Ratio and regression estimators are powerful tools in sampling techniques. They use relationships between variables to improve estimation precision, reducing variance compared to simple random sampling. These methods are crucial for accurate population parameter estimates in various fields.

Choosing between ratio and regression estimators depends on the relationship between variables. Ratio estimators work best for proportional relationships, while regression estimators handle linear but not necessarily proportional relationships. Both require strong correlations for effectiveness.

Ratio and Regression Estimators

Definitions and Key Concepts

• Ratio estimators utilize relationship between two variables expressed as a ratio to estimate population parameters
• Regression estimators leverage linear relationship between study variable and auxiliary variable to improve estimation precision
• Ratio estimator defined as ratio of sample means of two variables (used for proportional relationships)
• Regression estimator uses least squares regression line to adjust sample mean based on relationship with auxiliary variable
• Both estimators aim to reduce variance and increase precision compared to simple random sampling estimates
• Effectiveness depends on strength of correlation between study variable and auxiliary variable

Applications and Examples

• Ratio estimators used in business surveys to estimate total revenue based on number of employees
• Regression estimators applied in environmental studies to estimate forest biomass using satellite imagery data
• Ratio estimators employed to estimate total household income in a city using property tax records
• Regression estimators utilized in social science research to predict voter turnout based on demographic variables
• Both estimators valuable in stratified sampling designs where auxiliary information known for each stratum (census data)

Choosing Ratio vs Regression Estimators

Relationship Between Variables

• Ratio estimators most appropriate for strong, proportional relationships between study and auxiliary variables
• Regression estimators preferred for strong linear relationships that may not be strictly proportional
• Both require accurate, up-to-date information on population total or mean of auxiliary variable
• Strength of correlation between variables crucial for effectiveness of both estimators

Specific Use Cases

• Ratio estimators often employed in business and economic surveys (natural proportional relationships)
  • Estimating total sales based on floor space in retail stores
  • Calculating total crop yield based on cultivated land area
• Regression estimators common in environmental and social science research (linear but not necessarily proportional relationships)
  • Predicting air pollution levels based on traffic density
  • Estimating wildlife population using habitat characteristics

Bias and Variance of Estimators

Bias Considerations

• Ratio estimator bias approximately inversely proportional to sample size and directly proportional to population variance of ratio
• Regression estimators generally less biased than ratio estimators, especially when relationship not strictly proportional
• Bias in ratio estimators can be significant for small samples or weak correlations between variables
• Regression estimators more robust to departures from proportionality assumption

Variance Analysis

• Variance of ratio estimators estimated using Taylor series approximation or resampling methods (jackknife, bootstrap)
• Regression estimator variance depends on residual variance of regression model and variance of auxiliary variable
• Both estimators have smaller variances compared to simple random sampling with strong variable correlations
• Efficiency quantified using relative precision (ratio of variances compared to simple random sampling)
  • Relative precision > 1 indicates improved efficiency
  • Example: Relative precision of 1.5 means estimator is 50% more efficient than simple random sampling

Confidence Intervals for Estimators

Construction Methods

• Ratio estimator confidence intervals typically use estimated variance and assume normal distribution for large samples
• Regression estimator confidence intervals incorporate uncertainty in regression coefficients and population mean of auxiliary variable
• Bootstrap methods construct nonparametric confidence intervals for both estimators
  • Particularly useful for small sample sizes or non-normal distributions
  • Process: Resample with replacement, calculate estimate for each resample, determine percentiles for interval

Interpretation and Comparison

• Width of confidence interval for ratio estimators depends on sample size, correlation between variables, and chosen confidence level
• Degrees of freedom for t-distribution based intervals may need adjustment in complex sampling designs
• Comparing interval widths between simple random sampling and ratio/regression estimators demonstrates precision gains
  • Example: Ratio estimator confidence interval 20% narrower than simple random sampling interval indicates improved precision