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Chapter 6 (cont.) Relative Efficiency of Estimators

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. But First, Need to Address Bias.

Samuel
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Chapter 6 (cont.) Relative Efficiency of Estimators

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  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

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