1. 2015-7-3 www.uic.edu.hk/~xlpeng 1 Ratio Estimation and Regression Estimation (Chapter 4, Textbook, Barnett, V., 1991) 2.1 Estimation of a population ratio: The ratio estimator In some situations it is useful to estimate a (positive) ratio of two population characteristics: the totals, or means, of two (positive) variables X and Y.

2. The sample average of ratio unbiased for estimating the population mean Two obvious estimators of R are The ratio of the sample averages is widely used. 2015-7-3 www.uic.edu.hk/~xlpeng 2 but biased for estimating R

3. The bias in estimating R by r The bias in estimating R by r is the expectation of the following difference: (2.3) 2015-7-3 www.uic.edu.hk/~xlpeng 3

4. Discussion about the bias 2015-7-3 www.uic.edu.hk/~xlpeng 4 ≈

5. 2015-7-3 www.uic.edu.hk/~xlpeng 5 (2.5)

6. 2015-7-3 www.uic.edu.hk/~xlpeng 6 A (slightly) biased estimate of the true variance parameter For large n, an approximate 100 (1-α) % (symmetric) two-sided confidence interval for the population ratio R is: And the required sample size is

7. 2.2 Ratio estimation of a population mean or total 2015-7-3 www.uic.edu.hk/~xlpeng 7

8. Variance of ratio estimator 2015-7-3 www.uic.edu.hk/~xlpeng 8

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10. Example: (Food additive) A researcher was investigating a new food additive for cattle. Midway through the two-month study, he was interested in estimating the average weight for the entire herd of N = 500 steers. A simple random sample of n = 12 steers was selected from the herd and weighed. These data and prestudy weights are presented in the accompanying table for all cattle sampled. Assume the prestudy average = 880 pounds. Estimate the ratio of present weight to prestudy weight of the herd, and provide an estimate of the standard error for your answer. Which points have greatest influence on the estimate? 2015-7-3 www.uic.edu.hk/~xlpeng 10

11. 2015-7-3 www.uic.edu.hk/~xlpeng 11

12. 2015-7-3 www.uic.edu.hk/~xlpeng 12 Solution:

13. 2015-7-3 www.uic.edu.hk/~xlpeng 13 The estimate of the ratio R of the present weight to prestudy weight for the herd is: Solution:

14. Sample size determination 2015-7-3 www.uic.edu.hk/~xlpeng 14 Hence we should sample 94 steers to estimate R, the change in weight of herd after the study with error bound of 1%.

15. Example: (Sugar content) 2015-7-3 www.uic.edu.hk/~xlpeng 15 In a study to estimate the total sugar content of a truckload of oranges, a SRS of n = 10 oranges was juiced and weighted. The total weight of all the oranges, obtained by first weighing the truck loaded and then unloaded, was found to be 1800 pounds. Estimate Y, the total sugar content for the oranges and provide the standard error of the estimate.

16. 2015-7-3 www.uic.edu.hk/~xlpeng 16 Solution: The scatter plot shows a strong positive association between sugar content and weight, making the ratio estimator a reasonable choice. An estimate of Y is

17. 2015-7-3 www.uic.edu.hk/~xlpeng 17 Solution:

18. Example: (Promotional campaign) 2015-7-3 www.uic.edu.hk/~xlpeng 18 An advertising firm is concerned about the effect of a new regional promotional campaign on the total dollar sales for a particular product. A SRS of n = 20 stores is drawn from the N = 452 regional stores in which the product is sold. Quarterly sales data are obtained for the current three-month period and the three-month period prior to the new campaign. The pre-campaign sales for all stores X = 260, 256. Check the scatter plot to see if these stores are in two different size groups.

19. 2015-7-3 www.uic.edu.hk/~xlpeng 19 Example: (Promotional campaign)

20. 2015-7-3 www.uic.edu.hk/~xlpeng 20 Solution: (a) Without using the auxiliary information, the estimate of the average current three-month sales using ordinary estimator is

21. 2015-7-3 www.uic.edu.hk/~xlpeng 21 Solution: (b) When the total pre-campaign three-month sales is known to be X = 260256, the average pre-campaign three-month sales is Then the estimate of the average current three-month sales using ratio estimator is which represent an average increase of 7.1% of the current three-month sales from the pre-campaign three-month sales.

22. Solution: 2015-7-3 www.uic.edu.hk/~xlpeng 22 The ratio estimator here is much better than ordinary estimator since the current three-month sales y i is closely and positively related to the pre- campaign three-month sales x i with correlation coefficient 0.9986.

23. 2015-7-3 www.uic.edu.hk/~xlpeng 23

24. 2015-7-3 www.uic.edu.hk/~xlpeng 24 This examines when the variance of (2.10) could be less or greater than that of (1.9)

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26. 2.3 Regression estimation Condition (2.15.1) demands that X and Y be linearly related, but, if the linear relationship does not pass through the origin, then, it suggests considering an alternative estimator known as regression estimator. 2015-7-3 www.uic.edu.hk/~xlpeng 26

27. 2.3 Regression estimation 2015-7-3 www.uic.edu.hk/~xlpeng 27 A practicable simple linear regression model is (2.17). An ideal (perfect) linear relationship is (2.16) (2.18)

28. 2.3 Regression estimation 2015-7-3 www.uic.edu.hk/~xlpeng 28 Consider the average (mean) of either (2.16) or (2.17), (2.19)

29. 2.3 Regression estimation 2015-7-3 www.uic.edu.hk/~xlpeng 29 (2.20)

30. 2.3 Regression estimation 2015-7-3 www.uic.edu.hk/~xlpeng 30 (2.21)

31. 2.3 Regression estimation 2015-7-3 www.uic.edu.hk/~xlpeng 31 From (2.20), The minimum is obtained with Thus the most efficient regression estimator of is

32. 2.3 Regression estimation 2015-7-3 www.uic.edu.hk/~xlpeng 32 The optimal value of b of (2.22) suggests the obvious estimate: (2.24) (2.25) which enjoys the following asymptotic properties:

33. 2.3 Regression estimation 2015-7-3 www.uic.edu.hk/~xlpeng 33 Asymptotic properties: (2.27)

34. 2.4 Comparison of ratio and regression estimators 2015-7-3 www.uic.edu.hk/~xlpeng 34

35. 2015-7-3 www.uic.edu.hk/~xlpeng 35 2.4 Comparison of ratio and regression estimators