$	$s_h^2 = \frac{n_h}{n_h - 1} \hat{p}_h \hat q_n$
Weight	$W_h = \frac{N_h}{N}$	$f_h = \frac{n_h}{N_h}$

Stratified Sampling Estimation of Population Proportion $P$

1. Estimator for Population Proportion:

$$\hat{p}_{st} = \sum_{h=1}^L W_h \hat{p}_h = \sum_{h=1}^L W_h \cdot \frac{a_h}{n_h}$$

2. Expected Value:

$$E(\hat{p}_{st}) = P$$

3. Variance:

$$Var(\hat{p}_{st}) = \sum_{h=1}^L W_h^2 Var(\hat{p}_h) = \sum_{h=1}^L W_h^2 \left( \frac{1 - f_h}{n_h } \cdot \frac{N_h}{N_h - 1} P_h Q_h \right)$$

4. Estimated Variance:

$$\widehat{Var}(\hat{p}_{st}) = \sum_{h=1}^L W_h^2 \widehat{Var}(\hat{p}_h) = \sum_{h=1}^L W_h^2 \left( \frac{1 - f_h}{n_h } \cdot \frac{n_h}{n_h - 1} \hat{p}_h \hat{q}_h \right)$$

5. Confidence Interval (CI):

$$CI \quad d.r.$$

Stratified Sampling Estimation for Total $A$

1. Estimator for Population Total:

$$\hat{A}_{st} = N \left( \sum_{h=1}^L W_h \hat{p}_h \right) = \sum_{h=1}^L W_h \cdot \frac{a_h}{n_h}$$

2. Expected Value:

$$E(\hat{A}_{st}) = A$$

3. Variance:

$$Var(\hat{A}_{st}) = N^2 \sum_{h=1}^L W_h^2 Var(\hat{p}_h) = \sum_{h=1}^L W_h^2 \left( \frac{1 - f_h}{n_h } \cdot \frac{N_h}{N_h - 1} P_h Q_h \right)$$

4. Estimated Variance:

$$\widehat{Var}(\hat{A}_{st}) = N^2 \sum_{h=1}^L W_h^2 \widehat{Var}(\hat{p}_h) = \sum_{h=1}^L W_h^2 \left( \frac{1 - f_h}{n_h } \cdot \frac{n_h}{n_h - 1} \hat{p}_h \hat q_h \right)$$

5. Confidence Interval (CI):

$$CI \quad d.r.$$

Determining Sample Size

When given $n$, determine $n_h$ for each stratum

Use

strata.weight=function(Wh, S2h, Ch=NULL, allocation)

return(wh)

allocation = "Prop" or "Opt" or "Neyman"

Use

strata.size=function(n, Wh, S2h, Ch=NULL, allocation)

return(list(n=n, allocation=allocation, wh=wh, nh=ceiling(nh)))

The sample size for each stratum, $ n_h $, can be determined using different allocation methods. The general formula is:

$$n_h = W_h \cdot n$$

where $ W_h $ is the stratum weight and $ n $ is the total sample size.

1. Proportional Allocation (Prop):

The stratum weight $ W_h $ is proportional to the stratum size $ N_h $:

$$W_h = \frac{N_h}{N}$$

Thus, the sample size for stratum $ h $ is:

$$n_h = \frac{N_h}{N} \cdot n$$

This method ensures that the sample size in each stratum is proportional to the stratum’s size in the population.

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2. Optimal Allocation (Opt):

The stratum weight $ W_h $ is adjusted based on the stratum’s variability and cost. The formula is:

$$W_h = \frac{\frac{N_h S_h}{\sqrt{c_h}}}{\sum_{h=1}^L \frac{N_h S_h}{\sqrt{c_h}}}$$

Thus, the sample size for stratum $ h $ is:

$$n_h = \frac{\frac{N_h S_h}{\sqrt{c_h}}}{\sum_{h=1}^L \frac{N_h S_h}{\sqrt{c_h}}} \cdot n$$

This method minimizes the variance of the estimator by allocating more samples to strata with higher variability or lower costs.

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3. Neyman Allocation:

The stratum weight $ W_h $ is adjusted based on the stratum’s variability. The formula is:

$$W_h = \frac{\frac{N_h S_h}{\sqrt{c_h}}}{\sum_{h=1}^L \frac{N_h S_h}{\sqrt{c_h}}}$$

If the cost per unit is the same across all strata ($ c_h = c $), this simplifies to:

$$W_h = \frac{\frac{N_h S_h}{\sqrt{c}}}{\sum_{h=1}^L \frac{N_h S_h}{\sqrt{c}}} = \frac{N_h S_h}{\sum_{h=1}^L N_h S_h}$$

Thus, the sample size for stratum $ h $ is:

$$n_h = \frac{N_h S_h}{\sum_{h=1}^L N_h S_h} \cdot n$$

This method minimizes the variance of the estimator by allocating more samples to strata with higher variability.

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Summary

• Proportional Allocation: Simple and easy to implement, but does not account for variability.
• Optimal Allocation: Minimizes variance by considering both variability and cost.
• Neyman Allocation: A special case of optimal allocation when costs are equal across strata.

$$\boxed{ \begin{aligned} &\text{Proportional Allocation: } n_h = \frac{N_h}{N} \cdot n \\ &\text{Optimal Allocation: } n_h = \frac{\frac{N_h S_h}{\sqrt{c_h}}}{\sum_{h=1}^L \frac{N_h S_h}{\sqrt{c_h}}} \cdot n \\ &\text{Neyman Allocation: } n_h = \frac{N_h S_h}{\sum_{h=1}^L N_h S_h} \cdot n \quad (\text{when } c_h = c) \end{aligned} }$$

See stratified size.r

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When given $V,C,d,r$ of $\bar Y$ , determine $n$ and $n_{h}$

Use

strata.mean.size=function(Nh, S2h, Ch=NULL, allocation, method, bound, Ybar=NULL, alpha=NULL)

Step 1 Calculate $w_h$ with different allocation methods.

$$n_h = w_h n$$

$$w_h = \begin{cases}W_h & \text{prop} \\ \frac{Wh S_h /\sqrt {C_h} }{\sum_h W_h Sh/\sqrt {C_h} } & \text{opt} \\ \frac{W_hS_h}{\sum W_h S_h } & \text{Neyman}\end{cases}$$

Step 2
Calculate $n_{\min}$ :

$$n_{\min} = \frac{\sum_h W_h^2 S_h^2/w_h }{V + \frac 1 N \sum_h W_h S_h^2}$$

where

$$V = \begin{cases} V & V\\ C^2\bar Y ^2 & C \\ (d/z_{\alpha/2})^2 & d \\ (r\bar Y / z_{\alpha/2})^2 & r\end{cases}$$

$S_h^2, \bar Y$ are given from historical data.

