Unbiased Ratio Estimators In Stratified Sampling

Delving into the realm of statistical estimation, particularly within the framework of stratified sampling, unveils the intricacies of unbiased ratio estimators. These estimators play a crucial role in providing accurate estimates of population parameters when auxiliary information is available. Understanding their properties, applications, and limitations is paramount for researchers and practitioners alike.

Stratified Sampling: A Foundation for Precision

Stratified sampling is a powerful technique used to improve the precision of estimates when the population is heterogeneous. It involves dividing the population into subgroups, or strata, based on shared characteristics, and then drawing a random sample from each stratum. This ensures representation from each subgroup, leading to more accurate and reliable results compared to simple random sampling.

Why Stratify?

• Reduced Variability: Stratification reduces the variability within each stratum, leading to more precise estimates.
• Representation: Ensures adequate representation of all subgroups in the population.
• Administrative Convenience: Sampling can be organized and managed more efficiently within strata.

Ratio Estimation: Leveraging Auxiliary Information

Ratio estimation is a method that utilizes auxiliary information to improve the accuracy of estimates. This is particularly useful when there is a strong correlation between the variable of interest and an auxiliary variable. The ratio estimator uses the ratio of the sample mean of the variable of interest to the sample mean of the auxiliary variable to estimate the population ratio.

The Basic Idea:

Suppose we want to estimate the population mean of a variable Y. We also have information on an auxiliary variable X, which is correlated with Y. The ratio estimator is defined as:

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

Where:

• $\hat{R}$ is the estimated ratio.
• $\bar{y}$ is the sample mean of the variable of interest.
• $\bar{x}$ is the sample mean of the auxiliary variable.

Biasedness of the Traditional Ratio Estimator

The traditional ratio estimator, while widely used, is inherently biased. This bias arises because the expected value of the ratio of two random variables is not equal to the ratio of their expected values. In other words:

$E\left[\frac{\bar{y}}{\bar{x}}\right] \neq \frac{E[\bar{y}]}{E[\bar{x}]}$

This bias can be significant, especially when the sample size is small or when the correlation between X and Y is weak.

Unbiased Ratio Estimators: Addressing the Bias

To overcome the bias associated with the traditional ratio estimator, various unbiased ratio estimators have been developed. These estimators aim to correct for the bias, providing more accurate estimates of the population ratio. Several methods exist, each with its own assumptions and properties.

1. Hartley-Ross Estimator

The Hartley-Ross estimator is a classic approach to reducing bias in ratio estimation. It involves adjusting the traditional ratio estimator based on the sample covariance between the variable of interest and the auxiliary variable.

Formula:

The Hartley-Ross estimator is defined as:

$\hat{R}{HR} = \frac{\bar{y}}{\bar{x}} + \frac{1}{n(n-1)\bar{x}} \sum{i=1}^{n} (y_i - \bar{y})(x_i - \bar{x})$

Where:

• $n$ is the sample size.
• $y_i$ and $x_i$ are the individual observations of the variable of interest and the auxiliary variable, respectively.

Properties:

• The Hartley-Ross estimator is approximately unbiased, with the bias decreasing as the sample size increases.
• It is relatively simple to compute, making it a practical choice for many applications.

2. Quenouille Estimator (Jackknife Estimator)

The Quenouille estimator, also known as the jackknife estimator, is a more general technique for bias reduction. It involves creating multiple sub-samples by omitting one observation at a time and then calculating the estimate for each sub-sample. The final estimate is obtained by averaging the sub-sample estimates and applying a correction factor to reduce bias.

Procedure:

1. Divide the sample into n sub-samples, each with n-1 observations.

2. Calculate the ratio estimator for each sub-sample:

   $\hat{R}_i = \frac{\bar{y}_i}{\bar{x}_i}$

3. Compute the pseudo-values:

   $P_i = n\hat{R} - (n-1)\hat{R}_i$

4. The Quenouille estimator is the average of the pseudo-values:

   $\hat{R}{Q} = \frac{1}{n} \sum{i=1}^{n} P_i$

Properties:

• The Quenouille estimator is generally less biased than the traditional ratio estimator, especially for small sample sizes.
• It is computationally intensive, as it requires calculating the ratio estimator for each sub-sample.
• It is a versatile technique that can be applied to a wide range of estimators.

