Research Article | Open Access

Volume 2014 |Article ID 360549 | https://doi.org/10.1155/2014/360549

Angela Shirley, Ashok Sahai, Isaac Dialsingh, "On Improving Ratio/Product Estimator by Ratio/Product-cum-Mean-per-Unit Estimator Targeting More Efficient Use of Auxiliary Information", Journal of Probability and Statistics, vol. 2014, Article ID 360549, 8 pages, 2014. https://doi.org/10.1155/2014/360549

# On Improving Ratio/Product Estimator by Ratio/Product-cum-Mean-per-Unit Estimator Targeting More Efficient Use of Auxiliary Information

Accepted28 Aug 2014
Published23 Sep 2014

#### Abstract

To achieve a more efficient use of auxiliary information we propose single-parameter ratio/product-cum-mean-per-unit estimators for a finite population mean in a simple random sample without replacement when the magnitude of the correlation coefficient is not very high (less than or equal to 0.7). The first order large sample approximation to the bias and the mean square error of our proposed estimators are obtained. We use simulation to compare our estimators with the well-known sample mean, ratio, and product estimators, as well as the classical linear regression estimator for efficient use of auxiliary information. The results are conforming to our motivating aim behind our proposition.

#### 1. Introduction and Notation

This paper addresses the problem of efficiently estimating the population mean, using auxiliary information. A fairly large simple random sample of size is selected without replacement from, say, a large bivariate population of size (which could, reasonably, be thought to have come from a normal superpopulation), with the sampling fraction , , so that is negligible. Quite often, we have surveys where some auxiliary variable may be relatively less expensive to observe than the main variable . In order to have a survey estimate of the population mean of the main variable, assuming knowledge of the population mean of the auxiliary variable, the following estimators are well known.

The ratio estimator:

The product estimator: Here is the estimate of the ratio of the population means and is the estimate of the product of the population means, and being unweighted sample means of the two variables, respectively. Usually, the variability of is less than that of .

It is straightforward to derive first order approximations to the bias and mean square error of these estimators. Let and be the population coefficients of variation of and , respectively, where are the population variances of and , respectively. Let and , where the errors and can be positive or negative, so that . It is known that, for simple random sample without replacement, , , and where is the correlation coefficient between the variables (P. V. Sukhatme and B. V. Sukhatme [1]). Further, to validate our first order large sample approximations, we assume that the sample is large enough to make and so small that the terms involving and/or to a degree higher than two are negligible, an assumption which is not unrealistic.

Substituting the expressions for and in terms of and in (1) we have Assuming that , we expand to obtain Since as , we have that the ratio estimator is asymptotically unbiased up to . Similarly we have that the product estimator is asymptotically unbiased (Murthy [2]). Also, Thus up to order of approximation, if and only if , or if and only if , where . (It is worth noting here that because of long association with the experimental data, is guessable.) Similarly we have if and only if (Murthy [2]). Thus the ratio and product estimators are relatively more efficient than the usual unbiased estimator (u.u.e) sample mean when and , respectively. Consequently, fail to improve (by using auxiliary information) when .

Also we cannot ignore the classically well-known linear regression estimator, say : If we recall the ANOVA of linear regression analysis, we must remember that the residual sum of squares for is (Cochran [3]). Thus when is high (say or ), linear regression estimator is most likely to be more efficient than ratio/product estimators in using the auxiliary information (via auxiliary variable ). We aim at improving use of auxiliary information on when ; when ; when ; and when .

#### 2. Our Proposed Estimators

Because and are relatively more efficient than when and , respectively, we try the following single parameter linear combinations of and , as well as and to propose the estimators:(i)Shirley-Sahai-Dialsingh-ratio-cum-mean, say : (ii)Shirley-Sahai-Dialsingh-product-cum-mean, say : In (8) and (9), is the design parameter for our proposed estimators to be assigned an optimal value, for example, so as to minimize the first order of MSE, , as in our case. Note that when , and . As remarked earlier, quite often a good guess of is available from which we can give a suitable value to .

#### 3. Sampling Bias and Mean Square Error of the Proposed Estimators

We derive the first order approximation, , to the bias of . Using the notation introduced in Section 1 and substituting the expressions for and in terms of and in (8) we have It is realistic practically to suppose that so that is expandable. Then to the first order of approximation , the bias of is given by where , , and . , as ; therefore, is asymptotically unbiased up to .