Step 3 $$n_{h{\min}} = w_h n_{\min}$$

Given $V,C,d,r$ of $P$, determine $n$ and $n_h$

Use

strata.prop.size=function(Nh, Ph, Ch=NULL, allocation, method, bound, Ybar=NULL, alpha=NULL)

Here

$$S_h^2 = \frac{N_h}{N_h - 1} P_h Q_h$$

Given $V,C,d,r$ of Total $Y_T$, determine $n$ and $n_h$

Adjust the input bound parameter of calculating $n$ and $n_h$ of given parameters of $\bar Y$.
Use

strata.mean.size=function(Nh, S2h, Ch=NULL, allocation, method, bound, Ybar=NULL, alpha=NULL)

Population Total $(Y_T)\quad \hat y_T$	V	C	d	r
Population Mean $(\bar Y)\quad \bar y_{st}$	$\frac{V}{N^2}$	C	$\frac d N$	r

Given $V,C,d,r$ of Total $A$, determine $n$ and $n_h$

Adjust the input bound parameter of calculating $n$ and $n_h$ of given parameters of $\bar Y$.
Use

strata.prop.size=function(Nh, Ph, Ch=NULL, allocation, method, bound, Ybar=NULL, alpha=NULL)

Population Total $(A)\quad \hat a$	V	C	d	r
Population Mean $(P)\quad \hat p_{st}$	$\frac{V}{N^2}$	C	$\frac d N$	r

Design Efficiency - Comparison of Sampling Methods

Comparing the variance of your method versus Simple Random Sampling under the same sampling size, the design efficiency is defined as the fraction.

$$\text{Deff} = \frac{Var(\hat \theta_p)}{Var(\hat \theta_{SRS})}$$

Ratio Estimation and Regression Estimation

Notations
For population use UPPER CASE characters and for sample use lower case.

$$\begin{aligned} & S_y^2 = \frac{1}{N-1} \sum_{i=1}^{N} (Y_i - \bar{Y})^2 \quad &\text{($Y$ Total Variance)} \\ & S_x^2 = \frac{1}{N-1} \sum_{i=1}^{N} (X_i - \bar{X})^2 \quad &\text{($X$ Total Variance)} \\ & S_{yx} = \frac{1}{N-1} \sum_{i=1}^{N} (Y_i - \bar{Y})(X_i - \bar{X}) \quad &\text{(Y, X Total Covariance)} \\ & \rho = \frac{S_{yx}}{\sqrt{S_y^2 \cdot S_x^2}} = \frac{S_{yx}}{S_y \cdot S_x} \quad &\text{(Y, X Total Correlation)} \\ & C_y^2 = \frac{S_y^2}{\bar{Y}^2} \quad &\text{(Y Total Relative Variance)}\\ &C_x^2 = \frac{S_x^2}{\bar{X}^2} \quad &\text{(X Total Relative Variance)}\\ &C_{yx} = \rho \cdot \frac{S_y}{\bar{Y}} \cdot \frac{S_x}{\bar{X}} \quad &\text{(Y X Relative Covariance)} \end{aligned}$$

Estimation of Ratio

Ratio is defined as

$$R = \frac{\bar Y}{\bar X} = \frac{Y_T}{X_T}$$

1. Point Estimation$$\hat R = \frac{\bar y}{\bar x}$$
2. AUE $$\lim_{n\to \infty} E(\hat R) = R$$
3. Variance of Estimation

   Proposition :$$ MSE(\hat R) \overset{AUE}{\simeq} Var(\hat R) \overset{n\to \infty}\simeq \frac{1-f}{n\bar X^2} \frac{1}{N-1}\sum_{i=1}^N (Y_i - RX_i)^2$$
   Where: $$\begin{aligned}S_g^2 &\overset{0}{=} \frac{1}{N-1} \sum_{i=1}^{N} (Y_i - R X_i)^2\&\overset{1}{=} S_y^2 + R^2 S_x^2 - 2R S_{yx}\&\overset{2}{=} \bar Y^2 (C_y^2 + C_x^2 - 2C_{yx})\end{aligned}$$

4. Estimation of Variance
   Method 1 When $\bar X$ is given

$$\begin{aligned} \widehat{\text{Var}}_1(\hat{R}) &\overset{0}{=} \frac{1-f}{n} \cdot \frac{1}{\bar{X}^2} \cdot \frac{1}{n-1} \sum_{i=1}^{n} (y_i - \hat{R} x_i)^2\\ &\overset{1}{=} \frac{1-f}{n} \cdot \frac{1}{\bar{X}^2} \cdot (S_y^2 + \hat{R}^2 S_x^2 - 2 \hat{R} S_{yx})\end{aligned}$$

**Method 2** When $\bar X$ is unknown, we use $\bar x$ from the sample

$$\begin{aligned} \widehat{\text{Var}}_2(\hat{R}) &\overset{0}{=} \frac{1-f}{n} \cdot \frac{1}{\bar{X}^2} \cdot \frac{1}{n-1} \sum_{i=1}^{n} (y_i - \hat{R} x_i)^2\\ &\overset{1}{=} \frac{1-f}{n} \cdot \frac{1}{\bar{X}} \cdot (S_y^2 + \hat{R}^2 S_x^2 - 2 \hat{R} S_{yx})\end{aligned}$$

Note: When $\bar X$ is given, we can use both methods 1 and 2. When $\bar X$ is unknown, use method 2.

