The present paper utilizes auxiliary information to neutralize the effect of non-response for estimating the population mean. Improved ratio type estimators for population mean have been proposed and their properties are studied. These estimators are suggested for both single phase sampling and two phase sampling in presence of non-response. Empirical studies are conducted to validate the theoretical results and demonstrate the performance of the proposed estimators. The proposed estimators are shown to perform better than those used by Cochran (
The problem of non-response is inevitable in most sample surveys because the information cannot be obtained from all units selected in the survey due to various reasons. An estimator based on such incomplete information is biased and the final outcome may be misleading, when the respondents differ from non-respondents. In their seminal paper Hansen and Hurwitz (1946) considered a technique of sub-sampling the non-respondents in order to adjust for the non-response bias in a mail survey.
In sampling theory, it is well known that the efficiency of the estimators of unknown population parameters of the study variable can be increased by suitably using known information on an auxiliary variable. The ratio, product and regression methods of estimation are good examples in this context. Non-response adversely affects the estimate of population mean and population variance; in addition, many authors have suggested a number of estimators to estimate population parameter and their variance under the non-response for various situations. Cochran (1977) and Rao (1986) suggested a ratio method to estimate the population mean
Khare and Srivastava (1995) suggested an estimation procedure of the population mean using an auxiliary character in the presence of non-response, Khare and Srivastava (1995) proposed the studying of a conventional and alternative two phase sampling ratio, product and regression estimators in the presence of non-response. Khare and Srivastava (1997) proposed transformed ratio type estimators for the population mean in the presence of non-response. Okafor and Lee (2000) proposed a double sampling scheme for ratio and regression estimation with sub sampling; in addition, the non-respondent also deal with the non-response problem. Khare and Srivastava (1993). Singh and Kumar (2008) proposed a general class of population mean estimators in survey sampling using auxiliary information with sub sampling for the non-respondent. Singh
An interesting finding of all these papers was that the regression (difference) estimators were found to be best in terms of mean squared error (MSE); in addition, any ratio type estimator can at best attain the MSE of these regression (difference) estimators. In this paper, we have proposed some improvement over regression as well as ratio estimators proposed by various authors in their earlier works.
Hansen and Hurwitz (1946) considered mail surveys in the first attempt-, and personal interviews in the second attempt. In the Hansen & Hurwitz method, the population of size of
where
where
Let
The variance of
where
Similarly, In two phase sampling, we have
When
where
where
When
where
where
When
where
where
When
To the first order of approximation, the MSEs of the estimators
where optimum values of
When
To the first order of approximation, the MSEs of the estimators
where
When
To the first order of approximation, the MSE’s of the estimators
where
When
To the first order of approximation, the MSEs of the above estimators are given by
where optimum values of
Bhushan and Pandey (2017) proposed some improved regression type estimators under non-response in seven different strategies using Searls methodology (Searls, 1964). These estimators were an improvement over the corresponding regression estimators, which are BLUE, under non-response in seven different strategies stated as follow.
where
obviously,
The optimal values of
where
Bhushan and Pandey (2017) showed that the these estimators were better than conventional regression estimators. In this article, we propose some new ratio type estimators and compared these with the corresponding regression estimators given earlier. The proposed estimators are motivated by Cochran (1977), Khare and Srivastava (1995), Rao (1986), Okafor and Lee (2000), and Singh and Kumar (2008) under the one phase and two phase sampling using seven different strategies under non-response.