To compute the MSE of we have where and .

For large sample size, is minimum for . The optimal value of is thus . If a good guess of , say , is available, we use in our proposed estimator (8), so that We deduce the large sample approximation for bias of in a similar manner: where and are as before. , as ; therefore is asymptotically unbiased.

To compute the MSE of we have Up to , is a minimum for . The optimal value of is thus .

We use in our proposed estimator (9), where is the guess of . Thus,

#### 4. Comparison of the Estimators

Apparently, no algebraic comparison of mean square errors is feasible. We, therefore, have a numerical setup under simulation to do so. Knowing exactly is seldom tenable in practice. Consequently, we have to assume the availability of a guess value of , which we have called , defined by , where designates the quantum of relative under guess/overguess. We have taken the following values: , , , , , and . We have also assumed that the parent population is very large, envisaged to have come from a superpopulation which is bivariate normal with the following parameters, therefore having the same parametric values: Consequently we have used the software R [4] to calculate the MSEs of each of the following estimators: We use 10,000 replications of simulated sample sizes , 40, 60, 80, and 100. Hence we have compared the efficiencies of these estimators relative to by using Motivated by our desire to beat ratio/product estimators (implicitly, therefore also), we have, therefore, taken up the numerical simulation comparisons, for example, values of : 0.1 (0.1) 0.7. For positive, we compare , , and , while for negative we compare , , and .

#### 5. Results and Discussion

The results of our simulations are tabulated in the Appendix. For a given value of the relative efficiencies of , , and do not depend on ; they are, therefore, not included in the main body of the tables but are stated at the top for each value of . For 30, 40, 60, 80, and 100, for each value of , , , , , and , we have given the values of for 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, and 0.7 and of for −0.1, −0.2, −0.3, −0.4, −0.5, −0.6, and −0.7.

As apparent, our proposed estimators and are consistently significantly better than and (or , as the case may be).

For illustrative purposes, we highlight below the relative efficiency values for the various values of , for the cases when and . To lessen the obscurity in the results, we have rounded these values to two decimal places. We also include a column for the value of .

Tables 1 and 2 illustrate very well the relative betterment achieved by our proposed estimators vis-à-vis and (or , as the case may be). Notably, when is not greater than 1/2, our estimators are more efficient than even though or (as the case may be) is worse than which does not even use auxiliary information. Also when is significantly less than 1/2, our estimators are more efficient than even though is worse than (i.e., it fails to use auxiliary information rightly)!

 0.1 0.225 97.00 89.71 100.88 0.2 0.45 100.35 98.09 104.22 0.3 0.675 105.26 106.32 109.10 0.4 0.90 114.29 117.98 118.34 0.5 1.125 128.73 133.07 133.58 0.6 1.35 152.63 152.08 159.40 0.7 1.575 186.79 172.17 193.33
 −0.1 −0.225 97.66 89.14 100.73 −0.2 −0..45 100.42 98.05 104.14 −0.3 −0.675 106.48 108.23 110.39 −0.4 −0.90 114.87 119.11 119.43 −0.5 −1.125 127.32 131.54 132.05 −0.6 −1.35 150.37 150.36 155.88 −0.7 −1.575 186.53 172.80 194.68

#### 6. Conclusion

Our results conform to our motivating aim of achieving more efficient use of auxiliary information. Many other authors, such as Sahai [5] and Chami et al. [6], have suggested efficient variants of ratio and product estimators. In future work we are engaged in comparing these estimators and in trying even better estimators, like the proposed ones, which will be not only more efficient relatively, but also, possibly, more robust against the possible over/underguess of the key-population parameter .

#### Appendix

See Tables 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12.

