5.4 Estimate of bias of the alternative linear Estimators
5.4.2 Design-based Relative MSE of the proposed estimators compared
We utilized the conventional Hansen and Hurwitz Estimator (HHE) as the denominator in order to compare the performance of the proposed alternative linear estimators using the relative efficiency criteria defined by
𝑅𝐸 𝜏 𝑔,𝑐\𝜏 𝐻𝐻𝐸 = 0 or 𝑀𝑆𝐸(𝜏 𝑔,𝑐) = 𝑀𝑆𝐸(𝜏 𝐻𝐻)
Any alternative estimator is relatively more efficient than HHE in terms of minimum variance and hence, MSE if and only if
93 𝑅𝐸 𝜏 𝑔,𝑐\𝜏 𝐻𝐻𝐸 < 1 (100%)
or
𝑅𝐸 𝜏 𝑔,𝑐\𝜏 𝐻𝐻𝐸 < 𝑅𝐸 𝜏 𝐻𝐻\𝜏 𝐻𝐻𝐸
otherwise, HHE is relatively the most efficient estimator for the study population.
The efficiency of the proposed estimator given the conventional HHE for the PPS sampling design is presented in Tables 14, 15, 16 and 17 above for populations I to IV respectively.
Table 14: Design-based Relative efficiency of alternative estimators as compared with HHE for population I (measured by RE(𝜏 𝑔,𝑐\𝜏 𝐻𝐻) =𝑴𝑺𝑬(𝝉 𝑴𝑺𝑬(𝝉 𝒈,𝒄)
𝑯𝑯)< 𝟏 ) Rho RE(𝜏 𝐻𝐻\𝜏 𝐻𝐻) RE(𝜏 1\𝜏 𝐻𝐻) RE(𝜏 2\𝜏 𝐻𝐻) RE(𝜏 3\𝜏 𝐻𝐻) RE(𝜏 4\𝜏 𝐻𝐻)
0.000 100.0 57.0 57.0 57.0 57.0
0.100 100.0 56.1 56.9 57.0 57.0
0.162 100.0 56.1 56.7 57.0 57.0
0.500 100.0 64.2 56.9 56.0 56.3
0.900 100.0 90.4 82.9 76.9 72.2
1.000 100.0 100.0 100.0 100.0 100.0
From table 14 above, it is clear that RE(𝜏 𝐻𝐻\𝜏 𝑔,𝑐=1) < RE(𝜏 𝐻𝐻\𝜏 𝑔,𝑐), c = 2,3,4 and also, RE(𝜏 𝐻𝐻\𝜏 𝑔,𝑐=1) < RE(𝜏 𝐻𝐻\𝜏 𝐻𝐻) for population I at 𝜌 = 0.162 in terms of minimum variance. Thus, the estimator defined by c = 1 has minimum percentage relative MSE of 56.1% and this further confirm our postulation that the specification, c = 1 is only possible when ρ0. 𝑅𝐸(𝜏 𝑐)
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Table 15: Design-based Relative efficiency of alternative estimators as compared with HHE for population II (measured by RE(𝜏 𝑔,𝑐\𝜏 𝐻𝐻) =𝑴𝑺𝑬(𝝉 𝑴𝑺𝑬(𝝉 𝒈,𝒄)
𝑯𝑯)< 𝟏 ) Rho RE(𝜏 𝐻𝐻\𝜏 𝐻𝐻) RE(𝜏 1\𝜏 𝐻𝐻) RE(𝜏 2\𝜏 𝐻𝐻) RE(𝜏 3\𝜏 𝐻𝐻) RE(𝜏 4\𝜏 𝐻𝐻)
0.000 100.0 35.2 35.2 35.2 35.2
0.100 100.0 31.6 34.7 35.1 35.2
0.395 100.0 30.8 30.4 32.8 34.1
0.500 100.0 33.8 29.5 31.0 32.7
0.900 100.0 73.7 58.4 48.8 42.4
1.000 100.0 100.0 100.0 100.0 100.0
For population II, RE(𝜏 𝐻𝐻\𝜏 𝑔,𝑐=2) < RE(𝜏 𝐻𝐻\𝜏 𝑔,𝑐), c = 1,3,4 and also, RE(𝜏 𝐻𝐻\𝜏 𝑔,𝑐=2) <
RE(𝜏 𝐻𝐻\𝜏 𝐻𝐻) so that the estimator defined by c = 2 with minimum percentage relative MSE of 30.4% performed better than all the competing estimators including the conventional estimator at 𝜌 = 0.395 as shown on table 15. Again, we have postulated that this is possible when ρ<1, 𝜌2 < 1 and 𝐶𝑣𝑥 < 𝐶𝑣𝑦. It is also clear that as ρ shift upwards, say, ρ → 0.5 rather than ρ0, c=2 is best specified for a target population.
