Introduction

In sample surveys, auxiliary attribute is always used to increase the precision of estimated of population parameters. This can be done at either estimation or selection stage or both stages. The commonly used estimators, which make use of auxiliary variables, include ratio estimator, product estimator, regression and difference estimator. The classical ratio estimator is preferred when there is a high positive correlation between the variable of interest, Y and the auxiliary variable, X with the regression line passing through the origin. The classical product estimator, on the other hand is most preferred when there is a high negative correlation between Y and X while the linear regression estimator is most preferred when there is a high positive correlation between the two variables and the regression line of the study variable on the auxiliary has intercept on Y axis. The classical ratio and product estimators even though considered to be more useful in many practical situations have efficiencies which does not exceed that of the linear regression.

The use of auxiliary information has become indispensable for improving the exact of the estimators of population parameters like the mean and variance of the variable under study. A great variety of the techniques such as the ratio, product and regression methods of estimation are commonly known in this esteem. Keeping this fact in view, large number of estimators have been suggested in sampling literature. Some noteworthy contribution in this direction have been made by^{1,2,4,5,6,7,8,9,11,12,13,15,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34} and many others.

The weaknesses discovered in this research is that ²¹estimators are ratio-based, therefore they are only efficient when the correlation between study and auxiliary variables is positive. The efficiency of the estimator by ³³reduces as k approaches zero in the presence of negative correlation between study and auxiliary variables.

To address the weaknesses in the ^21,33estimators, the estimators were modified using power transformation technique so as to obtain estimators that are applicable when the correlation between the study and auxiliary variables is either positive or negative.

This study focuses on the modification of some ratio-based estimators using power transformation under simple random sampling in the presence of auxiliary variables and limited to the work of^21,33.

Methodology

Let U denotes a finite population consisting of N units {U₁, U₂,…………., U_N} Also, let (Y, X) denote the study variable and auxiliary variable taking values (y_i, x_i), (i=1,2,…….,N) respectively, on the i^th unit U_i of the population U. On the assumption that the population mean ? of X is known, the estimate of the population mean (?)  of Y is obtained by selecting a sample of size n (n < N) from the population U using Simple Random Sampling without Replacement (SRSWOR) scheme.

N: population size, n: Samplesize being selected from the entire population,

 f = n/N: is the sampling fraction



The population mean of study variable Y,


The population mean of the auxiliary variable X,


The sample mean of study variable Y,  


The sample mean of the auxiliary variable X,


The finite population variance of the auxiliary variable X,


The finite population variance of the Study variable Y,


The finite population covariance between Y and X.


The population coefficient of variation of Y


The population coefficient of variation of X.


Pearson’s moment correlation coefficient of X and Y.

Qi The population i^th Quartile of auxiliary variable.


The population coefficients of skewness auxiliary variable X.


The population coefficient of kurtosis of auxiliary variable X.


The population quartile deviation of auxiliary variable X.

Q₂N The population second quartile of auxiliary variable X.


The decile mean for auxiliary variable X. 


Error terms of the study and auxiliary variable.

Review of existing estimators

The conventional unbiased sample mean estimator is given by

The variance of  under SRSWOR sampling scheme is given by:

                                    

⁸Proposed conventional ratio estimator for the estimation of the population mean ? of the study variable, under the assumption that there is strong positive correlation between the study variable Y and auxiliary variable X. The proposed estimator is given by:

The bias and MSE respectively of this estimator is given by   

⁵Suggested the following exponential type ratio and product estimators for estimation of the population mean ? as:

                                                                                                         

The MSEs of the estimators are given by:

¹Proposed improved Ratio Estimator for the population mean using non-conventional measures of dispersion. The estimators are as follows:

The biases, related constants and the MSEs of the estimators are given by

²¹Modified ratio estimator of population Mean using quartile and skewness coefficient. The estimators are as follows:

The biases and the MSEs of the estimators are given by

³³Proposed a new alternative estimator by combining the ratio, product and exponential ratio type estimators using linear combination.The estimator is given as:

where, k = (0,1) is a suitably chosen constant to be determined.

