Introduction

In sampling theory, auxiliary information is used to increase the precision of estimate of an estimator when there is correlation between response variable y and auxiliary variable x. Authors have suggested different estimators note varying renowned parameters of an auxiliary information. In probability sampling, it is well confirmed that auxiliary variable may be qualitative form, such variable is term auxiliary attribute. Many authors in literature have suggested estimators using auxiliary variable, they include Cochran¹ who developed the ratio estimator to investigate problem of estimation of the population mean when auxiliary variable is present. Other researchers that developed estimators using auxiliary information include: : Audu et al. ¹, Tailor et al. ¹¹, Singh ^6,7, Kadilar and Cingi ³, Khoshnevisan et al.⁵, Perri ⁵, Yunusa et al.¹², Singh and Kumar ⁸.

When the auxiliary information are qualitative in nature, that is, auxiliary information in the form of attribute, such as colour of hair of individuals and their weight can be regarded as auxiliary attribute and study variable, sex and height of women in a locality may be regarded as auxiliary attribute and study variable etc. Several authors have developed estimators in this direction like Singh et al. ^9,10, Zaman¹³ and Zaman and Kadilar¹⁴.

Currently in this research, we have intended efficient regression type estimators of finite population mean, that gives precise estimate for the size of finite population mean in the presence of auxiliary attribute when the bi-serial correlation between study variable and auxiliary attribute is weak.

Materials and Methods

Sample mean y_m of simple random sampling is given as



Bias and variance of y_m is given by


Zaman and Kadilar ¹⁴ class of exponential ratio type estimators in the presence of auxiliary attribute as:


Bias and mean square error of the estimator y_zk are given by


Zaman¹³, an improved class of estimator for the estimation of population mean as



The MSE of the estimator is



Audu et al. ¹ modified class of estimators for the population mean of the study variable in the presence of auxiliary attribute as



bias and mean square error of the estimator t_pi and t_qi are given by





Suggested Estimators

By exploiting the idea of Audu et al.¹ and other estimators in literature, finite population mean based on the presence of auxiliary attribute for estimation of population mean of study variable are proposed





a_i, b_i, u_i and v_i are invariable to be determined, I =1,2,……..,..10 . The suggested estimators will be defined if and only if a_i ≠ 0, b_i ≠ 0, u_i ≠ 0, v_i ≠ 0, and y_h  ≠ 0. The estimator was obtained by incorporating unknowns into the estimators and taken the sample mean in the estimators of Audu et al.¹ as the average of the exponential ratio and product type estimators.

Table 1: Members of the proposed estimators Tri . Click here to view Table

Table 2: Members of the proposed estimators Tqi. Click here to view Table

To obtain the biases and MSEs of  T_ri  and T_qi , the following error terms are defined as



Such that





Expressing (16) and (17) in terms of error terms, we have



Simplify (19) and (20), then obtained



Where, θ_i = kP / (kP +l), i =1,2,…..,10.

Taking expectation of (21) and (22) and apply the results of (18) to obtain the biases of the T_ri  and T_qi  as



Squaring and taking expectation of (21) and (22) and apply the results of (18) to obtain the MSE of proposed estimators asT_ri  and T_qi as



Differentiating (25) with respect to a_i and b_i, equate to zero and solve for a_i and b_i simultaneously, we obtain 



Substituting the results in (25), we obtained the minimum MSE of T_ri  as



Differentiating (26) with respect to u_i and v_i, equate to zero and solve for u_i and v_i simultaneously, we obtain



Substituting the results in (26), we obtained the minimum MSE of T_qi  as



Efficiency Comparisons

The suggested estimators T_ri  and T_qi are more efficient than 



t_pi and t_qi, if the following condition are satisfied




Empirical Study

In this section, the suggested estimators T_ri  and T_qi  performance are assessed with that of the sample mean y_m , Audu et al. [1] estimators, t_pi and t_qi numerically considering two natural populations used as:

Population 1: Zaman [13]



N = 89, n = 20, ? = 3.3596, P = 0.1236, β_2(Ø) =3.492, C_y = 0.6008, C_Ø =2.6779, p_yØ = 0.766,

Population 2: Zaman [13]



N = 111, n = 30, ? = 29.279, P = 0.117, β_2(Ø) =3.898, C_y = 0.872, C_Ø =2.758, p_yØ = 0.797,

Table 3: MSEs and PREs of suggested estimators and existing ones using population 1

Table 4: MSEs and PREs of suggested estimators and existing ones using population 2.