3. Mickey's Unbiased Ratio Estimator

Mickey's unbiased ratio estimator is specifically designed for stratified sampling. It provides an unbiased estimate of the population ratio by considering the individual strata and their respective sample sizes.

Formula:

$\hat{R}{M} = \frac{\sum{h=1}^{H} N_h \bar{y}h}{\sum{h=1}^{H} N_h \bar{x}_h}$

Where:

• $H$ is the number of strata.
• $N_h$ is the population size of stratum h.
• $\bar{y}_h$ is the sample mean of the variable of interest in stratum h.
• $\bar{x}_h$ is the sample mean of the auxiliary variable in stratum h.

Properties:

• Mickey's estimator is unbiased, making it a preferred choice when unbiasedness is crucial.
• It requires knowledge of the population sizes of each stratum.
• It is particularly effective when the relationship between X and Y varies across strata.

Unbiased Ratio Estimators in Stratified Sampling: A Deeper Dive

When applying unbiased ratio estimators in stratified sampling, it's crucial to consider how the stratification affects the estimation process. The goal is to leverage the benefits of stratification while minimizing bias in the ratio estimation.

Applying Hartley-Ross in Stratified Sampling

To apply the Hartley-Ross estimator in stratified sampling, we can calculate the estimator separately for each stratum and then combine the results.

Procedure:

1. Calculate the Hartley-Ross estimator for each stratum:

   $\hat{R}_{HR,h} = \frac{\bar{y}_h}{\bar{x}h} + \frac{1}{n_h(n_h-1)\bar{x}h} \sum{i=1}^{n_h} (y{hi} - \bar{y}h)(x{hi} - \bar{x}_h)$

2. Combine the stratum estimates using appropriate weights (e.g., stratum sizes):

   $\hat{R}{HR,strat} = \frac{\sum{h=1}^{H} N_h \hat{R}_{HR,h} \bar{x}h}{\sum{h=1}^{H} N_h \bar{x}_h}$

Applying Quenouille (Jackknife) in Stratified Sampling

The Quenouille estimator can also be adapted for stratified sampling. The procedure involves creating sub-samples within each stratum and then combining the results.

Procedure:

1. For each stratum, divide the sample into n_h sub-samples, each with n_h-1 observations.

2. Calculate the ratio estimator for each sub-sample within each stratum:

   $\hat{R}{h,i} = \frac{\bar{y}{h,i}}{\bar{x}_{h,i}}$

3. Compute the pseudo-values for each stratum:

   $P_{h,i} = n_h\hat{R}h - (n_h-1)\hat{R}{h,i}$

   Where $\hat{R}_h$ is the ratio estimator for stratum h using the entire sample from that stratum.

4. The Quenouille estimator for the stratified sample is:

   $\hat{R}{Q,strat} = \frac{\sum{h=1}^{H} N_h \left(\frac{1}{n_h} \sum_{i=1}^{n_h} P_{h,i}\right) \bar{x}h}{\sum{h=1}^{H} N_h \bar{x}_h}$

Comparison of Estimators

Each of these unbiased ratio estimators has its own strengths and weaknesses. The choice of estimator depends on the specific characteristics of the data and the research objectives.

• Hartley-Ross: Simple and computationally efficient, but may not be as effective in reducing bias for small sample sizes or weak correlations.
• Quenouille (Jackknife): More robust to bias, but computationally intensive, especially for large samples.
• Mickey's: Unbiased by design, but requires knowledge of stratum sizes and may not be as efficient if the relationship between X and Y is inconsistent across strata.

Variance Estimation

Estimating the variance of unbiased ratio estimators is crucial for assessing the precision of the estimates. The variance estimation methods vary depending on the specific estimator used.

Variance Estimation for Hartley-Ross

The approximate variance of the Hartley-Ross estimator is given by:

$Var(\hat{R}{HR}) \approx \frac{1}{n \bar{x}^2} \left[ S_y^2 - 2\hat{R}{HR} S_{xy} + \hat{R}_{HR}^2 S_x^2 \right]$

Where:

• $S_y^2$ is the sample variance of Y.
• $S_x^2$ is the sample variance of X.
• $S_{xy}$ is the sample covariance between X and Y.

