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In chapter three, we utilized the law of direct proportion to establish that the selection probabilities, pi, is a realization of positive correlation between the study

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variables y and the measure of size x, which is an advancement on the classical ratio-estimator to obtain the Hansen-Hurwitzβ€Ÿs ratio-estimator.

The selection probabilities, piβ€Ÿs provided the normed-size measure for estimating population total under the PPSWR sampling scheme while the generalized selection probabilities 𝑝𝑖,𝑔,π‘βˆ— provided a linear transformation that utilized the cth (c=1,2,3,4) moment in correlation coefficient to develop a class of alternative linear estimators. We have shown that for efficiency, the relationship between the statistical properties namely coefficient of variation, skewness and kurtosis of the study variables and measure of size variables and correlation coefficient is expressed by 𝜌1 = 𝐢𝑉𝐢𝑉π‘₯

𝑦 <

1, 𝜌2 < 1, 𝜌3 =𝛾𝛾𝑦

π‘₯ < 1 ; 𝛾π‘₯ β‰  0 and 𝜌4 =𝐾𝐾𝑦

π‘₯ < 1; 𝐾π‘₯ β‰  0.

When c = 1, we showed that 𝜌1 > 𝐢𝑉𝐢𝑉π‘₯

𝑦 < 1 along with the conditions namely 𝜌2 < 1, 𝜌3 =𝛾𝛾𝑦

π‘₯ < 1and 𝜌4 =𝐾𝐾𝑦

π‘₯ < 1 must hold true for the estimator defined by c = 1 to be utilized. This agrees with Cochran(1977) who showed that the ratio estimator is most efficient among other competing estimators when 𝜌1 >2𝐢𝑉𝐢𝑉π‘₯

𝑦 < 1. However, this estimator can only be specified when ρ→0. This again agreed with the positions of Rao(1966), Bansal and singh(1985), Amahia et al(1989), Grewal(1999) among other scholars. We note here that this condition is only true for a linear estimator.

The study have also shown that when there is moment in ρ such that ρ takes a value 0.25<ρ<0.50 or some neighbourhood and ρ2β†’0, then the estimator defined by c

= 2 is best suitable for the target population. If there is further moment in ρ such that 𝜌3 = 𝛾𝛾𝑦

π‘₯ < 1 satisfying 0 < 𝜌3 < 1, then an estimator defined by c = 3 would be the best in term of MSE and relative efficiency. Empirical results have shown that this happens when 0.5<ρ<0.7 and its neighbourhood. Similarly, when 𝜌4 =𝐾𝐾𝑦

π‘₯<1 and 0.7<ρ<0.99, then the estimator defined by c = 4 is the best for the target population.

In situation where negatively correlated variables are encountered, direct transformation of measure of size variables could not provide the desired estimator.

Thus, taking cognizance of the law of inverse proportion and further transformation from inverse to direct proportion by 𝑝𝑖 = 1 π‘₯1 π‘₯𝑖

𝑁 𝑖

𝑖=1 οƒžπ‘π‘– =𝑧𝑍𝑖, the correlation structure

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changed from negative to positive correlation. By this transformation, a Modified Hansen-Hurwitz Estimator (MHHE) or Modified Horvitz-Thompson Estimator (MHTE) has been proposed for any observed case of negative correlation.

It is worth to note here that the MHHE or MHTE determined by 𝑝𝑖 = 1 π‘₯1 π‘₯𝑖

𝑁 𝑖 𝑖=1

possess the properties of harmonic mean which is mostly used when it is desirable to assign lower weights to higher values and higher weights to lower values. The selection probabilities and the derived generalized selection probabilities 𝑝𝑖,𝑔,π‘βˆ— can be utilized in the class of alternative linear estimator by observing the conditions similar with those estimators involving positively correlated variables.