5. Confidence Interval
   CI1, CI2, CI3

placeholder for confidence interval

Use ratio.r

ratio = function(y.sample, x.sample, N=NULL, auxiliary=FALSE, Xbar=NULL, alpha)
#when auxiliary = false , Xbar =null ; when auxiliary = true, Xbar = Xbar

Ratio Estimation of Population Mean $\bar Y$ and Total $Y_T$

SRSF (Simple Random Sampling with Fixed Ratio Estimation) of Population Mean $\bar Y$

1. Estimator for the Population Mean:

$$\bar{y}_R = \frac{\bar{y}}{\bar{x}} \cdot \bar{X} = \hat{R} \cdot \bar{X}$$

2. Expected Value of the Estimator:

$$E(\bar{y}_R) = E(\hat{R}) \cdot \bar{X} \approx R \cdot \bar{X} = \bar{Y} \quad (\text{AUE})$$

3. Variance of the Estimator:

$$\text{Var}(\bar{y}_R) = \bar{X}^2 \cdot \text{Var}(\hat{R})$$

4. Estimated Variance of the Estimator:

$$\widehat{\text{Var}}(\bar{y}_R) = \bar{X}^2 \cdot \widehat{\text{Var}}_1(\hat{R})$$

5. Confidence Interval:

$$\text{CI} = \left[ \bar{X} \cdot \text{left}, \, \bar{X} \cdot \text{right} \right]$$

In ratio.r use:

ratio.mean=function(y.sample, x.sample, N=NULL, Xbar, alpha)

Example:

mean.simple.result=srs.mean(N, y.sample, alpha)

mean.ratio.result=ratio.mean(y.sample, x.sample, N, Xbar, alpha)

var.result=c(mean.simple.result$ybar.var, mean.ratio.result$ybarR.var)
deff.result=deff(var.result)

rownames(deff.result)=c("Simple", "Ratio")
print(deff.result)

SRSF Estimation of Population Total $ Y_T $

1. Estimator for the Population Total:

$$\hat{Y}_R = N \cdot \bar{y}_R$$

2. Approximately Unbiased Estimator (AUE):

$$E(\hat{Y}_R) \approx Y_T$$

3. Variance of the Estimator:

$$\text{Var}(\hat{Y}_R) = N^2 \cdot \text{Var}(\bar{y}_R)$$

4. Estimated Variance of the Estimator:

$$\widehat{\text{Var}}(\hat{Y}_R) = N^2 \cdot \widehat{\text{Var}}(\bar{y}_R)$$

5. Confidence Interval:

$$\text{CI} = \left[ N \cdot \text{left}, \, N \cdot \text{right} \right]$$

use:

ratio.total=function(y.sample, x.sample, N, Xbar, alpha)

Example

total.simple.result=srs.total(N, y.sample, alpha)

total.ratio.result=ratio.total(y.sample, x.sample, N, Xbar, alpha)

var.result=c(total.simple.result$ytot.var, total.ratio.result$ytotal.var)
deff.result=deff(var.result)
rownames(deff.result)=c("Simple", "Ratio")
print(deff.result)

Design Efficiency

Ratio and Regression Estimation are called complex estimation methods, while Simple Random Sampling is called simple estimation method. When comparing complex methods to simple methods, design efficiency is defined as the fraction.

$$\text{Deff} = \frac{Var(\bar y_R)}{Var(\bar y)} = \begin{cases} <1 & \bar y_R \text{ is more efficient}\\ \geq 1 & \bar y \text{ is more efficient} \end{cases}$$

When $$\rho > \frac{C_x}{2C_y}$$
$\bar y_R$ if more efficient then $\bar y$.
When $Y$ and $X$ are highly correlated, $\bar y_R$ is more efficient than $\bar y$.

Determining Sample Size

Step 1
When given bound $(V,C,d,r)$ of $\bar Y$, determine the simple sample size $n_{\text{simple}}$
Using the function size.mean
Step 2
Determine the ratio sample size $n_{R}$

$$n_R = \text{Deff} \cdot n_{\text{simple} }$$

Use deff=function(var.result) to calculate the design efficiency and use deff.size=function(deff.result, n.simple) to calculate the size.

Example

mean.simple.result=srs.mean(N, y.sample, alpha)
mean.ratio.result=ratio.mean(y.sample, x.sample, N, Xbar, alpha)

var.result=c(mean.simple.result$ybar.var, mean.ratio.result$ybarR.var)
deff.result=deff(var.result)

n.simple=size.mean(N, Mean.his=NULL, Var.his=var(y.sample), method="d", bound=0.05, alpha)$size
size.result=deff.size(deff.result, n.simple)

rownames(size.result)=c("Simple", "Ratio")
print(size.result)

Regression Estimation of Population Mean $\bar Y$ and Total $Y_T$

The Linear Regression Estimator is defined as

$$\bar y_{lr} = \bar y + \beta(\bar X - \bar x)$$

$$\hat Y_{lr} = N \bar y_{lr}$$

Normally $\beta$ is either constant or the regression coefficient $B$ of $X$ on $Y$.

When $\beta = 1$, we obtain the difference estimator Diff $$\bar y_d = \bar y + (\bar X - \bar x)$$

When $\beta = 0$, it degenerates to the simple estimator $\bar y$.
When $\beta = {\bar y}/{\bar x} = \hat R$, we obtain the ratio estimator $\bar y_R$.

Regression Estimation of Population Mean $\bar Y$

Case 1: $\beta = \beta_0$ is constant

1. Estimator for the Population Mean

$$\bar{y}_{lr}(\beta_0) = \bar{y} + \beta_0 (\bar{X} - \bar{x})$$

2. Unbiased Estimator

$$E(\bar{y}_{lr}) = \bar{Y} + \beta_0 (\bar{X} - E(\bar{x})) = \bar{Y} \quad \text{(UE)}$$

3. Variance of the Estimator

$$\text{Var}(\bar{y}_{lr}) = \frac{1-f}{n} \left( S_y^2 + \beta_0^2 S_x^2 - 2 \beta_0 S_{yx} \right)$$

Minimum Variance Condition

$$\text{Minimum when } \beta_0 = B = \frac{S_{yx}}{S_x^2}\Rightarrow \text{Var}_{\text{min}} = \frac{1-f}{n} S_e^2$$

Here $B$ is the population regression coefficient of $X$ on $Y$. $$B = \frac{S_{yx}}{S_x^2}$$

$$\text{Var}_{\text{min}}(\bar y_{lr}) = \frac{1-f}{n} S_y^2(1-\rho^2)$$

$$S_e^2 \triangleq S_y^2(1-\rho^2),\qquad \rho = \frac{S_{yx}}{Sy S_x}$$

4. Estimated Variance of the Estimator

$$\widehat{\text{Var}}(\bar{y}_{lr}) = \frac{1-f}{n} \left( s_y^2 + \beta_0^2 s_x^2 - 2 \beta_0 s_{yx} \right)$$

5. Confidence Interval

$$\left[ \bar{y}_{lr} \pm z_\alpha \sqrt{\widehat{\text{Var}}(\bar{y}_{lr})} \right]$$

---

Case 2: $\beta = \hat b$ is the sample regression coefficient of $x$ and $y$

$$\beta = \hat b = \frac{s_{yx}}{s_x^2}$$

1. Estimator for the Population Mean

$$\bar{y}_{lr} = \bar{y} + \hat{b} (\bar{X} - \bar{x})$$

2. Approximate Unbiased Estimator

$$E(\bar{y}_{lr}) \approx \bar{Y} \quad \text{(AUE)}$$

3. Mean Squared Error (MSE) and Variance

$$\text{MSE}(\bar{y}_{lr}) \approx \text{Var}(\bar{y}_{lr}) \approx \frac{1-f}{n} S_e^2$$

This is the theoretically minimum variance estimator.