The Bias and MSE of the proposed estimator are given by:

where, the optimum value of k is 

The Proposed Estimators

Having studied the estimators of ^21,33 and identified some weaknesses, the following proposed exponential-type estimators for estimating population mean ? under Simple Random Sampling without Replacement (SRSWOR) were suggested based on the motivation from the works of ^4,34. The proposed estimators are as given in (33) and (34).

are real constants.

Properties of the proposed estimators

In this section, the bias and MSE of the estimator proposed in this paper are derived and discussed.

the first and second moment of £₁, i = 1,2 is

Theorem 1.1: To O(n^-1) ,  bias of the proposed estimators η₁^(*) is:

Proof : Express (33) and (34) in terms of £_i , we hav

Simplifying (38) and (39) up O(n^-1), we have

Subtract ? from both sides of (40) and (41), take expectation and apply the results of (35), theorem 1.1 is proved.

Theorem 1.2 : To O(n^-1) ,  MSE of the proposed estimators η₁^(*) is:

Proof : Subtract ? from both sides of (40) and (41), we have:

Square both sides of (44) and (45) then simplify up to O(n^-1), we get

Take expectation of (46), (47) and apply the results of (35), theorem 1.2 is proved.

Efficiency Comparison

In this section, conditions for the efficiency of the new estimators over some existing related estimators established were established.

Theorem 1.3: Estimator η_i^(*) is more efficient than η₁ if (48) and (49) is satisfied.

Proof : Minus (42) and (43) from (2), theorem 1.3 is proved.

Theorem 1.4: Estimator η_i^(*)  is more efficient than η₁ if (50) and (51) is satisfied.

Proof : Minus (42) and (43) from (4), theorem 1.4 is proved.

Theorem 1.5: Estimator η_i^(*)  is more efficient than η₁ if (52) and (53) is satisfied.

Proof: Minus (42) and (43) from (29) and (30), theorem 1.5 is proved.

Theorem 1.6: Estimator η_i^(*) is more efficient than η₁ if (54) and (55) is satisfied.

Proof: Minus (42) and (43) from (32), theorem 1.6 is proved.

Test for the Consistency of the Modified Estimators

In this section, the consistencies of the modified estimators η₁^(*), and η₂^(*) were established.

Proof: Let f(x) and g(x) be continuous function, then

As n → N, n = N. Using the results of (56), (57) and (58), we have

Hence, the estimators η₁^(*) and η₂^(*) are consistent.

6. Empirical Study                                                                                                                                             

In this section, real life data was conducted to examine the superiority of the proposed estimators over the existing estimators considered in the study. Natural dataset, population 1, 2, 3 and 4 as used is given in^32,14,11,16.

Population 1: The data is defined as follows:

Population 2: The data is defined as follows:

Population 3: The data is defined as follows:

Population 4: The data is defined as follows:

Table 1: Mean Square Errors of the Proposed Estimators and Existing Estimators Using the Population 1,2,3,4

Table 1 above show the numerical results of the Mean Square Errors (MSEs) of the estimators η_i, i = 0,1,2,......14 and η_i^(*) , i = 1,2  using four natural data sets of all the subjects examined, the two proposal have a minimum MSE for all data sets. This implies that the proposed methods have shown a high level of efficiency on others considered in the study, and can produce better estimate of the population parameters than the existing estimators.

Table 2: Percentage Relative Efficiencies of the Proposed Estimators and Existing Estimators Using the Population 1,2,3,4

Table 2 above show the numerical results of the Percentage Relative Efficiencies (PREs) of the estimators η_i, i = 0,1,2,......14 and η_i^(*) , i = 1,2 using four natural data sets of all the subjects examined, the two proposal have the highest PREs for all data sets. This implies that the proposed methods have shown a high level of efficiency on others considered in the study, and can produce better estimate of the population parameters than the existing estimators.

where var(?) is the variance of sample mean, MSE_min (η_i^(*))  is the mean square error values of the proposed estimator in  section 3 and MSE (η_i) is the mean square error values of the existing estimators mentioned in section 2.

