In estimation theory, the researchers have put their efforts to develop some estimators of population mean which may give more precise results when adopting ordinary least squares (OLS) method or robust regression techniques for estimating regression coefficients. But when the correlation is negative and the outliers are presented, the results can be distorted and the OLS-type estimators may give misleading estimates or highly biased estimates. Hence, this paper mainly focuses on such issues through the use of non-conventional measures of dispersion and a robust estimation method. Precisely, we have proposed generalized estimators by using the ancillary information of non-conventional measures of dispersion (Gini’s mean difference, Downton’s method and probability-weighted moment) using ordinary least squares and then finally adopting the Huber M-estimation technique on the suggested estimators. The proposed estimators are investigated in the presence of outliers in both situations of negative and positive correlation between study and auxiliary variables. Theoretical comparisons and real data application are provided to show the strength of the proposed generalized estimators. It is found that the proposed generalized Huber-M-type estimators are more efficient than the suggested generalized estimators under the OLS estimation method considered in this study. The new proposed estimators will be useful in the future for data analysis and making decisions.

For obtaining proficient estimators in sampling theory, a multiplicity of techniques has been used and the commonly one is the simple random sampling without replacement (SRSWOR) to obtain an estimator for the population mean, when auxiliary information is not available. But when auxiliary information is available and even has a relationship with study variable, there are lots of methods by which this auxiliary information can be incorporated viz., ratio, product, difference and regression, etc. Utilizing this auxiliary information for parameters will increase the estimation efficiency. The utilization of auxiliary information has been made in a number of ways for achieving the improved estimates of population parameters. Some latest uses of auxiliary information are provided in [

The rest of the paper is organized as follows. In Section 2 shows the generalized estimator, outliers present, negative correlation exist and the adaptation of the OLS method with the expressions of Bias and the mean squared error (MSE) derived up to the second degree of approximation. The generalized estimators based on adopting Huber M estimation instead of OLS and their bias and MSE equations are proposed in Section 3. Efficiency comparisons between the proposed and existing estimators are considered in Section 4. The results of the numerical examples are reported in Section 5. Discussion is devoted to Section 6, and the paper is concluded in the last section.

Let

Reference [

Reference [

where

Using Taylor expansion of order 2 of

Therefore, the bias of the estimator is

The MSE of the proposed estimator in

Product Estimators |
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Note:

The main issue on which we focus in the present study is the proposition of a generalized class of ratio and product estimators that are suitable for data with the existence of outliers. To deal with this situation, we have adopted the Huber M-estimation technique to the developed generalized class of estimators, displayed in

In adopting the Huber M-estimates, the outlier’s negative effect is reduced and valid results are obtained; hence, valid inferences will be drawn from the results. The compromise between

where

with respect to

Then, using the Taylor expansion of order 2 of

Hence, the bias of the estimator is

and the MSE of

Substituting the different values of

Product Estimators |
Ratio Estimators |
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Note:

The efficiencies of the generalized estimators using ancillary information when OLS is adopted are compared with the generalized estimator using the same ancillary information but with Huber M-estimation. For

Since,

When the conditions given in

In this section, we consider three real data populations and their descriptive statistics are summarized in

Parameter | Pop. 1 | Pop. 2 | Pop. 3 | Parameter | Pop. 1 | Pop. 2 | Pop. 3 |
---|---|---|---|---|---|---|---|

20 | 30 | 70 | 0.3943 | 0.13711 | 14.73 | ||

8 | 6 | 14 | −0.9199 | −0.8552 | 0.611 | ||

19.55 | 384.2 | 1251.8 | 13.55 | 55.27 | 1269.5 | ||

18.8 | 67.267 | 248.21 | 5.104 | 60.208 | 659.79 | ||

6.9441 | 59.402 | 226.1 | 4.789 | 59.087 | 563.74 | ||

−47.352 | −472.607 | 5053.116 | 5.3122 | 60.907 | 690.94 | ||

0.3552 | 0.15588 | 18.06 | −0.8617 | −5.5446 | 3.7786 | ||

7.4128 | 9.2324 | 36.56 | −0.4917 | −2.2267 | 2.2058 |

We applied to these data different class members of estimators using both proposed methods with the same auxiliary information; OLS and Huber M-estimation technique. The bias, mean squared error and percent relative efficiency (PRE) of some product types estimators for populations 1, 2 and 3 are given in

Estimators | Bias (OLS) | MSE (OLS) | Estimators | Bias |
MSE |
PRE |
---|---|---|---|---|---|---|

