Nordheim Statistics 411 Spring 2015 (Feb 17) The “anatomy” of ratio estimation Consider the following “artificially” simple problem. Suppose I have a population of 5 (N=5) junior executives in banks in a given city. Given below are their salaries (in thousand of dollars) in 2012-13. Y: 132 102 116 142 108 µYF=120 I wish to take a simple random sample of size 2 (n=2) to estimate the population mean salary. There are 10 possible samples of size 2, each with an equal probability of occurrence (according to SRS theory). They are: (132,102) (132,116) (132,142) (132,108) (102,116) (102,142) (102,108) (116,142) (116,108) (142,108) € y : 117 124 137 120 109 122 105 129 112 125 I entered the 10 sample means into R as “ybar2”. (These are the SRS estimates of the population mean salary for each of the 10 samples.) > mean(ybar2) [1] 120 > sd(ybar2) [1] 9.626353 Thus, the SRS estimate is unbiased (as expected). The “sd” value provides a measure of the variability of the estimates. ----------- Now consider using some “auxiliary” information to sharpen the estimation. Suppose that we have the population data on the salaries for the same 5 junior executives in 2010-2011 (two years earlier). They are, (in the same order): X: 108 78 98 122 94 µXF=100 To compute the ratio estimates for each of the 10 possible samples, we need to compute the values for € x (for each of the samples). (108,78) (108,98) (108,122) (108,94) (78,98) (78,122) (78,94) (98,122) (98,94) (122,94) € x : 93 103 115 101 88 100 86 110 96 108 I entered these 10 sample means into R as “xbar2” --- in exactly the same order as the “ybar2”. Then: > Rhat=yvar2/xbar2 > Rhat [1] 1.258065 1.203883 1.191304 1.188119 1.238636 1.220000 1.220930 [8] 1.172727 1.166667 1.157407 > yrat=Rhat*100 > yrat These are the ratio estimates for each of the 10 possible samples of size 2. [1] 125.8065 120.3883 119.1304 118.8119 123.8636 122.0000 122.0930 [8] 117.2727 116.6667 115.7407> mean(yrat) [1] 120.1774 Here you can see that the ratio estimate is NOT unbiased. (The bias is not too large, however.) > sd(yrat) [1] 3.26118 Here you can see that the variability of the estimates is a good deal smaller than with SRS. In many situations we perform ratio estimation largely to achieve a substantial reduction in variance, as illustrated in this example. However, this works only if the relationship between the auxiliary variable (X in this case) and the response (Y) is a strong one. 80 90 100 110 120110 120 130
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