In stratified sampling, the variance can be estimated by summing the stratum-specific variances:

$Var(\hat{R}{HR,strat}) \approx \sum{h=1}^{H} \left( \frac{N_h}{N} \right)^2 \frac{1}{n_h \bar{x}h^2} \left[ S{y,h}^2 - 2\hat{R}{HR,h} S{xy,h} + \hat{R}{HR,h}^2 S{x,h}^2 \right]$

Variance Estimation for Quenouille (Jackknife)

The variance of the Quenouille estimator can be estimated using the pseudo-values:

$Var(\hat{R}{Q}) = \frac{1}{n(n-1)} \sum{i=1}^{n} (P_i - \hat{R}_{Q})^2$

In stratified sampling:

$Var(\hat{R}{Q,strat}) = \sum{h=1}^{H} \frac{1}{n_h(n_h-1)} \sum_{i=1}^{n_h} (P_{h,i} - \bar{P}_h)^2$

Where $\bar{P}h = \frac{1}{n_h} \sum{i=1}^{n_h} P_{h,i}$

Variance Estimation for Mickey's Estimator

The variance of Mickey's estimator is given by:

$Var(\hat{R}{M}) = \sum{h=1}^{H} \frac{N_h^2 (1 - \frac{n_h}{N_h})}{n_h(n_h - 1)} \left( \frac{\sum_{i=1}^{n_h} (y_{hi} - \hat{R}{M}x{hi})^2}{(\sum_{h=1}^{H} N_h \bar{x}_h)^2} \right)$

Practical Considerations

When implementing unbiased ratio estimators in stratified sampling, several practical considerations should be taken into account:

• Sample Size: Ensure that the sample size within each stratum is large enough to provide reliable estimates. Small sample sizes can lead to unstable estimates and increased variability.
• Correlation: The effectiveness of ratio estimation depends on the strength of the correlation between the variable of interest and the auxiliary variable. Choose auxiliary variables that are highly correlated with the variable of interest.
• Stratification: Carefully consider the stratification criteria to ensure that the strata are homogeneous and that the variability within each stratum is minimized.
• Computational Resources: The Quenouille estimator can be computationally intensive, especially for large samples. Ensure that you have adequate computational resources to perform the calculations.

Real-World Applications

Unbiased ratio estimators in stratified sampling have numerous applications in various fields:

• Agriculture: Estimating crop yields using satellite imagery as an auxiliary variable. Stratifying by farm size or geographic region can improve the accuracy of the estimates.
• Economics: Estimating income levels using housing values as an auxiliary variable. Stratifying by socioeconomic status or geographic area can provide more precise estimates.
• Environmental Science: Estimating pollution levels using population density as an auxiliary variable. Stratifying by industrial activity or land use can improve the accuracy of the estimates.
• Healthcare: Estimating disease prevalence using age or gender as auxiliary variables. Stratifying by demographic characteristics can provide more precise estimates.

Example: Estimating Average Income

Let's consider an example where we want to estimate the average income of households in a city using stratified sampling. We stratify the city into three regions: high-income, middle-income, and low-income. We also have data on housing values, which is correlated with income.

We can use Mickey's unbiased ratio estimator to estimate the overall average income:

$\hat{R}{M} = \frac{\sum{h=1}^{3} N_h \bar{y}h}{\sum{h=1}^{3} N_h \bar{x}_h} = \frac{(5000 \times 100000) + (10000 \times 50000) + (15000 \times 25000)}{(5000 \times 500000) + (10000 \times 250000) + (15000 \times 100000)} = \frac{1.375 \times 10^9}{6.5 \times 10^9} \approx 0.2115$

To estimate the overall average income, we multiply the ratio by the overall average housing value:

$\text{Overall Average Housing Value} = \frac{\sum_{h=1}^{3} N_h \bar{x}h}{\sum{h=1}^{3} N_h} = \frac{6.5 \times 10^9}{30000} \approx 216666.67$

$\text{Estimated Average Income} = \hat{R}_{M} \times \text{Overall Average Housing Value} = 0.2115 \times 216666.67 \approx 45888.33$

Therefore, the estimated average income of households in the city is approximately $45,888.33.

Conclusion

Unbiased ratio estimators in stratified sampling provide a powerful toolkit for researchers and practitioners seeking to improve the accuracy and reliability of their estimates. By understanding the properties, applications, and limitations of these estimators, we can make informed decisions about which method to use in different scenarios. The key to success lies in carefully considering the characteristics of the data, the research objectives, and the available resources. With proper implementation, unbiased ratio estimators can lead to more accurate and meaningful insights, contributing to better decision-making in a wide range of fields.