4. Estimated Variance of the Estimator

$$\widehat{\text{Var}}(\bar{y}_{lr}) = \frac{1-f}{n} s_e^2 = \frac{1-f}{n} \cdot \frac{n-1}{n-2} \left( s_y^2 - \frac{s_{yx}^2}{s_x^2} \right)$$

where

$$s_e^2 = \frac{n-1}{n-2} \left( s_y^2 - \frac{s_{yx}^2}{s_x^2} \right)$$

In regression.r :

regression.mean=function(y.sample, x.sample, N=NULL, Xbar, alpha, method="Min", beta0=NULL)

Regression Estimation of Population Total $Y_T$

Notice that $$\text{mean }\quad \bar y_{lr} \overset{N}{\longrightarrow} \hat y_{lr} \quad \text{total}$$

regression.total=function(y.sample, x.sample, N=NULL, Xbar, alpha, method="Min", beta0=NULL)

Comparison of Simple, Ratio, and Regression Estimation

Their corresponding variances are:

$$\begin{aligned} \text{Var}(\bar y) &= \frac{1-f}{n} \cdot S_y^2 \\ \text{Var}(\bar y_R) &\approx \frac{1-f}{n} \cdot (S_y^2 + R^2 S_x^2 - 2 R\rho S_yS_x) \\ \text{Var}(\bar y_{lr}) &\approx \frac{1-f}{n} \cdot S_y^2 (1-\rho^2) \end{aligned}$$

The condition for the regression estimator to be more efficient than the ratio estimator is:

$$(B-R)^2 \geq 0$$

When $n$ is not large, the estimations might be biased. Real-life experiments show that when $n$ is small, the regression estimator can be more biased than the ratio estimator.

Example: Comparing Simple, Ratio, and Regression Estimation of Population Total $Y_T$

total.simple.reult=srs.total(N, y.sample, alpha)
print(total.simple.result)

total.ratio.result=ratio.total(y.sample, x.sample, N, Xbar, alpha)
print(total.ratio.result)

total.reg.result=regression.total(y.sample, x.sample, N, Xbar, alpha, method="Min", beta0=NULL)
print(total.reg.result)

var.result=c(total.simple.reult$ytot.var, total.ratio.result$ytotal.var, total.reg.result$Var)
deff.result=deff(var.result)
rownames(deff.result)=c("Simple", "Ratio", "Regression")
print(deff.result)

Determining Sample Size

The design efficiency is defined as the fraction.

$$\text{Deff} = \frac{Var(\bar y_{lr})}{Var(\bar y)}$$

Given the bound $(V,C,d,r)$ of $\bar y$, determine the simple sample size $n_{\text{simple}}$, then

$$n_{lr} = \text{Deff} \cdot n_{\text{simple}}$$

Which is similar to ratio estimation.

Stratified Ratio and Regression Estimation

Two approaches for stratified estimation.

1. Separated Estimation First estimate for each stratum, then take weighted average or sum.
2. Combined Estimation First take the weighted average or sum, then estimate for the combined sample.

Stratified Ratio Estimation

For the $ h $-th stratum ($ h = 1, \dots, L $):

Notations	Population $\qquad \overset{SRS}{\longrightarrow}$	Sample
	$$\begin{pmatrix}Y_{h1} & \cdots & Y_{hN_h} \X_{h1} & \cdots & X_{hN_h}\end{pmatrix}$$	$$\begin{pmatrix}y_{h1} & \cdots & y_{hn_h} \x_{h1} & \cdots & x_{hn_h}\end{pmatrix}$$
Mean	$$\bar{Y}_h\quad \bar X_h$$	$$\bar y_h \quad \bar{x}_h$$
Var, Cov, $\rho$	$$S_{yh}^2, S_{xh}^2, S_{yxh}, \rho_h$$	$$s_{yh}^2, s_{xh}^2, s_{yxh}, \hat \rho_h$$
Seperate Ratio Estimation for each stratum	$$R_h = \frac{\bar{Y}_h}{\bar{X}_h}$$	$$\hat R_h = \frac{\bar y_h}{\bar x_h}$$
Combined Ratio Estimation	$$R_c = \frac{\bar{Y}}{\bar{X}}$$	$$\hat R_c = \frac{\bar y_{st}}{\bar x_{st}}$$

Separate Ratio Estimation of Population Mean $\bar Y$›

1. Estimator for the Population Mean

$$\bar{y}_{RS} = \sum_h W_h \bar{y}_{Rh} = \sum_h W_h \left( \frac{\bar{y}_h}{\bar{x}_h} \cdot \bar{X}_h\right)$$

Notice that $$\bar y_{Rh} = \frac{\bar{y}_h}{\bar{x}_h} \cdot \bar{X}_h$$ is the ratio estimator of the $ h $-th stratum.

2. Approximate Unbiasedness

$$E(\bar{y}_{RS}) \approx \bar{Y} \quad (\text{AUE})$$

3. Variance of the Estimator

$$\text{Var}(\bar{y}_{RS}) \approx \sum_h W_h^2 \frac{1-f_h}{n_h} \left( S_{y_h}^2 + R_h^2 S_{x_h}^2 - 2R_h S_{yxh} \right)$$

4. Estimated Variance of the Estimator

$$\widehat{\text{Var}}(\bar{y}_{RS}) \approx \sum_h W_h^2 \frac{1-f_h}{n_h} \left( s_{y_h}^2 + \hat{R}_h^2 s_{x_h}^2 - 2\hat{R}_h s_{yxh} \right)$$

where $ \hat{R}_h = \frac{\bar{y}_h}{\bar{x}_h} $

In stra ratio.r

separate.ratio.mean=function(Nh, y.sample, x.sample, stra.index, Xbarh, alpha)

Combined Ratio Estimation of Population Mean $\bar Y$

1. Estimator for the Population Mean

$$\bar{y}_{RC} = \frac{\bar{y}_{st}}{\bar{x}_{st}} \cdot \bar{X} = \hat{R}_c \cdot \bar{X}$$