 97.00734 100.3539 105.2556 114.2872 128.7345 152.627 186.7908 89.71481 98.09601 106.3240 117.9799 133.0705 152.0778 172.174 value −0.1 100.8797 104.1726 109.0713 118.1939 133.1713 157.8555 191.6527 −0.08 100.8806 104.1888 109.0944 118.2593 133.3251 158.3010 192.2725 −0.06 100.8808 104.2015 109.109 118.3068 133.4430 158.6793 192.7517 −0.04 100.8801 104.2107 109.1151 118.3364 133.5250 158.9895 193.0882 −0.02 100.8787 104.2164 109.1125 118.3480 133.5709 159.2308 193.2804 0 100.8764 104.2186 109.1014 118.3416 133.5805 159.4025 193.3276 0.02 100.8733 104.2173 109.0818 118.3172 133.5539 159.5041 193.2295 0.04 100.8694 104.2125 109.0536 118.2749 133.4912 159.5355 192.9865 0.06 100.8646 104.2043 109.0169 118.2147 133.3924 159.4964 192.5998 0.08 100.8591 104.1925 108.9717 118.1366 133.2576 159.3871 192.0710 0.1 100.8527 104.1772 108.9179 118.0408 133.0872 159.2078 191.4025
 98.56872 101.6871 106.8427 116.5161 131.2267 153.2434 195.3283 89.4024 97.99826 107.7805 119.1094 133.8761 150.5256 175.4088 value −0.1 100.763 104.0057 110.1035 119.1169 133.9862 155.4611 197.6259 −0.08 100.7616 104.02 110.1465 119.1571 134.1563 155.7931 198.4408 −0.06 100.7593 104.0309 110.1807 119.2293 134.29 156.0572 199.1088 −0.04 100.7563 104.0385 110.206 119.2836 134.3872 156.2525 199.6271 −0.02 100.7525 104.0426 110.2224 119.3198 134.4476 156.3785 199.9932 0 100.7479 104.0433 110.2299 119.3379 134.4712 156.435 200.2054 0.02 100.7425 104.0406 110.2285 119.3379 134.4579 156.4217 200.2627 0.04 100.7362 104.0345 110.2182 119.3198 134.4078 156.3387 200.1649 0.06 100.7292 104.025 110.199 119.2837 134.3209 156.1862 199.9125 0.08 100.7214 104.0121 110.171 119.2295 134.1974 155.9647 199.5065 0.1 100.7127 103.9959 110.134 119.1573 134.0375 155.6746 198.9489
 99.78738 102.7511 106.9806 115.7236 133.4915 154.5581 191.2787 90.7815 98.23825 106.5209 117.4447 135.1337 150.7982 172.6729 value −0.1 101.0542 104.2512 109.2088 117.8291 135.2587 155.6597 192.4683 −0.08 101.0596 104.2689 109.2346 117.8792 135.4549 155.9771 193.1069 −0.06 101.0642 104.2832 109.2519 117.9111 135.6138 156.2254 193.6036 −0.04 101.068 104.2939 109.2606 117.925 135.7354 156.4038 193.9563 −0.02 101.0711 104.3011 109.2606 117.9206 135.8192 156.5118 194.1633 0 101.0733 104.3048 109.2521 117.8982 135.8653 156.5493 194.2238 0.02 101.0747 104.3051 109.2349 117.8576 135.8735 156.516 194.1373 0.04 101.0754 104.3017 109.2092 117.7989 135.8438 156.4121 193.9044 0.06 101.0753 104.2949 109.1749 117.7222 135.7762 156.2379 193.5261 0.08 101.0743 104.2846 109.132 117.6275 135.6709 155.9937 193.004 0.1 101.0726 104.2707 109.0806 117.515 135.5281 155.6804 192.3406
 99.83416 102.7266 106.6169 115.6179 131.7743 154.1441 194.0998 91.45779 97.66782 105.0443 116.7154 133.2987 150.0163 174.2752 value −0.1 101.3714 103.9561 108.1498 117.1127 133.4017 154.9814 194.8863 −0.08 101.3831 103.9678 108.1557 117.1581 133.5593 155.3240 195.5633 −0.06 101.3939 103.9759 108.1534 117.186 133.6809 155.6000 196.0938 −0.04 101.4039 103.9806 108.1429 117.1966 133.7663 155.8088 196.4753 −0.02 101.4131 103.9819 108.1240 117.1897 133.8154 155.9498 196.7060 0 101.4214 103.9796 108.0970 117.1654 133.828 156.0227 196.7849 0.02 101.4289 103.974 108.0617 117.1238 133.8042 156.0272 196.7115 0.04 101.4355 103.9648 108.0181 117.0648 133.744 155.9634 196.4864 0.06 101.4414 103.9522 107.9664 116.9885 133.6476 155.8314 196.1104 0.08 101.4463 103.9361 107.9065 116.8949 133.5149 155.6315 195.5853 0.1 101.4505 103.9165 107.8385 116.7842 133.3464 155.3643 194.9135