Conclusion

By considering the results obtained from the empirical study on the efficiency of the suggested estimators over some exists related estimators considered in the study. From the empirical study, the results revealed that the suggested estimators η₁^(*)  and η₂^(*)  have minimum mean square error and higher percentage relative efficiency compared to other estimators considered in the numerical computations carried out in the study. In the other words, the suggested estimators η₁^(*) and η₂^(*)  have higher chance of producing estimate that is closer to the true value of the population mean than other estimators considered in the literature of this study.

Acknowledgment

The authors are profoundly grateful to the editors for the corrections and guidance made on this research.

Conflict of Interest

The authors declare no conflict of interest.

Funding Sources

The authors received no financial support for the research, authorship and publication of this article.

References

1. Abid M., Abbas N., Sherwani R. A. K. and Nazir H. Z., Improved ratio estimators of the population mean using non-conventional measure of dispersion. Pakistan Journal of Statistics and Operations research, 12(2), 353-367. (2016).
2. Ahmed A., Adewara A. A. and Singh R. V. K., Class of Ratio Estimators with Known Functions of Auxiliary Variables for Estimation Finite Population Variance. Asian Journal of Mathematics and Computer Research, 12(1), 63-70. (2016).
3. Ahmed A. and Singh R. V. K., Improved Exponential Type Estimators for Estimating Population Variance in Survey sampling. International Journal of Advance Research, 3(4), 1-16. (2015).
4. Audu A. and Singh R. V. K., Exponential-type regression compromised imputation class of estimators, Journal of Statistics and Management System, 24(6), 1253-1266. (2021). DOI:10.1080/09720510.2020.1814501
5. Bahl S. and Tuteja R.K., Ratio and Product type exponential estimator. Information and Optimization Sciences, 12, 159-163. (1991).
6. Choudhury S. and Singh B. K., An e?cient class of dual to product-cum- dual to ratio estimators of ?nite population mean in sample surveys, Global Journal of Science Frontier Research Mathematics and Decision Sciences, 12(3), Version 1.0, 25-33. (2012).
7. Choudhury S. and Singh B. K., A class of chain ratio-cum-dual to ratio type estimator with two auxiliary characters under double sampling in sample surveys, Statistics in Transition New Series, 13 (3), 519-536. (2012).
8. Cochran W.G., The estimation of the yields of the cereal experiments by sampling for the ratio of grain to total produce. The Journal of Agricultural Science, 30, 262-275. (1940).
9. Diana G., Giordan M. and Perri P.F., An improved class of estimators for the population means, Statistical Methods and Application, 20, 123-140. (2011).
10. Kadilar C. and Cingi H., A new estimator using two auxiliary variables. Applied Mathematics and Computation, 162, 901-908. (2005).
11. Kadilar C. and Cingi H., Ratio estimators in simple random sampling, Applied Mathematics and Computation, 151, 893-902. (2004).
12. Kadilar C. and Cingi H., Improvement in estimating the population mean in simple random sampling, Applied Mathematics Letters, 19 (1), 75-79. (2006).
13. Kadilar C. and Cingi H., An improvement in estimating the population mean by using the correlation coe?cient, Hacettepe Journal of Mathematics and Statistics, 35(1), 103-109. (2006).
14. Saddam D.A., Abdullahi J. and Nura U., Enhanced Mean Ratio Estimators of Auxiliary Variables Based on the Linear Mixture of Variances. CBN Journal of Applied Statistics (JAS), 9(2), Article 1. (2018).
15. Shabbir J., Haq A. and Gupta S., A new different-cum-exponential type estimator of finite population mean in simple random sampling. Revista Columbiana de Estadistica, 37, 197-209. (2014).