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Table 16: Design-based Relative efficiency of alternative estimators as compared with HHE for population III (measured by RE(𝜏 𝑔,𝑐\𝜏 𝐻𝐻) =𝑴𝑺𝑬(𝝉 𝑴𝑺𝑬(𝝉 𝒈,𝒄)
𝑯𝑯)< 𝟏 ) Rho RE(𝜏 𝐻𝐻\𝜏 𝐻𝐻) RE(𝜏 1\𝜏 𝐻𝐻) RE(𝜏 2\𝜏 𝐻𝐻) RE(𝜏 3\𝜏 𝐻𝐻) RE(𝜏 4\𝜏 𝐻𝐻)
0 100 44.2 44.2 44.2 44.2
0.1 100 39.3 43.4 44.1 44.1
0.3 100 40.4 39.2 42 43.4
0.5 100 48 38.8 38.8 40.5
0.9 100 84.3 73 64.7 58.4
1 100 100 100 100 100
Population III is analysed under two conditions of correlation, that is, at the actual correlation coefficient of 𝜌 = −0.32 and the correlation coefficient of 𝜌 = 0.55 realized after transformation of the measure of size variable. Examining the results on table 16 above, it is observed that the estimator defined by c = 2 with minimum percentage relative MSE of 39.24% performed better than all the competing estimators including the conventional estimator at 𝜌 = −0.32 . The result is the same when 𝜌 = 0.55 is considered. However, at 𝜌 = 0.55, minimum percentage relative MSE is obtained at two points namely, c=2 and c=3 as shown on table 16. Thus, RE(𝜏 𝐻𝐻\𝜏 𝑔,𝑐=2)= RE(𝜏 𝐻𝐻\𝜏 𝑔,𝑐=3) < RE(𝜏 𝐻𝐻\𝜏 𝑔,𝑐=1) and RE(𝜏 𝐻𝐻\𝜏 𝑔,𝑐=3) <
RE(𝜏 𝐻𝐻\𝜏 𝑔,𝑐=4) < RE(𝜏 𝐻𝐻\𝜏 𝐻𝐻).
Based on this result, we can infer as follows:
i. When 0.3 < ρ < 0.5 or neighbourhood, the estimator defined by c =2, that is, 𝜏 𝑔,𝑐=2 would be relatively more efficient than all other estimators;
ii. When ρ is slightly greater than 0.5, the estimator changes from c=2 to c=3 as evidenced in our study which c = 3 at ρ = 0.55.
This suggests that ρ = 0.55 is perhaps a boundary point for which two estimators defined by c=2 and c=3 performed best for population III.