 100.3458 102.4952 108.1571 117.9797 130.9376 154.1079 194.4502 90.15273 96.99257 106.6954 119.0344 131.7312 150.3096 174.1364 value −0.1 101.0155 103.6111 109.4732 119.0467 131.8139 155.0562 195.1446 −0.08 101.0193 103.6156 109.4997 119.0954 131.9356 155.3614 195.8685 −0.06 101.0224 103.6166 109.5173 119.1661 132.0222 155.5981 196.4477 −0.04 101.0246 103.6142 109.526 119.2189 132.0735 155.7656 196.8795 −0.02 101.026 103.6084 109.5259 119.2537 132.0895 155.8634 197.1620 0 101.0265 103.5992 109.5169 119.2704 132.0701 155.8914 197.2938 0.02 101.0263 103.5865 109.499 119.2690 132.0154 155.8494 197.2745 0.04 101.0252 103.5704 109.4722 119.2496 131.9254 155.7375 197.104 0.06 101.0233 103.5510 109.4366 119.212 131.8303 155.5561 196.7832 0.08 101.0206 103.5281 109.3922 119.1565 131.7603 155.3057 196.3135 0.1 101.0170 103.5018 109.3389 119.083 131.7357 154.9868 195.6971
 97.65509 100.4233 106.4819 114.8732 127.3161 150.3700 186.5302 89.13544 98.04504 108.2301 119.1056 131.536 150.3559 172.7989 value −0.1 100.7439 104.0964 110.2180 119.2223 131.6342 155.0711 192.8206 −0.08 100.7418 104.1117 110.2696 119.3008 131.7849 155.371 193.4796 −0.06 100.7389 104.1235 110.3126 119.3606 131.9022 155.6019 193.9968 −0.04 100.7351 104.1319 110.347 119.4016 131.9858 155.7635 194.3700 −0.02 100.7305 104.1368 110.3728 119.4239 132.0357 155.8551 194.5975 0 100.7252 104.1383 110.3899 119.4275 132.0516 155.8766 194.6782 0.02 100.719 104.1363 110.3984 119.4122 132.0337 155.8278 194.6118 0.04 100.7119 104.1309 110.3982 119.3781 131.9819 155.7090 194.3986 0.06 100.7041 104.122 110.3893 119.3253 131.8963 155.5205 194.0396 0.08 100.6955 104.1096 110.3718 119.2538 131.777 155.2627 193.5363 0.1 100.6860 104.0938 110.3456 119.1637 131.6243 154.9363 192.8910
 98.81814 100.6730 107.4055 116.5276 128.5121 148.6026 187.6882 91.74324 97.31070 107.6696 119.5246 131.7449 148.3706 171.5955 value −0.1 101.2921 103.7255 109.9459 119.5251 131.8443 152.1479 190.7650 −0.08 101.3028 103.733 109.9884 119.5365 131.9973 152.3256 191.3691 −0.06 101.3128 103.7371 110.0222 119.6137 132.1165 152.4345 191.8342 −0.04 101.322 103.7377 110.0472 119.6726 132.2019 152.4742 192.1584 −0.02 101.3303 103.735 110.0636 119.7132 132.2532 152.4448 192.3400 0 101.3379 103.7289 110.0711 119.7354 132.2704 152.3461 192.3784 0.02 101.3447 103.7194 110.0700 119.7392 132.2535 152.1787 192.2734 0.04 101.3506 103.7064 110.0601 119.7247 132.2025 151.9428 192.0254 0.06 101.3558 103.6901 110.0415 119.6918 132.1174 151.6391 191.6355 0.08 101.3602 103.6703 110.0141 119.6406 131.9984 151.2684 191.1055 0.1 101.3638 103.6472 109.9781 119.5711 131.8457 150.8318 190.4377
 99.45873 102.7277 107.7386 113.9839 131.7095 153.2816 192.1807 90.94703 99.18717 107.1685 115.5287 133.5669 149.5516 173.2163 value −0.1 101.1575 104.661 109.4991 116.2884 133.6678 154.36 193.0719 −0.08 101.1648 104.6883 109.5359 116.3009 133.821 154.6835 193.7006 −0.06 101.1714 104.7122 109.5642 116.2955 133.9376 154.9408 194.1851 −0.04 101.1771 104.7325 109.5841 116.2724 134.0173 155.1312 194.5234 −0.02 101.1821 104.7493 109.5955 116.2316 134.0601 155.2542 194.7139 0 101.1862 104.7626 109.5984 116.173 134.0659 155.3096 194.7557 0.02 101.1895 104.7724 109.5929 116.0968 134.0346 155.2971 194.6486 0.04 101.192 104.7786 109.5789 116.0031 133.9663 155.2168 194.3932 0.06 101.1937 104.7813 109.5564 115.8918 133.8612 155.0689 193.9905 0.08 101.1946 104.7805 109.5255 115.7631 133.7193 154.8538 193.4425 0.1 101.1947 104.7761 109.4862 115.6172 133.571 154.572 192.7515
