1. Chapter 6 (cont.) Relative Efficiency of Estimators
Compare the variances of this chapter’s 3 estimators of the population mean (ratio, regression, difference)
Compare these variances to that of the sample mean from a SRS

2. But First, Need to Address Bias
Generally, it’s inappropriate to compare variances of biased estimators
The bias becomes negligible if the relationship between x and y falls along a straight line through the origin (next slide)

3. Approx. of Relative Bias of r

4. Relative Efficiency
How do we tell which one is best for a particular sampling situation?
Cannot always answer definitively, but there are some guidelines.
One such guideline: relative efficiency.

5. Relative Efficiency - 2

6. Relative Efficiency - 3

7. Relative Efficiency-4

8. Relative Efficiency-5
In ratio estimation, the y values are frequently updated x values (for example, 1st quarter earnings this year compared to 1st quarter earnings last year).
In such situations cv(y) is frequently very close in value to cv(x)

9. Relative Efficiency-6

10. Relative Efficiency-7
Thus, is always more efficient than as an estimator of (However, can have serious bias problems unless the regression of y on x is truly linear.

11. Relative Efficiency-8

12. Relative Efficiency-9
So the regression estimator is more efficient than the ratio estimator unless , in which case they are equivalent.

13. Relative Efficiency-10

14. Relative Efficiency-11
If the variation in x and y values is about the same, then the difference estimator is more efficient than when the correlation between x and y is greater than ½.

15. Relative Efficiency-12
The regression estimator will be equivalent to the difference estimator when b1 = 1. Otherwise, the regression estimator will be more efficient than the difference estimator.

16. Relative Efficiency-13 Summary