We propose improved ratio type estimators using Searls methodology (Searls, 1964), Searls (1964) proposed a technique to improve the conventional estimators by multiplying a tuning constant term
optimum value of
The proposed estimator under Strategy I, when
The proposed estimator under Strategy II, when
The proposed estimator under Strategy III, when
The proposed estimator under Strategy IV, when
The proposed estimator under Strategy V, when
The proposed estimator under Strategy VI, when
The proposed estimator under Strategy VII, when
where
It is interesting to note that simultaneous optimization with respect to the characterizing scalars
The only way ascertain (
In order to have a better understanding about the efficiency of the proposed estimators we have conducted a comprehensive empirical study on three populations and compared the proposed estimators with the existing estimators. The percentage relative efficiency (PRE) is calculated as
The first population considered by Srivastava (1993,
The second population considered by Khare and Kumar (2011). For the population of 96 villages of rural areas under Police Station, Singur, District Hooghly from district census Handbook (1981), published by the government of India, the data on the number of cultivators
Third population considered from Srivastava (1993, p.50). The data belongs to the data on physical growth of upper-socio-economic group of 95 school children of Varanasi under an ICMR study. The first 25% (i.e., 24 children) units have been considered as non-response units. The values of the parameters related to the study variate
From perusal of above results it is observed that the new ratio type estimators proposed
In this section, simulation is conducted to evaluate the performance of the proposed class of estimators with respect to traditional estimators. For this study we have generated a population size
From the above computational results as shown in Tables 1–3
Mean squared error and percentage relative efficiency of the estimators with respect to
Estimator | |||||
---|---|---|---|---|---|
1 | 2 | 3 | 4 | ||
10160.17 (100.00) | 10636.88 (100.00) | 11113.60 (100.00) | 11590.32 (100.00) | ||
10054.08 (101.05) | 10520.67 (101.10) | 10986.79 (101.15) | 11452.47 (101.20) | ||
Strategy I | 4288.77 (236.90) | 4745.94 (224.13) | 5195.20 (213.92) | 5637.74 (205.58) | |
4269.75 (237.95) | 4722.66 (225.23) | 5167.32 (215.07) | 5604.92 (206.78) | ||
4247.51 ( | 4696.51 ( | 5137.05 ( | 5570.35 ( | ||
Strategy II | 4298.93 (236.34) | 4775.65 (222.73) | 5252.36 (211.59) | 5729.08 (202.31) | |
4279.82 (237.39) | 4752.08 (223.83) | 5223.87 (212.74) | 5695.19 (203.51) | ||
4259.43 ( | 4729.86 ( | 5199.82 ( | 5669.32 ( | ||
Strategy III | 10065.76 (100.94) | 10448.08 (101.81) | 10830.39 (102.61) | 11212.71 (103.37) | |
9961.63 (101.99) | 10335.93 (102.91) | 10709.93 (103.77) | 11083.65 (104.57) | ||
9959.45 ( | 10331.38 ( | 10702.83 ( | 11073.81 ( | ||
Strategy IV | 4204.53 (241.65) | 4586.84 (231.90) | 4969.16 (223.65) | 5351.47 (216.58) | |
4186.25 (242.70) | 4565.09 (233.00) | 4961.16 (224.80) | 5321.89 (217.78) | ||
4164.88 ( | 4540.74 ( | 4916.11 ( | 5291.00 ( | ||
Strategy V | 6727.54 (151.02) | 7182.84 (148.09) | 7638.14 (145.50) | 8093.44 (143.21) | |
6680.87 (152.07) | 7123.29 (149.32) | 7554.66 (147.11) | 7977.40 (145.29) | ||
6662.06 ( | 7104.86 ( | 7546.58 ( | 7987.23 ( | ||
Strategy VI | 6741.11 (150.72) | 7217.83 (147.37) | 7694.55 (144.43) | 8171.26 (141.84) | |
6694.24 (151.77) | 7164.13 (148.47) | 7633.55 (145.58) | 8102.50 (143.04) | ||
6677.77 ( | 7146.59 ( | 7614.95 ( | 8082.86 ( | ||
Strategy VII | 6646.71 (152.86) | 7029.02 (151.33) | 7411.34 (149.95) | 7793.66 (148.71) | |
6601.14 (153.91) | 6978.08 (152.43) | 7354.73 (151.10) | 7731.08 (149.91) | ||
6583.19 ( | 6957.40 ( | 7331.14 ( | 7704.39 ( |
Mean squared error and percentage relative efficiency of the estimators with respect to
Estimator | |||||