2. Approximate Unbiasedness

$$E(\bar{y}_{RC}) \approx \bar{Y} \quad (\text{AUE})$$

3. Variance of the Estimator

$$\text{Var}(\bar{y}_{RC}) = \sum_h W_h^2 \frac{1-f_h}{n_h} \left( S_{y_h}^2 + R_h^2 S_{x_h}^2 - 2R_h S_{yx_h} \right)$$

4. Estimated Variance of the Estimator

$$\widehat{\text{Var}}(\bar{y}_{RC}) = \sum_h W_h^2 \frac{1-f_h}{n_h} \left( s_{y_h}^2 + \hat{R}_c^2 s_{x_h}^2 - 2\hat{R}_c s_{yx_h} \right)$$

where:

$$\hat{R}_c = \frac{\bar{y}_{st}}{\bar{x}_{st}}$$

In stra ratio.r

combined.ratio.mean=function(Nh, y.sample, x.sample, stra.index, Xbar, alpha)

Stratified Regression Estimation

Separate Regression Estimation of Population Mean $\bar Y$

Case I: When $\beta_h$ is constant

1. Estimator for the Population Mean

$$\bar{y}_{lrS} = \sum_hW_h \bar y_{lrh }=\sum_h W_h \left( \bar{y}_h + \beta_h (\bar{X}_h - \bar{x}_h) \right)$$

Notice that $$\bar y_{lrh} = \bar{y}_h + \beta_h (\bar{X}_h - \bar{x}_h)$$ is the regression estimator of the $ h $-th stratum.

2. Unbiasedness

$$E(\bar{y}_{lrS}) = \bar{Y} \quad (\text{UE})$$

3. Variance of the Estimator

$$\text{Var}(\bar{y}_{lrS}) = \sum_h W_h^2 \frac{1-f_h}{n_h} \left( S_{y_h}^2 + \beta_h^2 S_{x_h}^2 - 2\beta_h S_{yx_h} \right)$$

Minimum Variance Condition
When $ \beta_h = B_h = \frac{S_{yx_h}}{S_{x_h}^2} $:

$$\text{Var}_{\text{min}} = \sum_h W_h^2 \frac{1-f_h}{n_h} S_{eh}^2$$

where:

$$S_{eh}^2 = S_{y_h}^2 (1 - \rho_h^2)$$

4. Estimated Variance of the Estimator

$$\widehat{\text{Var}}(\bar{y}_{lrS}) = \sum_h W_h^2 \frac{1-f_h}{n_h} \left( s_{y_h}^2 + \hat{\beta}_h^2 s_{x_h}^2 - 2\hat{\beta}_h s_{yx_h} \right)$$

---

Case II: When $ \beta_h = \hat{b}h = \frac{S{yx_h} }{S_{x_h}^2} $ (Regression Coefficient)

1. Estimator for the Population Mean

$$\bar{y}_{lrS} = \sum_h W_h \left( \bar{y}_h + \hat{b}_h (\bar{X}_h - \bar{x}_h) \right)$$

2. Asymptotically Unbiased Estimator

$$E(\bar{y}_{lrS}) \approx \bar{Y} \quad (\text{AUE})$$

3. Variance of the Estimator

$$\text{Var}(\bar{y}_{lrS}) \approx \sum_h W_h^2 \frac{1-f_h}{n_h} S_{y_h}^2 (1 - \rho_h^2)$$

4. Estimated Variance of the Estimator

$$\widehat{\text{Var}}(\bar{y}_{lrS}) \approx \sum_h W_h^2 \frac{1-f_h}{n_h} \frac{n_h-1}{n_h-2} \left( s_{y_h}^2 - \frac{s_{yx_h}}{s_{x_h}^2} \right)^2$$

In stra regression.r

seperate.regression.mean=function(Nh, y.sample, x.sample, stra.index, Xbarh, alpha, method = "Min", beta0 = NULL)

Combined Regression Estimation of Population Mean $\bar Y$

Case I: When $\beta$ is constant

1. Estimator for the Population Mean

$$\bar{y}_{lrC} = \bar{y}_{st} + \beta (\bar{X} - \bar{x}_{st})$$

2. Unbiasedness

$$E(\bar{y}_{lrC}) = \bar{Y} \quad (\text{UE})$$

3. Variance of the Estimator

$$\text{Var}(\bar{y}_{lrC}) = \sum_h W_h^2 \frac{1-f_h}{n_h} \left( S_{yh}^2 + \beta^2 S_{xh}^2 - 2\beta S_{yxh} \right)$$

Minimum Variance Condition
When $$ \beta = B_c = \frac{\sum_h W_h^2 \frac{1-f_h}{n_h} S_{yxh}}{\sum_h W_h^2 \frac{1-f_h}{n_h} S_{xh}^2} $$

The Variance achieves its minimum.

5. Estimated Variance of the Estimator

$$\widehat{\text{Var}}(\bar{y}_{lrC}) = \sum_h W_h^2 \frac{1-f_h}{n_h} \left( s_{yh}^2 + \hat{\beta}^2 s_{xh}^2 - 2\hat{\beta} s_{yxh} \right)$$

---

Case II

When $$ \beta = \hat{b}c = \frac{\sum_h W_h^2 \frac{1-f_h}{n_h} S{yx_h}}{\sum_h W_h^2 \frac{1-f_h}{n_h} S_{x_h}^2} $$

1. Estimator for the Population Mean

$$\bar{y}_{lrC} = \bar{y}_{st} + \hat{b}_c (\bar{X} - \bar{x}_{st})$$

2. Approximate Unbiasedness

$$E(\bar{y}_{lrC}) \approx \bar{Y} \quad (\text{AUE})$$

3. Variance of the Estimator

$$\text{Var}(\bar{y}_{lrC}) \approx \sum_h W_h^2 \frac{1-f_h}{n_h} \left( S_{y_h}^2 + B_c^2 S_{x_h}^2 - 2B_c S_{yx_h} \right)$$

4. Estimated Variance of the Estimator

$$\widehat{\text{Var}}(\bar{y}_{lrC}) \approx \sum_h W_h^2 \frac{1-f_h}{n_h} \left( s_{y_h}^2 + \hat{b}_c^2 s_{x_h}^2 - 2\hat{b}_c s_{yx_h} \right)$$