The interesting features of the developed estimators are that their bias, MSEβ€Ÿs and MSEβ€Ÿs coincide with the Raoβ€Ÿs Estimator (RE) when ρ=0 and the conventional HHE or HTE in cases of PPSWR or Ο€PS respectively when ρ=1. This provided boundaries for the linear estimators in PPS sampling scheme different from those defined by Sahoo(1995) estimator which had extended the boundaries but reduced the magnitude of negative correlation by restricting the estimators to instances of strong negative correlation.

The derived expression for determining approximate value of c is another useful means of defining an efficient estimator for a target population. Empirical evidence have shown that the optimum value of c lies between Min pi and Max pi.

The main aim of developing a general class of linear estimator is as a result of the fact of the non-existence of a uniformly most efficient estimator (UMEE) in the parameter space on one hand and the fact that no single estimator can be efficient for all populations and at all conditions. Thus, the class of alternative linear estimators defined by the generalized selection probabilities 𝑝𝑖,π‘”βˆ— = 1βˆ’πœŒπ‘π‘+ πœŒπ‘π‘π‘–, c = 1,2,3,4 for 0<ρ<1 provided the best estimators of population total for any target population.

The implication of the results above is that one estimator, say HHE, ACRE, RE among others cannot be said to be the best for all populations at all times. Thus different populations may have different estimators depending on their correlation coefficient and how it relates with the characteristics of the study populations.

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The results of our empirical studies using sample of size two, that is, n =2 for the four study populations provided practical evidence of the behaviour of the developed class of linear estimators in PPS sampling schemes. Under the sampling design, we can conveniently infer that when ρ=0, the Raoβ€Ÿs estimator would always be the best among other competing estimators. As ρ moves slightly from zero, Raoβ€Ÿs estimator increased in bias and hence MSE and the anticipated MSE or MSE thereby suggesting the estimator defined by c = 1. As ρ moves closer to the 0.5, another estimator defined by c= 2 becomes the best among other competing estimators.

Similarly, as ρ moves slightly away from 0.5 but not so strong, the estimator defined by c = 3 becomes the best. Furthermore, the estimator defined by c = 4 would be the best when it is clear that there is very strong correlation between the variables of study, especially when ρ→1..

Certainly, the proposed estimators form a class of linear estimators bounded by Raoβ€Ÿs estimator by the left and HHE by the right so that all other estimators defined by c = 1, 2, 3 and 4 are found within this class. Therefore, for a given population, the proposed linear estimators provide the best estimators for use in PPS sampling than utilizing the conventional estimator or a specified alternative estimator that are rigidly specified by fixed order of ρ.

The behaviour of the proposed estimators under the Rao-Hartley and Cochran scheme when n=5 is consistent with our earlier findings for n=2 in both PPS and Ο€PS sampling schemes thereby suggests that increasing sample size would not change the estimators for the target populations. However, apart from uniform distribution for which all estimators are equal in performance, empirical evidence have shown that for theoretical populations that are normally distributed, estimators with c=3 or c = 4 performed better than other estimators in terms of MSE or MSE. However, for skewed distributions such as chi-squared and gamma distributions, estimators defined by c = 1 or 2 are best specified when ρ→0 or somewhat moderate. However, as ρ→1, estimators defined by c=3 or c=4 are best specified. Furthermore, the Grewals estimator is only best under super-population model than sampling design and utilizing this estimator would require transformation of c into c* =1/c as shown in chapter three.

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Generally, the discussions above have shown that no single estimator is efficient for all populations and conditions and a rough idea of the magnitude of the correlation between the study and auxiliary variables and hence the size measures would provide insight into which estimator would be best for the target population.

Secondly, the idea of the ratio of coefficients of variations, skewness and kurtosis as related with the correlation coefficient would help in the specification of estimators.

Whereas the information of the target populations is not available to the survey statistician, this study have shown that among the estimators in the class defined by c=1,2,3 and 4 there is the one estimator that is best for estimating population total.

Thus, in this era of information technology, it would be easier to identify such estimator when the suggested estimators are run simultaneously.