---|---|---|---|---|---|
1 | 2 | 3 | 4 | ||
1220.94 (100.00) | 1316.63 (100.00) | 1412.31 (100.00) | 1508.00 (100.00) | ||
1178.98 (103.56) | 1267.96 (103.84) | 1356.47 (104.12) | 1444.51 (104.40) | ||
Strategy I | 225.71 (540.93) | 245.57 (536.15) | 265.30 (532.34) | 284.93 (529.25) | |
224.23 (544.49) | 243.83 (539.99) | 263.26 (536.46) | 282.58 (533.65) | ||
223.29 ( | 242.63 ( | 261.79 ( | 280.80 ( | ||
Strategy II | 301.36 (405.14) | 397.05(331.61) | 492.74 (286.62) | 588.43 (256.28) | |
298.74 (408.69) | 392.51(335.44) | 485.76 (290.74) | 578.50 (260.67) | ||
297.73 ( | 391.21 ( | 484.18 ( | 576.64 ( | ||
Strategy III | 1144.29 (106.70) | 1163.33 (113.18) | 1182.37 (119.45) | 1201.41 (125.52) | |
1107.36 (110.26) | 1125.18 (117.01) | 1142.98 (123.56) | 1160.76 (129.91) | ||
1106.75 ( | 1123.95 ( | 1141.10 ( | 1158.21 ( | ||
Strategy IV | 224.72 (312.90) | 243.76 (291.08) | 262.79 (277.11) | 281.84 (267.40) | |
223.25 (546.88) | 242.04 (543.98) | 260.80 (541.53) | 279.54 (539.46) | ||
222.32 ( | 240.86 ( | 259.36 ( | 277.81 ( | ||
Strategy V | 380.12 (226.28) | 400.02 (223.52) | 419.93 (221.57) | 439.83 (220.11) | |
375.95 (324.76) | 395.33 (333.05) | 414.54 (340.69) | 433.63 (347.76) | ||
374.68 ( | 393.68 ( | 412.61 ( | 431.48 ( | ||
Strategy VI | 455.79 (184.51) | 551.48 (162.44) | 647.16 (149.51) | 742.85 (141.01) | |
449.81 (271.43) | 542.75 (242.58) | 635.18 (222.35) | 727.11 (207.40) | ||
448.60 ( | 541.30 ( | 633.50 ( | 725.19 ( | ||
Strategy VII | 379.14 (227.20) | 398.18 (224.87) | 417.22 (223.21) | 436.26 (221.97) | |
374.99 (325.59) | 393.61 (334.50) | 412.21 (342.62) | 430.78 (350.06) | ||
373.74 ( | 392.05 ( | 410.31 ( | 428.53 ( |
Mean squared error and percentage relative efficiency of the estimators with respect to
Estimator | |||||
---|---|---|---|---|---|
1 | 2 | 3 | 4 | ||
0.2067 (100.000) | 0.2464 (100.000) | 0.2860 (100.000) | 0.3256 (100.000) | ||
0.2066 (100.054) | 0.2462 (100.064) | 0.2857 (100.075) | 0.3253 (100.085) | ||
Strategy I | 0.0664 (311.050) | 0.0853 (288.830) | 0.1040 (274.840) | 0.1227 (265.220) | |
0.0664 ( | 0.0852 ( | 0.1040 ( | 0.1227 ( | ||
0.0664(311.060) | 0.0853 (288.840) | 0.1040 (274.860) | 0.1227 (265.250) | ||
Strategy II | 0.0871 (237.290) | 0.1267 (194.380) | 0.1663 (171.900) | 0.2060 (158.070) | |
0.0871 ( | 0.1267 ( | 0.1663 ( | 0.2058 ( | ||
0.0871 (237.310) | 0.1267 (194.410) | 0.1663 (171.950) | 0.2059 (158.130) | ||
Strategy III | 0.1857 (111.340) | 0.2043 (120.610) | 0.2228 (128.350) | 0.2414 (134.890) | |
0.1856 ( | 0.2041 ( | 0.2227 ( | 0.2412 ( | ||
0.1856 (111.390) | 0.2042 (120.670) | 0.2227 (128.410) | 0.2412 (134.960) | ||
Strategy IV | 0.0660 (312.890) | 0.0846 (291.080) | 0.1032 (277.110) | 0.1217 (267.400) | |
0.0660 ( | 0.0846 ( | 0.1031 ( | 0.1217 ( | ||
0.0660 (312.910) | 0.0846 (291.090) | 0.1032 (277.130) | 0.1217 (267.430) | ||
Strategy V | 0.0913 (226.280) | 0.1102 (223.520) | 0.1290 (221.570) | 0.1479 (220.110) | |
0.0913 ( | 0.1101 ( | 0.1288 ( | 0.1475 ( | ||
0.0913 (226.300) | 0.1102 (223.550) | 0.1290 (221.600) | 0.1479 (220.150) | ||
Strategy VI | 0.1120 (184.510) | 0.1516 (162.440) | 0.1913 (149.510) | 0.2309 (141.010) | |
0.1120 (184.568) | 0.1516 (162.503) | 0.1912 ( | 0.2307 ( | ||
0.1120 184.540) | 0.1516 (162.480) | 0.1912 (149.560) | 0.2308 (141.080) | ||
Strategy VII | 0.0910 (227.200) | 0.1095 (224.870) | 0.1281 (223.210) | 0.1467 (221.970) | |
0.0909 ( | 0.1095 ( | 0.1280 ( | 0.1466 ( | ||
0.0909 (227.230) | 0.1095 (224.890) | 0.1281 (223.240) | 0.1467 (222.010) |
Percentage relative efficiency (PRE) of the proposed estimators with respect to
Estimator | PRE | |
---|---|---|
100 | ||
100.086 | ||
Strategy I | 165.163 | |
166.775 | ||
Strategy II | 142.838 | |
143.0174 | ||
Strategy III | 120.771 | |
120.795 | ||
Strategy IV | 168.449 | |
168.503 | ||
Strategy V | 148.608 | |
149.331 | ||
Strategy VI | 130.484 | |
130.495 | ||
Strategy VII | 153.491 | |
153.893 | ||