In stra regression.r

combined.regression.mean = function(Nh, y.sample, x.sample, stra.index, Xbar, alpha, method = "Min", beta0 = NULL)

Estimation of Population Total $Y_T$

Notice that $$\text{mean }\quad \bar Y\overset{N}{\longrightarrow} Y_T \quad \text{total}$$

In stra ratio.r

separate.ratio.total=function(Nh, y.sample, x.sample, stra.index, Xbarh, alpha)

combined.ratio.total=function(Nh, y.sample, x.sample, stra.index, Xbar, alpha)

In stra regression.r

seperate.regression.total = function(Nh, y.sample, x.sample, stra.index, Xbarh, alpha, method = "Min", beta0 = NULL)

combined.regression.total = function(Nh, y.sample, x.sample, stra.index, Xbar, alpha, method = "Min", beta0 = NULL)

Determining Sample Size

$$\text{Deff} = \frac{Var(\bar y_{\text{prop}})}{Var(\bar y_{\text{st}})} \implies n_{\text{prop}} = \text{Deff} \cdot n_{\bar y_{st}}$$

Where given bound $(V,C,d,r)$ of $\bar y_{st}$, calculate sample size $n_{\bar y_{st}}$
$\bar y _{\text{prop}}$ is estimated by the methods (RS, RC, lrS, lrC).

Double Sampling

Double Sampling or Two-phase Sampling is a method with two phases.

1. First, sample from the population to obtain a big sample to obtain auxiliary information. In this course, the first-phase sampling is always SRS.
2. Second, sample with a small size. In this book, the second-phase is always sampled from the first-phase.

Process

Population:

$$Y_1, \dots, Y_N$$

Step 1:

• Sampling Method: SRS (Simple Random Sampling)
• Sample Drawn:

$$y_1', \dots, y_{n'}' \quad \text{(First Sample)}$$

Step 2:

• Second Sample:

$$y_1, \dots, y_n \quad \text{(Second Sample)}$$

Estimation:

• Estimator:

$$\hat{\theta} = \hat{\theta}(y_1, \dots, y_n)$$

Expectation and Variance Decomposition

Expectation of the Estimator

$$\begin{aligned} E(\hat{\theta}) &= E\left( \hat{\theta}(y_1, \dots, y_n) \right)\\ &= E_1 \left[ E_2 \left( \hat{\theta} (y_1,\ldots,y_n) \big| y_1', \dots, y_{n'}' \right) \right]\\ &= E_1 \left[ E_2 (\hat \theta)\right]\end{aligned}$$

Variance of the Estimator

The variance of the estimator $ \hat{\theta} $ can be decomposed as:

$$\text{Var}(\hat{\theta}) = \text{Var}_1 \left( E_2 \left( \hat{\theta} \mid y_1', \dots, y_{n'}' \right) \right) + E_1 \left( \text{Var}_2 \left( \hat{\theta} \mid y_1', \dots, y_{n'}' \right) \right)$$

Double Stratified Sampling

Sampling Process

Step 1

SRS sample from the population to obtain the first-phase samples. For known $N$ and given $n'$:

$$(Y_1,\ldots,Y_N) \overset{SRS}{\longrightarrow} (y_1',\ldots,y_{n'}' )$$

Step 2

Stratify the first-phase samples $(y_1',\ldots,y_{n'}' )$ into $L$ strata. The unit count for stratum $h$ is $n_h'$.
The samples are: $(y_{n_1'}',\ldots, y_{n_h'}'),\quad h = 1,\ldots, L$

Step 3

Estimate the stratum weight of stratum $h$, since $W_h = \frac{N_h}{N}$ is unkown.
Using samples from the first-phase, we have:

$$w_h' = \frac{n_h'}{n'},\quad h = 1,\ldots ,L$$

Step 4

Perform a stratified sampling from the first-phase samples $(y_1',\ldots,y_{n'}' )$ to obtain the second-phase samples.

$$(y_{n_1'}',\ldots, y_{n_h'}') \longrightarrow (y_{n_1},\ldots, y_{n_{n_h}})$$

---

Two-Phase Sampling Formulas

1. Second-phase sampling proportion:

$$v_h = \frac{n_h}{n_h'}$$

• $ n_h $: Size of the second-phase sample.
• $ n_h’ $: Size of the first-phase sample.

2. Second-phase Sample Mean for $h$-th stratum ($ \bar{y}_h $):

$$\bar{y}_h = \frac{1}{n_h} \sum_{j=1}^{n_h} y_{hj}$$

• $ y_{hj} $: Value of the target variable $ y $ for the $ j $-th unit in the second-phase sample of stratum $ h $.

3. Second-phase Variance for $h$-th stratum ($ S_h^2 $):

$$S_h^2 = \frac{1}{n_h - 1} \sum_{j=1}^{n_h} (y_{hj} - \bar{y}_h)^2$$

• $ \bar{y}_h $: Sample mean of the target variable $ y $ for the second-phase sample of stratum $ h $.

---

Double Stratified Sampling Estimation of Population Mean $ \bar{Y}$

1. Estimator for the Population Mean:

$$\bar{y}_{stD} = \sum_{h=1}^{L} w_h' \cdot \bar{y}_h$$

2. Unbiased Estimation:

$$E(\bar{y}_{stD}) = \bar{Y} \quad (\text{UE})$$

3. Variance of the Estimator:

$$\text{Var}(\bar{y}_{stD}) = \left( \frac{1}{n'} - \frac{1}{N} \right) S^2 + \sum_{h=1}^{L} \frac{1}{n_h'} (v_h' - 1) w_h' S_h^2$$

4. Estimated Variance of the Estimator:

$$\widehat{\text{Var}}(\bar{y}_{stD}) = \sum_{h=1}^{L} \left( \frac{1}{n_h} - \frac{1}{n_h'} \right) w_h' s_h^2 + \left( \frac{1}{n'} - \frac{1}{N} \right) \sum_{h=1}^{L} w_h' \left( \bar{y}_h - \bar{y}_{stD} \right)^2$$

In two phase stra.r

twophase.stra.mean1=function(N=NULL, nh.1st, nh.2nd, ybarh, s2h, alpha)

twophase.stra.total1=function(N, nh.1st, nh.2nd, ybarh, s2h, alpha)

Double Ratio Estimation and Double Regression Estimation

Sampling Process

$Y$ is the target property and $X$ is the auxiliary property.