16. Singh D. and Chaudhary F.S., Theory and Analysis of Sample Survey Designs. New Age International Publisher. (1986).
17. Singh H. P. and Tailor R., Use of known correlation coe?cient in estimating the ?nite population mean, Statistics in Transition, 6(4), 555-560. (2003).
18. Singh, H.P. and Tailor R., Estimation of ?nite population mean with known coe?cient of variation of an auxiliary character, Statistica Anno LXV, (3), 301-313. (2005).
19. Singh H.P. and Solanki R.S., An efficient class of Estimators for the Population Mean using Auxiliary Information, Communication in Statistics - Theory and Methods, 42, 145-163. (2013).
20. Singh R., Mishra P., Audu A. and Khare S., Exponential Type Estimator for Estimating Finite Population Mean. Int. J. Comp. Theo. Stat. 7(1), 37-41. (2020).
21. Sirait H., Sukuno, Sundari S. and Kalfin, Ratio estimator of population mean using   quantile and skewness coefficient, International Journal of Advanced Science and Technology, Vol. 29(6), 3289-3295. (2020).
22. Solanki R. S., Singh H. P. and Rathour A., An alternative estimator for estimating ?nite population mean using auxiliary information in sample surveys, ISRN Probability and Statistics, 1-14. (2012).
23. Subramani J. and Kumarapandiyan G., Estimation of population mean using coe?cient of variation and median of an auxiliary variable, International Journal of Probability and Statistics, 1(4), 111 -118. (2012a).
24. Subramani J. and Kumarapandiyan G., Modi?ed ratio estimators using known median and coe?cient of kurtosis. American Journal of Mathematics and Statistics, 2(4), 95-100. (2012b).
25. Subramani J. and Kumarapandiyan G., Estimation of population mean using known median and coe?cient of skewness. American Journal of Mathematics and      Statistics, 2(5), 101-107. (2012c).
26. Subramani J. and Kumarapandiyan G., A class of modi?ed ratio estimators using deciles of an auxiliary variable. International Journal of Probability and Statistics, 2(6):101-107. (2012d).
27. Subzar M., Maqbool S., Raja T. A. and Shabeer M., A New Ratio Estimators for Estimation of Population Mean Using Conventional Location Parameters, World Applied Sciences Journal, 35(3), 377-384. (2017).
28. Subzar M., Maqbool S., Raja T. A., Mir S. A., Jeeiane M. M. and Bhat, M. A., Improve Family of ratio type Estimators for Estimating Population Mean Using Conventional and non-conventional Location Parameter. Investigation Operational.   38(5), 510-524. (2018a).
29. Subzar, M., Maqbool, S., Raja, T. A., & Abid, M., Ratio Estimators for Estimating Population in Simple Random Sampling Using Auxiliary Information. Applied Mathematics & Information Sciences Letters, 6(3), 123-130. (2018b).
30. Subzar M., Maqbool S., Raja T. A., Pal S. K. and Sharma P., Efficient Estimators of Population Mean Using Auxiliary Information under Simple Random Sampling. Statistics in Transition New Series, 19(2), 219-238. (2018c).
31. Yadav S. K. and Kadilar C., E?cient family of exponential estimators for the population mean, Hacettepe Journal of Mathematics and Statistics, 42(6), 671-677.(2013).
32. Yadav S. K. and Zaman T., Use of some conventional and non-conventional parameters for improving the efficiency of ratio-type estimators. Journal of Statistics and Management Systems. https://doi.org/10.1080/09720510.2020.1864939. (2021).
33. Zakari Y., Muhammad I. and Sani N.M., Alternative Ratio-Product type estimator in Simple Random Sampling. Communication in Physical Sciences, 5(4), 418-426. (2020).
34. Zaman T., Generalized exponential estimators for the finite population mean. Statistics in transition, 21(1), 159-168. (2020).