Ratio Method of Estimation

Overview

• Utilized when a study variable is correlated with an auxiliary variable.
• Auxiliary variable is readily available, requiring no extra effort or resources.

Key Concepts

• Ratio Estimator:

  • Denoted as ( R = \frac{Y}{X} )
  • Both (Y) and (X) are sample estimates.
  • Can estimate ratios by dividing totals or means.

• Estimating Population Mean and Total:

  • Population mean can be estimated using the formula: [ \text{small } \bar{y} = \frac{\text{small } \bar{y}}{\text{small } \bar{x}} \cdot \text{Capital } \bar{X} ]
  • Where Capital ( \bar{X} ) is known for auxiliary variable.

Example

• Registered births example:
  • If the ratio of registered births to total population is 2, it indicates that the population is double the number of registered births.
  • Multiply total registered births by the ratio to estimate total population.

Bias of the Ratio Estimator

• The ratio estimator is biased, but bias decreases with larger sample sizes.
• For large sample sizes:
  • ( \text{expected value of } R_{cap} \approx R )

Mathematical Proof

1. Ratio Estimator:
   • ( R_{cap} = \frac{\text{small } \bar{y}}{\text{small } \bar{x}} )
2. Subtracting R:
   • ( R_{cap} - R = \frac{\text{small } \bar{y}}{\text{small } \bar{x}} - R )
3. Large Sample Size Approximation:
   • For large samples, replace small ( \bar{x} ) with capital ( \bar{X} ).
4. Expected Value:
   • ( E[R_{cap}] - R = 0 )
   • Thus, ( E[R_{cap}] \approx R ) when sample size is large.

Conclusion

• When the sample size is large, the ratio estimator becomes approximately unbiased.