Step 1

SRS sample from the population to obtain the first-phase samples. For known $N$ and given $n'$:

$$\begin{pmatrix}Y_1 &\cdots &Y_N\\X_1 & \cdots &X_N \end{pmatrix} \overset{SRS}{\longrightarrow} \begin{pmatrix}y_1' &\cdots &y_{n'}'\\x_1 & \cdots &x_{n'}' \end{pmatrix}$$

Step 2

Since the auxiliary information $\bar X$ is unknown, use the first-phase samples to estimate $\bar X$:

$$\bar X = \frac{1}{n'} \sum_{j=1}^{n'} x_{j}'$$

Step 3

SRS from the first-phase samples to obtain the second-phase samples:

$$\begin{pmatrix}y_1' &\cdots &y_{n'}'\\x_1 & \cdots &x_{n'}' \end{pmatrix} \overset{SRS}{\longrightarrow} \begin{pmatrix}y_1 &\cdots &y_{n}\\x_1 & \cdots &x_{n} \end{pmatrix}$$

Notations for second-phase samples:

$$\bar y , \bar x, s_y^2, s_x^2, s_{yx}$$

Double Ratio Estimation of Population Mean $ \bar{Y}$

1. Estimator for the Population Mean:

$$\bar{y}_{RD} = \hat{R} \cdot \bar{x}' = \frac{\bar{y}'}{\bar{x}'} \cdot \bar{x}'$$

2. Asymptotically Unbiased Estimation:

$$E(\bar{y}_{RD}) {\approx} \bar{Y}\quad \text{AUE}$$

3. Variance of the Estimator:

$$\text{Var}(\bar{y}_{RD}) = \left( \frac{1}{n'} - \frac{1}{N} \right) S_y^2 + \left( \frac{1}{n} - \frac{1}{n'} \right) (S_y^2 + R^2 S_x^2 - 2RS_{yx})$$

4. Estimated Variance of the Estimator:

$$\widehat{\text{Var}}(\bar{y}_{RD}) = \left( \frac{1}{n'} - \frac{1}{N} \right) s_y^2 + \left( \frac{1}{n} - \frac{1}{n'} \right) \left( s_y^2 + \hat{R}^2 s_x^2 - 2\hat{R}s_{yx} \right)$$

In twophase ratio.r

twophase.ratio.mean=function(N=NULL, n.1st, xbar.1st, y.sample, x.sample, alpha)

twophase.ratio.total=function(N, n.1st, xbar.1st, y.sample, x.sample, alpha)

Double Regression Estimation of Population Mean $ \bar{Y}$

Case 1: When $ \beta $ is a Constant ($ \beta = \beta_0 $, i.e., $ \beta = 1 $)

1. Estimator for the Population Mean:

$$\bar{y}_{lrD} = \bar{y} + \beta (\bar{x}' - \bar{x})$$

2. Unbiasedness:

$$E(\bar{y}_{lrD}(\beta_0)) = \bar{Y} \quad (\text{UE})$$

3. Variance of the Estimator:

$$\text{Var}(\bar{y}_{lrD}(\beta_0)) = \left( \frac{1}{n'} - \frac{1}{N} \right) S_y^2 + \left( \frac{1}{n} - \frac{1}{n'} \right) \left( S_y^2 + \beta_0^2 S_x^2 - 2\beta_0 S_{yx} \right)$$

4. Estimated Variance of the Estimator:

$$\widehat{\text{Var}}(\bar{y}_{lrD}(\beta_0)) = \left( \frac{1}{n'} - \frac{1}{N} \right) s_y^2 + \left( \frac{1}{n} - \frac{1}{n'} \right) \left( s_y^2 + {\beta}_0^2 s_x^2 - 2{\beta}_0 s_{yx} \right)$$

Here is the Markdown representation of the given mathematical expressions:

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Case II: When $ \beta$ is the regression coefficient of the second-phase sample

$$\beta = \hat{b} = \frac{S_{yx}}{S_x^2}$$

1. Estimator for the Population Mean

$$\bar{y}_{lrD} = \bar{y} + \hat{b} (\bar{x}' - \bar{x})$$

2. Asymptotically Unbiased Estimation

$$E(\bar{y}_{lrD}) \approx \bar{Y} \quad (\text{AUE})$$

3. Variance of the Estimator

$$\text{Var}(\bar{y}_{lrD}) = \left( \frac{1}{n'} - \frac{1}{N} \right) S_y^2 + \left( \frac{1}{n} - \frac{1}{n'} \right) S_y^2 (1 - \rho^2)$$

4. Estimated Variance of the Estimator

$$\widehat{\text{Var}}(\bar{y}_{lrD}) = \left( \frac{1}{n'} - \frac{1}{N} \right) s_y^2 + \left( \frac{1}{n} - \frac{1}{n'} \right) s_e^2$$

where:

$$s_e^2 = \frac{n-1}{n-2} \left( s_y^2 - \frac{s_{yx}^2}{s_x^2} \right)$$

In twophase regression.r

twophase.regression.mean=function(N=NULL, n.1st, xbar.1st, y.sample, x.sample, alpha, beta0=NULL)

twophase.regression.total=function(N=NULL, n.1st, xbar.1st, y.sample, x.sample, alpha, beta0=NULL)

Cluster Sampling

The population is formed by clusters. Cluster Sampling is to sample clusters and examine all the smaller units within the clusters.

Cluseter Sampling Estimation of Population Mean $ \bar{Y}$

Sampling Process

Population is formed by $N$ clusters:

$$\boxed{Y_{11},\ldots ,Y_{1M_1}}_1\quad \cdots\quad \boxed{Y_{i1},\ldots ,Y_{iM_i}}_i\quad \cdots\quad \boxed{Y_{N1},\ldots ,Y_{NM_N}}_N$$

SRS from the cluster indices:

$$(1,\ldots,N) \overset{SRS}{\longrightarrow} (1,\ldots,n)$$

We obtain the samples:

$$\boxed{y_{11},\ldots,y_{1m_1}}_1\quad \cdots\quad \boxed{y_{i1},\ldots,y_{im_i}}_i\quad \cdots\quad \boxed{y_{n1},\ldots,y_{nm_n}}_n$$

For a given $n$, the sample rate: $$f = \frac n N$$

Clusters with the same size $(M_i = M = m_i)$

Notations

UPPER CASE: population; lower case: sample.

