Almost Unbiased Ratio cum Product Estimator for Finite Population Mean with Known Median in Simple Random Sampling

Keywords: Bias, mean squared error, natural populations, product estimator, ratio estimator, simple random sampling

Introduction:

In sampling theory, different procedures are used to obtain the efficient estimator of the population mean. The commonly used method is to obtain the estimator of the population mean is simple random sampling without replacement when there is no auxiliary variable is available. There are methods that use auxiliary information of the study characteristics. If the auxiliary variable is correlated with study variable, number of estimators are widely available in the literature. 
Objective: This study deals with a new ratio cum product estimator is developed for the estimation of population mean of the study variable with the known median of the auxiliary variable in simple random sampling. 

Materials and Methods:

The bias and mean squared error of proposed estimator are derived and compared with that of the existing estimators by analytically and numerically. 

Results:

The proposed estimator is less biased and mean squared error is less than that of the existing estimators and from the numerical study, under some known natural populations, the bias of proposed estimator is approximately zero and the mean squared error ranged from 6.83 to 66429.21 and percentage relative efficiencies ranged from 103.65 to 2858.75. 

Conclusion:

The proposed estimator under optimum conditions is almost unbiased and performs better than all other existing estimators. 

INTRODUCTION

The main aim of sampling theory is to obtain precise results about the parameters of study variable on the basis of random samples. One can use the parameters of auxiliary variable such as mean, median, variance, the coefficient of variation, correlation coefficient etc. are used to improve the efficiency of the estimators of study variable. Let in a finite population having N units { 1 , 2 , 3 … .  }. Y be the study variable and X be the auxiliary variable which is correlated with Y. 
If = { 1 , 2 , 3 . . . .  }, and = { 1 , 2 , 3 . . . .  } be n sample values. When the population mean of the auxiliary variable is known, different estimators such as ratio, product and linear regression estimators are proposed in the literature. Among all these estimators the ratio estimator and its modifications are widely used for the estimation of the mean of the study variable. This work is focused on improving the efficiency of some ratio cum product estimators in the literature by obtaining the optimal value. Cochran (1940), Singh, Tailor and Kakran (2004), Kadilar andCingi (2004, 2006), Subramani (2013), Subramani andKumarapandiyan (2012a, 2012b), Subramani andMaster Ajith (2016a, 2016b), Upadhyaya and Singh (1999), Yan and Tian (2010), Sisodia and Dwivedi (1981), etc. suggested different class of estimators of the population mean with known parameters of the auxiliary variable. 
Before discussing further the various estimators, the notations to be used in this article are listed here. The simple random sample mean without replacement is used only when there is no additional information of the study variable is available. In simple random sampling the estimator ̅ is an unbiased estimator for the population mean Y and its variance is 

