Dual of Ratio Estimators of Finite Population Mean Obtained on Using Linear Transformation to Auxiliary Variable

Abstract

The efficiencies of the ratio-type estimators have been increased by using linear transformation on auxiliary variable in the literature. But such type of estimators requires the additional knowledge of unknown population parameters, which restricts their applicability. Keeping in view such restrictions, we have proposed two unbiased estimators of population mean of study variable on applying linear transformation to auxiliary variable by using its extreme values in the population that are generally available in practice. The comparison of the proposed estimators with the existing ones have been done with respect to their variances. It has also been shown that the proposed estimators have greater applicability and are more efficient than the mean per unit estimator even when the existing estimators are less efficient. We have also shown that under some known conditions the choice of most efficient estimators among the considered ones can be made for a given population. The theoretical results obtained are shown diagrammatically and have been verified numerically by taking some empirical populations.

Families of estimators for ratio and product of study characters using mean and proportion of auxiliary character in presence of non-response

“…It may be easily verified that f 11 (1) and f 11 (2) minimize the value of MSE (T 11 ). So, the mean square error of the proposed families of estimators T 11 will be minimum if f 11 (1) and f 11 (2) are calculated by the equations (42) and (43).…”

“…Here (ρ, ρ 13 , ρ 23 ) and (ρ (2) , ρ 13 ′ , ρ 23 ′ ) are the correlation coefficients between (y 1 , y 2 ), (y 1 , x), (y 2 , x)for the entire and non responding group the population while ( ρ 10 , ρ 20 , ρ 30 ) and ( ρ 10 ′ , ρ 20 ′ , ρ 30 ′ ) are the biserial correlation coefficients between (y 1 , ∅), (y 2 , ∅), (x, ∅) for the entire and non responding group of population. Now following the strategies of (Khare & Sinha 2012 a, Sinha 2014 and Sinha & Kumar 2014), we have suggested two wider families of estimators for estimating the ratio and product of two population means using proportion and mean of auxiliary character for two different cases as follows: Case I -In this case we assumed that there is incomplete information on study characters y i (i= 1, 2) as well as auxiliary character x due to non-response, however the population proportion (∅ ̅ N ) and mean (X ̅ ) are known in advance.…”

“…So, the mean square error of the proposed families of estimators T 11 will be minimum if f 11 (1) and f 11 (2) are calculated by the equations (42) and (43). The minimum mean square error of T 11 using f 11 (1) and f 11 (2) respectively from the equations (42) and (43) is as follows…”

Generalized Modified Ratio Estimator for Estimation of Finite Population Mean

“…More detailed discussion about the ratio estimator and its modification can be found in Abdia and Shahbaz (2006), Ahmad et al (2009), Al-Jararha and AlHaj Ebrahem (2012), Bhushan (2012, Cochran (1977), Dalabehera and Sahoo (1994), David and Sukhatme (1974), Goodman and Hartley (1958), Gupta and Shabbir (2008), Jhajj et al (2006), Cingi (2003, 2004), Khoshnevisan et al (2007), Koyuncu and Kadilar (2009), Kulkarni (1978), Murthy (1967), Naik andGupta (1991), Olkin (1958), Pathak (1964), Perri (2007), Ray and Sahai (1980), Reddy (1973), Robinson (1987), Sen (1993), Shabbir and Yaab (2003), Sharma and Tailor (2010), Singh and Chaudhary (1986), Singh (2003), Singh and Espejo (2003), Singh and Agnihotri (2008), Singh and Solanki (2012), Singh andTailor (2003, 2005), Singh et al (2004, Sisodia and Dwivedi (1981) , Solanki et al (2012), Srivenkataramana (1980, Sharma (2009), Tin (1965), Upadhyaya and Singh (1999) and Yan and Tian (2010).…”

Vol. 12, No. 2 (Full Issue)

“…More detailed discussion about the ratio estimator and its modification can be found in Abdia and Shahbaz (2006) (2012), Cochran (1977), Dalabehera and Sahoo (1994), David and Sukhatme (1974), Goodman and Hartley (1958), Gupta and Shabbir (2008), Jhajj et al (2006), Cingi (2003, 2004), Khoshnevisan et al (2007), Kadilar (2009), Kulkarni (1978), Murthy (1967), Naik and Gupta (1991), Olkin (1958), Pathak (1964), Perri (2007), Ray and Sahai (1980), Reddy (1973), Robinson (1987), Sen (1993), Shabbir and Yaab (2003), Sharma and Tailor (2010), Singh and Chaudhary (1986), , Singh and Espejo (2003), Singh and Agnihotri (2008), Singh and Solanki (2012), Tailor (2003, 2005), Singh et al (2004Singh et al ( , 2008, Sisodia and Dwivedi (1981) , Solanki et al (2012), Srivenkataramana (1980), Tailor and Sharma (2009), Tin (1965), Upadhyaya and Singh (1999) and Yan and Tian (2010).…”