$$\begin{aligned} \overline{Y}_{i} &= \frac{1}{M} \sum_{j=1}^{M} Y_{ij} & \text{(cluster mean)} \\ \overline{\overline{Y}} &= \frac{1}{MN} \left( \sum_{i=1}^{N} \sum_{j=1}^{M} Y_{ij} \right) & \text{(unit mean)} \\ \overline{Y} &= \frac{1}{N} \left( \sum_{i=1}^{N} \frac{1}{M} \sum_{j=1}^{M} Y_{ij} \right) & \text{(mean by cluster)}\\ S_{w}^{2} &= \frac{1}{N(M-1)} \sum_{i=1}^{N} \sum_{j=1}^{M} \left( Y_{ij} - \overline{Y}_{i} \right)^{2} & \text{(Within-cluster variance)} \\ S^{2} &= \frac{1}{NM-1} \sum_{i=1}^{N} \sum_{j=1}^{M} \left( Y_{ij} - \overline{Y} \right)^{2} & \text{(Total variance)} \\ S_{b}^{2} &= \frac{M}{N-1} \sum_{i=1}^{N} \left( \overline{Y}_{i} - \bar{Y} \right)^{2} & \text{(Between-cluster variance)} \end{aligned}$$

$\bar y_i , \bar{\bar{y}}, \bar y , s_w^2, s^2,S_b^2$ can be defined similiarly.

Proposition:

• Decomposition of population variance $S^2$:

$$\begin{aligned}S^2 &= \frac{1}{NM-1} \left[(N-1)S_b^2 + N(M-1) S_2^2\right]\\&=\frac{N-1}{NM-1}S_b^2 + \frac{N(M-1)}{NM-1}S_w^2\end{aligned}$$

• Decomposition of sample variance $s^2$:

$$\begin{aligned}s^2 &=\frac{n-1}{nM-1}s_b^2 + \frac{n(M-1)}{nM-1}s_w^2\end{aligned}$$

Estimation of the unit mean $\overline{\overline{Y} }$

1. Estimation

$$\overline{\overline{y}} = \frac{1}{nM} \sum_{i=1}^n \sum_{j=1}^M y_{ij} = \left( \frac{1}{M} \right) \bar{y}$$

$$\left( = \frac{1}{n} \sum_{i=1}^n \bar{y}_i = \text{mean}(\bar{y}_1, \ldots, \bar{y}_n) \right)$$

2. Unbiased

$$E(\overline{\overline{y}}) = \overline{\overline{Y}} \quad (\text{UE})$$

3. Variance of Estimation

$$\text{Var}(\overline{\overline{y}}) = \frac{1-f}{nM} S_b^2$$

4. Estimation of Variance

$$\widehat{\text{Var}}(\overline{\overline{y}}) = \frac{1-f}{nM} S_b^2$$

In cluster srs.r

cluster.srs.mean = function(N, M.ith, ybar.ith, s2.ith, alpha)

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Estimation of Population Variance $S^2$

Proposition
Recall:

$$\begin{aligned}S^2 &=\frac{N-1}{NM-1}S_b^2 + \frac{N(M-1)}{NM-1}S_w^2\end{aligned}$$

$$\begin{aligned}s^2 &=\frac{n-1}{nM-1}s_b^2 + \frac{n(M-1)}{nM-1}s_w^2\end{aligned}$$

We have:

1. $s_b^2$ is an Unbiased Estimator of $S_b^2$
2. $s_w^2$ is an Unbiased Estimator of $S_w^2$
3. $s^2$ is NOT an Unbiased Estimator of $S^2$

The Unbiased Estimator of $S^2$ is:

$$\begin{aligned}S^2 &\approx \frac{N-1}{NM-1}s_b^2 + \frac{N(M-1)}{NM-1}s_w^2 \quad &(N\text{ is given})\\S^2& \approx \frac{1}M s_b^2 + \frac{M-1}M s_w^2 \quad &(N = +\infty)\end{aligned}$$

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Design Efficiency

Definition
Within Cluster Correlation Coefficient:

$$\begin{aligned}\rho_c &\overset{\text{def}}{=} \frac{2\sum_{i=1}^N \sum_{j<k}^M (Y_{ij} - \overline{\overline{Y}})(Y_{ik} - \overline{\overline{Y}})}{(M-1)(NM-1) S^2}\\&= 1 - \frac{NM S_w^2}{(NM-1) S^2}\end{aligned}$$

Note that:

$$\rho_c \in \left[ -\frac{1}{M-1}, 1 \right]$$

To Calculate the Design Efficiency, we need to calculate the variance of our cluster estimator versus the variance of SRS.

1. Variance of the Cluster Estimator

$$\begin{aligned}\text{Var}(\overline{\overline{y}}) &= \frac{1-f}{nM} S_b^2\\ &=\frac{1-f}{nM} \cdot \frac{NM-1}{M(N-1)} S^2 (1 + (M-1)\rho_c) \quad &(N \text{ is given})\\ &=\frac{1-f}{nM} S^2 (1 + (M-1)\rho_c) \quad & (N = +\infty)\end{aligned}$$

Now lets tackle $\rho_c$, the estimation of $\rho_c$ is:

$$\begin{aligned}\rho_c &= 1 - \frac{NMs_2^2}{(NM-1)\hat S^2} \quad & N \text{ is given}\\ & = \frac{s_b^2 - s_w^2}{s_b^2 + (M-1)s_w^2} \quad & N = +\infty\end{aligned}$$

2. Variance for SRS from a population of $NM$ with sample size $nM$

$$\text{Var}(\overline{y}_{SRS}) = \frac{1 - f}{nM} S^2$$

Hence the Design Efficiency can be derived as:

$$\widehat{\text{Deff}} = \frac{\text{Var}(\overline{\overline{y}})}{\text{Var}(\overline{y}_{SRS})} = \begin{cases} \frac{NM-1}{M(N-1)} (1 + (M-1)\hat\rho_c) & \text{ $N$ is limitied} \\ 1 + (M-1)\hat\rho_c & \text{ $N = +\infty$} \end{cases}$$

Determining the Sample Size

Given $(V,C,d,r)$ for $\bar y_{SRS}$, we can determine the sample size $n_{SRS}$, therefore:

$$n_{\min} = \widehat{\text{Deff}} \cdot n_{SRS}$$

The minimum number of clusters is:

$$n_{\text{cluster}} = \frac{n_{\min}}M$$

Clusters with different sizes

1. If $M_i$ are close enough, use the mean $\bar M$ as a proxy for $M$:

$$\bar M = \frac 1 N \sum_{i=1}^N M_i$$

2. When $M_i$ are widely apart, use the stratified method for each cluster to obtain a similar stratum size. Then use the mean as a proxy.