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Table 17: Design-based Relative efficiency of alternative estimators as compared with HHE for population IV (measured by RE(𝜏 𝑔,𝑐\𝜏 𝐻𝐻) =𝑴𝑺𝑬(𝝉 𝑴𝑺𝑬(𝝉 𝒈,𝒄)
𝑯𝑯)< 𝟏 ) Rho RE(𝜏 𝐻𝐻\𝜏 𝐻𝐻) RE(𝜏 1\𝜏 𝐻𝐻) RE(𝜏 2\𝜏 𝐻𝐻) RE(𝜏 3\𝜏 𝐻𝐻) RE(𝜏 4\𝜏 𝐻𝐻)
0 100 12.5 12.5 12.5 12.5
0.1 100 12.2 12.2 12.5 12.5
0.5 100 22.6 15.8 12.6 11.7
0.8 100 32.9 25.5 21.6 18.8
0.9 100 48.5 36.9 31.6 28.5
1 100 100 100 100 100
Again, for population IV, the percentage relative is analysed at the correlation coefficient of 𝜌 = −0.775 and the correlation coefficient of 𝜌 = 0.91 realized after transformation of the measure of size variable. Looking at results on table 17 above, it is clear that RE(𝜏 𝐻𝐻\𝜏 𝑔,𝑐=4) < RE(𝜏 𝐻𝐻\𝜏 𝑔,𝑐), c = 1,2,3 and also, RE(𝜏 𝐻𝐻\𝜏 𝑔,𝑐=4) <
RE(𝜏 𝐻𝐻\𝜏 𝐻𝐻) for population IV so that the estimator defined by c = 4 is relatively more efficient with percentage relative MSE of 18.8% when 𝜌 = −0.775 and 28.5%
when 𝜌 = 0.91 . It is also clear from results on table 16 that estimators defined by c
=4 is relatively more efficient than all other estimators for population IV.
By these results, it will be convenient to state that the specification parameter of an estimator c, changes with ρ. Thus,
i. As ρ0, estimator defined by c =1 would be appropriate;
ii. As ρ0.5, estimator defined by c = 2 would be preferred;
iii. As 0.5<ρ<.75, estimator defined by c=3 would be preferred while iv. As ρ1, estimator defined by c = 4 would be preferred.
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The results described above are further displayed on figures 1 to 4 below, showing the relative performances of the proposed estimators in the parameter space, 𝜏 𝑔,𝑐 c=1,2,3 and 4.
Figures 1 to 4 below presents the graphical view of the alternative estimators as compared with the Hansen and Hurwitz estimator for the four study populations.
On figure 1 above, the behaviour of the estimators in the parameters space with respect to MSE defined by the relative efficiency (RE) is presented. Again, It is clear that the
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estimator defined by c = 1 (with green coloured line), otherwise, the ACRE is uniformly most efficient (UME) estimator for population I when 0<ρ≤0.162 and neighbourhood. However, when 0.16<ρ≤0.50 and its neighbourhood, estimators defined by c = 2 and c = 3 performed equally better than other estimators. However, for ρ > 0.50, the estimator defined by c = 4 performed better than all other estimators.
For population II, results displayed on figure 2 above that for the values of ρ =0.395 and its neighbourhood, the estimator defined by c=2 (in grey colour) is most efficient for the 0.39 <ρ<0.5.
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It is also noticeable here that the estimator defined by c = 1 performed better than all other estimators if 0.01 <ρ<0.39 were assumed for this study population. As ρ>0.5, the estimator defined by c = 4 performed better than all other estimators.
For population III, the result displayed on figure 3 shows that the estimator defined by c = 1 is most efficient for 0 <ρ<0.28 or neighbourhood. However, when 0.30 <ρ<0.50 or its neighbourhood, estimator defined by c = 2 (with grey lines) performed best than all other estimators. As ρ>0.60, the estimator defined by c = 4 performed better than all other estimators in this class. Similarly, under linear transformation, two estimators
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namely, c=2 and c=3 performed equally well for population III under the derived value of ρ=0.55
In the case of population IV, the results displayed on figure 4 above shows that the estimator defined by c = 4 (with yellow line) is relatively most efficient than all other estimators including the conventional estimator throughout the parameter space
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defined by the correlation coefficient ρ. It is worthy to state here that all the values of correlation considered for this population is very high.
A closer look at figures 1 to 4 shows that the efficiency of estimators is changing along moments in the correlation coefficient. Thus, for populations that are weakly correlated, the estimators defined by c = 1 is sufficient. It is also noticeable that there are certain points in the moments in ρ in which two estimators could perform best and these points are the adjoining points, otherwise, boundary point between two estimators.
By these results, it is noticeable that there is no single estimator that is uniformly most efficient in the parameter space especially when correlation coefficient is weak.
Even when correlation coefficient is high, there are points whereby other estimators perform equally well or even better than other estimators. This suggests the need to identify the conditions that bring about the change in estimators at varying levels of correlation coefficient.
5.4.3 Expected Mean squared Error of Alternative Linear Estimators as