Table 3 and 4 show the Mean Square Errors and Percentage Relative Efficiencies of the sample mean, y_m , Audu et al. [1], t_pi  and t_qi , and suggested estimators, T_ri and T_qi estimators, considering two data sets respectively. The results revealed that the suggested estimators T_ri and T_qi have minimum MSEs and higher PREs as compared to the sample mean, Audu et al.[1] estimators.

Results and Discussion

An improved classes of regression type estimators of finite population mean are suggested. Table 3 shows MSEs and PREs of the suggested and some existing estimators using dataset 1. The result shows that the suggested estimators have minimum MSEs and higher PREs compared to the conventional estimators and Audu et al. [1] estimators. Table 4 shows MSEs and PREs of the suggested and some existing estimators using dataset 2. The result shows that the suggested estimators have minimum MSEs and higher PREs compared to the conventional estimators and Audu et al. [1] estimators.

Conclusion

In this research, we proposed an improved modified regression estimators for the estimation of population mean in the presence of auxiliary attribute. The results of the empirical study revealed that the proposed estimators are more efficient than sample mean and Audu et al. [1] estimators. This implies that the proposed estimators have great chance of producing precise estimate.

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. Audu A., Abdulazeez S.A., Danbaba A., Ahijjo Y.M., Gidado A. and Yunusa M.A.,   Modified Classes of Regression-type Estimators of Population Mean in the presence of Auxiliary Attribute. Asian Research Journal of Mathematics; 18(1): 65-89. DOI:      10.9734/ARJOM/2022/v18i130355, (2022).
   CrossRef
2. Cochran W.G., Sampling techniques. New York, NY: John Wiley and Sons. (1977).
3. Kadilar C. and Cingi H. A., New estimator using two auxiliary variables. Applied Mathematics and Computation; 16(2): 901–908, (2005).
   CrossRef
4. Khoshnevisan M., Singh R., Chauhan P., Sawan N. and Smarandache F. A., General family of estimators for estimating population mean using known value of some population parameter(s). Far East Journal of Theoretical Statistics; 22(2): 181–191,     (2007).
5. Perri P.F., Improved ratio-cum-product type estimators. Statist Trans; 8(2): 51- 69,    (2007).
6. Singh M.P., On the estimation of ratio and product of the population parameters.     Sankhya B; 2(7): 231-328, (1965).
7. Singh M.P., Ratio cum product method of estimation. Metrika; 1(2): 34–42, (1967)
   CrossRef
8. Singh R. and Kumar M. A., Note on transformations on auxiliary variable in survey sampling. MASA; 6(1): 17-19, (2011).
   CrossRef
9. Singh R., Chauhan P., Sawan N. and Smarandache F., Auxiliary information and a priori values in construction of improved estimators. Renaissance High Press. DOI: 10.9734/AJPAS/2021/v14i230323, (2007).
   CrossRef
10. Singh R., Chauhan P. and Sawan N., Ratio-product type exponential for estimating finite population mean using information on auxiliary attribute. Journal of Statistics and        Management Systems; 81(7): 679-1067, (2013).
11. Tailor R., Chouhan S. and Garg N., A ratio-cum-product estimator of population mean   in stratified random sampling using two auxiliary variables. STATISTICA, anno           LXXII;            3, (2012).
    CrossRef
12. Yunusa M.A., Audu A., Musa N., Beki D.O., Rashida A., Bello A.B. and Hairullahi M.U., Logarithmic ratio-type estimator of population coefficient of variation. Asian      Journal of Probability and Statistics; 14(2): 13-22, (2021).
    CrossRef
13. Zaman T., Generalized exponential estimators for the finite population mean. Statistics in transition; 21(1): 159-168, (2020).
    CrossRef
14. Zaman T. and Kadilar C., Novel family of exponential estimators using information of  auxiliary attribute. Journal of Statistics and Management Systems; 34(7): 978-1078,(2019).