−2.3771 | 1.6106 | −1.8668 | 0.7114 | 226.42 | ||

−2.4088 | 1.5775 | −1.8917 | 0.6918 | 228.02 | ||

−2.3566 | 1.6323 | −1.8506 | 0.7242 | 225.39 | ||

−1.5171 | 2.6620 | −1.1914 | 1.3935 | 191.03 | ||

−1.5300 | 2.6441 | −1.2015 | 1.3811 | 191.45 | ||

−1.5087 | 2.6737 | −1.1848 | 1.4016 | 190.76 | ||

38.4376 | 377.0613 | 30.1856 | 358.6372 | 105.14 | ||

50.0254 | 604.8798 | 39.2856 | 581.4802 | 104.02 | ||

33.3341 | 293.7308 | 26.1777 | 277.4980 | 105.85 | ||

−4.2879 | 0.3314 | −3.3674 | 0.2526 | 131.19 | ||

−4.1799 | 0.3649 | −3.2826 | 0.2397 | 152.22 | ||

−4.3625 | 0.3111 | −3.4259 | 0.2643 | 117.71 | ||

−3.0517 | 0.9925 | −2.3965 | 0.3829 | 259.20 | ||

0.0000 | 0.0222 | 0.0000 | 189.34 | |||

2.8738 | 9.9511 | 2.4017 | 7.1854 | 138.49 | ||

2.8738 | 10.0245 | 2.4453 | 7.2475 | 138.32 | ||

2.8738 | 9.9038 | 2.3736 | 7.1453 | 138.60 |

C | Bias (OLS) | MSE (OLS) | Estimators | Bias |
MSE |
PRE |
---|---|---|---|---|---|---|

−5.9238 | 13.5806 | −4.1528 | 6.0296 | 225.23 | ||

−5.9763 | 13.5042 | −4.1896 | 5.9846 | 225.65 | ||

−5.8915 | 13.6278 | −4.1302 | 6.0574 | 224.98 | ||

−4.1322 | 16.3379 | −2.8968 | 7.7157 | 211.75 | ||

−4.1577 | 16.2966 | −2.9147 | 7.6896 | 211.93 | ||

−4.1164 | 16.3635 | −2.8858 | 7.7318 | 211.64 | ||

11.1437 | 51.7997 | 7.8122 | 34.0438 | 152.16 | ||

11.3635 | 52.4662 | 7.9663 | 34.5788 | 151.73 | ||

11.0109 | 51.3992 | 7.7191 | 33.7227 | 152.42 | ||

240.8527 | 3164.9230 | 168.8476 | 3009.8200 | 105.15 | ||

413.8980 | 8705.1820 | 290.1594 | 8446.6120 | 103.06 | ||

191.0471 | 2079.1650 | 133.9318 | 1953.8410 | 106.41 | ||

−11.2283 | 7.1426 | −7.8715 | 2.7633 | 258.48 | ||

0.0000 | 0.3196 | 0.0000 | 0.0289 | 1106.54 | ||

7.3025 | 36.8603 | 5.5805 | 22.3234 | 165.12 | ||

7.3811 | 36.9896 | 5.6439 | 22.4222 | 164.97 | ||

7.2543 | 36.7810 | 5.5417 | 22.2628 | 165.21 |

C | Bias (OLS) | MSE (OLS) | Estimators | Bias |
MSE |
PRE |
---|---|---|---|---|---|---|

−88997.86 | 530757.70 | −70474.42 | 385872.40 | 137.547 | ||

−99525.91 | 513770.30 | −78811.23 | 373267.40 | 137.641 | ||

−86045.96 | 535602.30 | −68136.91 | 389488.20 | 137.514 | ||

−37111.39 | 621117.30 | −29387.27 | 454633.30 | 136.619 | ||

−38823.93 | 617958.80 | −30743.36 | 452187.70 | 136.660 | ||

−36587.99 | 622085.00 | −28972.80 | 455383.10 | 136.607 | ||

27775.66 | 749649.60 | 21994.61 | 556155.30 | 134.791 | ||

29362.17 | 753008.40 | 23250.92 | 558853.70 | 134.742 | ||

27297.32 | 748638.90 | 21615.83 | 555343.70 | 134.806 | ||

97169.20 | 906210.30 | 76945.03 | 683829.70 | 132.520 | ||

119817.80 | 961581.60 | 94879.67 | 729773.20 | 131.764 | ||

91556.54 | 892813.50 | 72500.55 | 672769.40 | 132.707 | ||

−325572.50 | 258658.60 | −257810.00 | 212251.60 | 121.864 | ||

0.00 | 86714.26 | 0.00 | 63933.60 | 135.632 | ||

−60111.50 | 557794.30 | −44961.00 | 406163.10 | 137.333 | ||

−70477.93 | 529514.60 | −51796.09 | 384946.00 | 137.556 | ||

−57329.50 | 564985.00 | −43054.12 | 411603.50 | 137.264 |

C | Bias (OLS) | MSE (OLS) | Estimators | Bias |
MSE |
PRE |
---|---|---|---|---|---|---|