 100.2239 102.4000 108.7921 116.2027 131.1301 154.2985 190.0055 91.07041 96.95651 107.8218 117.6336 132.9936 150.1114 172.3139 value −0.1 101.2429 103.7526 110.1224 117.7630 133.1038 155.1954 191.5790 −0.08 101.2519 103.7577 110.1661 117.8328 133.2751 155.5554 192.1719 −0.06 101.2600 103.7593 110.2009 117.8855 133.4115 155.8493 192.6230 −0.04 101.2673 103.7574 110.2269 117.921 133.5129 156.0764 192.9303 −0.02 101.2738 103.752 110.2441 117.9394 133.579 156.236 193.0923 0 101.2795 103.743 110.2523 117.9404 133.6098 156.3278 193.1084 0.02 101.2843 103.7305 110.2516 117.9243 133.6052 156.3516 192.9785 0.04 101.2883 103.7144 110.2421 117.891 133.5652 156.3072 192.7031 0.06 101.2915 103.6948 110.2237 117.8404 133.4899 156.1948 192.2835 0.08 101.2939 103.6717 110.1964 117.7727 133.3793 156.0147 191.7216 0.1 101.2954 103.6451 110.1602 117.6880 133.2338 155.7673 191.0199
 100.3811 102.6419 107.7698 118.1677 130.9040 152.8819 196.5500 90.55378 97.02887 106.4837 119.4323 131.4591 150.0207 173.0612 value −0.1 101.0559 103.5352 109.2741 119.4377 131.5400 154.2308 196.5671 −0.08 101.0611 103.5394 109.2983 119.4612 131.6587 154.4565 197.2139 −0.06 101.0654 103.5402 109.3137 119.5365 131.7426 154.6114 197.7604 −0.04 101.0689 103.5377 109.3204 119.5936 131.7916 154.6951 198.1543 −0.02 101.0716 103.5318 109.3184 119.6323 131.8055 154.7074 198.3935 0 101.0734 103.5226 109.3076 119.6528 131.7843 154.6483 198.4771 0.02 101.0745 103.5100 109.2881 119.6548 131.7281 154.5179 198.4046 0.04 101.0748 103.4941 109.2599 119.6385 131.6170 154.3166 198.1764 0.06 101.0742 103.4748 109.2229 119.6039 131.5610 154.0450 197.7935 0.08 101.0729 103.4522 109.1772 119.5509 131.5105 153.7037 197.2577 0.1 101.0707 103.4262 109.1229 119.4797 131.4656 153.2938 196.5715

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### References

1. P. V. Sukhatme and B. V. Sukhatme, Sampling Theory of Surveys: With Applications, Asia Publishing House, Bombay, India, 2nd edition, 1970.
2. M. N. Murthy, “Product method of estimation,” The Indian Journal of Statistics A, vol. 26, pp. 69–74, 1964. View at: Google Scholar | MathSciNet
3. W. G. Cochran, Sampling Techniques, John Wiley & Sons, New York, NY, USA, 3rd edition, 1977. View at: MathSciNet
4. R Development Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, 2008.
5. A. Sahai, “An efficient variant of the product and ratio estimators,” Statistica Neerlandica, vol. 33, no. 1, pp. 27–35, 1979. View at: Publisher Site | Google Scholar | MathSciNet
6. P. S. Chami, B. Sing, and D. Thomas, “A two-parameter ratio-product-ratio estimator using auxiliary information,” ISRN Probability and Statistics, vol. 2012, Article ID 103860, 15 pages, 2012. View at: Publisher Site | Google Scholar

Copyright © 2014 Angela Shirley et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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