−0.000054 | 1.355528 | −0.510384 | 0.565790 | 239.58 | ||

−0.000054 | 1.385855 | −0.517200 | 0.582487 | 237.92 | ||

−0.000053 | 1.336132 | −0.505977 | 0.555208 | 240.65 | ||

−0.000034 | 0.687007 | −0.325739 | 0.266521 | 257.77 | ||

−0.000035 | 0.694834 | −0.328502 | 0.268823 | 258.47 | ||

−0.000034 | 0.681941 | −0.323938 | 0.265057 | 257.28 | ||

−0.000033 | 0.649607 | −0.312200 | 0.256197 | 253.56 | ||

−0.000033 | 0.657102 | −0.314960 | 0.258173 | 254.52 | ||

−0.000033 | 0.644760 | −0.310402 | 0.254945 | 252.90 | ||

−0.000053 | 1.314725 | −0.501068 | 0.543618 | 241.85 | ||

−0.000053 | 1.345963 | −0.508216 | 0.560562 | 240.11 | ||

−0.000052 | 1.294792 | −0.496454 | 0.532913 | 242.97 | ||

−0.000068 | 2.016678 | −0.642786 | 0.962165 | 209.60 | ||

0.000000 | 0.022234 | 0.000000 | 190.24 | |||

−0.077200 | 0.974100 | −0.492400 | 0.374700 | 259.99 | ||

−0.079400 | 0.995600 | −0.500500 | 0.384300 | 259.07 | ||

−0.075800 | 0.960400 | −0.487200 | 0.368600 | 260.53 |

C | Bias (OLS) | MSE (OLS) | Estimators | Bias |
MSE |
PRE |
---|---|---|---|---|---|---|

−0.0048 | 3.6054 | −1.7758 | 2.0236 | 178.17 | ||

−0.0048 | 3.6340 | −1.7915 | 2.0208 | 179.83 | ||

−0.0048 | 3.5879 | −1.7661 | 2.0255 | 177.14 | ||

−0.0034 | 2.7823 | −1.2387 | 2.2718 | 122.47 | ||

−0.0034 | 2.7920 | −1.2463 | 2.2662 | 123.20 | ||

−0.0033 | 2.7764 | −1.2340 | 2.2753 | 122.02 | ||

−0.0030 | 2.6390 | −1.1190 | 2.3673 | 111.48 | ||

−0.0030 | 2.6473 | −1.1263 | 2.3610 | 112.13 | ||

−0.0030 | 2.6339 | −1.1145 | 2.3711 | 111.08 | ||

−0.0044 | 3.3767 | −1.6443 | 2.0572 | 164.14 | ||

−0.0045 | 3.4032 | −1.6601 | 2.0522 | 165.83 | ||

−0.0044 | 3.3605 | −1.6346 | 2.0603 | 163.10 | ||

−0.0091 | 7.7629 | −3.3645 | 3.0123 | 257.71 | ||

0.0000 | 0.3196 | 0.0000 | 1106.54 | |||

−0.0229 | 3.3017 | −1.6173 | 2.0729 | 159.28 | ||

−0.0233 | 3.3268 | −1.6331 | 2.0673 | 160.92 | ||

−0.0226 | 3.2864 | −1.6077 | 2.0764 | 158.27 |

C | Bias (OLS) | MSE (OLS) | Estimators | Bias |
MSE |
PRE |
---|---|---|---|---|---|---|

34754.07 | 279084.10 | 16230.63 | 220918.80 | 126.329 | ||

41013.26 | 287533.90 | 20298.58 | 224986.00 | 127.801 | ||

33080.66 | 276796.50 | 15171.61 | 219860.00 | 125.897 | ||

10544.88 | 244079.20 | 2820.75 | 207512.40 | 117.622 | ||

11167.77 | 245058.40 | 3087.21 | 207778.80 | 117.942 | ||

10356.90 | 243782.30 | 2741.71 | 207433.40 | 117.523 | ||

−4197.67 | 215835.90 | 1583.38 | 206279.50 | 104.633 | ||

−4341.94 | 215361.50 | 1769.31 | 206465.50 | 104.309 | ||

−4152.14 | 215980.90 | 1529.35 | 206225.40 | 104.731 | ||

−862.47 | 204732.60 | 19361.70 | 224062.50 | 91.373 | ||

4499.39 | 205382.10 | 29437.49 | 234139.80 | 87.718 | ||

−1866.06 | 204896.90 | 17189.93 | 221890.40 | 92.341 | ||

285027.40 | 578580.50 | 217264.80 | 421936.90 | 137.125 | ||

0.00 | 63933.60 | 0.00 | 86714.26 | 73.729 | ||

13326.80 | 266965.70 | −1823.70 | 215546.30 | 123.855 | ||

15910.33 | 279679.70 | −2771.50 | 221197.60 | 126.439 | ||

12656.01 | 263997.50 | −1619.37 | 214328.30 | 123.174 |

From

Based on the above discussion and numerical study, we can conclude that adopting Huber M instead of OLS, especially when outliers are presented, has superiority in precision (see

The authors would like to thank the editor in chief and worthy referees for valuable suggestions for giving the final shape of the manuscript.