N -

V(𝑦 ̅ 𝑠𝑟𝑠 ) = ( 1-f n ) 𝑆 𝑦 2 = ( 1-f n ) 𝑌 ̅ 2 𝐶 𝑦 2 (1) 
Where 
𝑌 ̅ = 1 𝑁 ∑ 𝑌 𝑖 , 𝑁 𝑖=1 𝑆 𝑦 2 = 1 𝑁-1 ∑ (𝑌 𝑖 -𝑌 ̅ ) 2 , 𝑁-1 𝑖=1 𝐶 𝑦 = 𝑆 𝑦 𝑌 ̅ ,
The auxiliary variable and study variable are positively correlated, ratio estimator and their modifications are used. The ratio estimator (Cochran, 1940) is given 
𝑌 ̅ ̂𝑅 = 𝑦 ̅ 𝑥̅ X ̅ = 𝑅 ̂𝑋 ̅ Where 𝑅 ̂= 𝑦 ̅ 𝑥̅ , 𝑦 ̅ = 1 𝑛 ∑ 𝑦 𝑖 𝑛 𝑖=1 , 𝑥̅ = 1 𝑛 ∑ 𝑥 𝑖 𝑛 𝑖=1
the bias and mean squared error of ratio estimator up to first order approximations are 
B(𝑌 ̅ ̂𝑅) = ( 1-f n ) 𝑌 ̅ [𝐶 𝑥 2 -𝜌𝐶 𝑥 𝐶 𝑦 ] MSE(𝑌 ̅ ̂𝑅) = ( 1-f n ) 𝑌 ̅ 2 [𝐶 𝑦 2 + 𝐶 𝑥 2 -2𝜌𝐶 𝑥 𝐶 𝑦 ] (2) 
Where 
𝐶 𝑥 = 𝑆 𝑥 𝑋 ̅ , 𝜌 = 𝑆 𝑥𝑦 𝑆 𝑥 𝑆 𝑦 , 𝑆 𝑥𝑦 = 1 𝑁-1 ∑ (𝑌 𝑖 -𝑌 ̅ ) 𝑁 𝑖=1 (𝑋 𝑖 -𝑋 ̅ ), 𝑆 𝑥 2 = 1 𝑁-1 ∑ (𝑋 𝑖 -𝑋 ̅ ) 2 𝑁 𝑖=1
The modified ratio estimator for the population mean ̅ with known median of the auxiliary variable is 
𝑌 ̅ ̂𝑀𝑅 = 𝑦 ̅ ( 𝑋 ̅ + 𝑀 𝑑 𝑥̅ + 𝑀 𝑑 )
The bias and mean squared error of modified ratio estimator are given by 
B(𝑌 ̅ ̂𝑀𝑅 ) = ( 1-f n ) 𝑌 ̅ [𝜃 2 𝐶 𝑥 2 -𝜃ρC x C y ] MSE(𝑌 ̅ ̂𝑀𝑅 ) = ( 1-f n ) 𝑌 ̅ 2 [𝐶 𝑦 2 + 𝜃 2 𝐶 𝑥 2 -2𝜃ρC x C y ] (3) 
where 
𝜃 = 𝑋 ̅ 𝑋 ̅ +𝑀 𝑑
The auxiliary variable and study variable are negatively correlated, the product estimator and its modifications are used. The product estimator (Murthy, 1964) is given by 
𝑌 ̅ ̂𝑝 = 𝑦 ̅ 𝑥̅ 𝑋 ̅ .
The bias and mean squared error of the product estimator are given by 
B (𝑌 ̅ ̂𝑝) = ( 1-f n ) 𝑌 ̅ [ρC x C y ] MSE (𝑌 ̅ ̂𝑝) = ( 1-f n ) 𝑌 ̅ 2 [𝐶 𝑦 2 + 𝐶 𝑥 2 + 2ρC x C y ] (4) 
The modified product estimator with known median of the auxiliary variable is defined by www.tucds.edu.np 
𝑌 ̅ ̂𝑀𝑝 = 𝑦 ̅ ( 𝑥̅ +𝑀 𝑑 𝑋 ̅ +𝑀 𝑑 )
The bias and mean squared error of the modified product estimator are given by 
B(𝑌 ̅ ̂𝑀𝑝 ) = ( 1-f n ) 𝑌 ̅ [𝜃𝜌𝐶 𝑥 𝐶 𝑦 ] MSE(𝑌 ̅ ̂𝑀𝑝 ) = ( 1-f n ) 𝑌 ̅ 2 [𝐶 𝑦 2 + 𝜃 2 𝐶 𝑥 2 + 2𝜃ρC x C y ] (5) 
where = ̅ ̅ +  

RESULTS

In this section, a ratio cum product estimator for the population mean by using the known median of an auxiliary variable is proposed and also derived the bias and the mean squared errors of the proposed estimators. Compare the mean squared error of the proposed estimator with the above mentioned estimators by algebraically and numerically with the help of three natural populations. The detailed expressions are given below 

Proposed estimator

The proposed ratio cum product estimator for the population mean by using the known population median of the auxiliary variable is given by 
𝑌 ̅ ̂𝑝𝑟 = 𝛼𝜆 1 𝑦 ̅ ( 𝑋 ̅ +𝑀 𝑑 𝑥̅ +𝑀 𝑑 ) + (1 -𝛼)𝜆 2 𝑦 ̅ ( 𝑥̅ +𝑀 𝑑 𝑋 ̅ +𝑀 𝑑 ) (6) 
where 
𝜆 1 = 𝑆 𝑦 𝑆 𝑦 +𝐾 1 𝐶 𝑦 𝜆 2 = 𝑆 𝑦 𝑆 𝑦 +𝐾 2 𝐶 𝑦
where K1 and K2 are constants 

The bias and mean squared error of the proposed estimator

To obtain the bias and mean squared error of the proposed estimator, 
Consider 𝑒 0 = 𝑦 ̅-𝑌 ̅ 𝑌 ̅ , 𝑒 1 = 𝑥̅ -𝑋 ̅ 𝑋 ̅ ,𝜃 = 𝑋 ̅ 𝑋 ̅ +𝑀 𝑑 𝐸(𝑒 0 ) = 𝐸(𝑒 1 ) = 0, 𝐸(𝑒 0 2 ) = ( 1-𝑓 𝑛 ) 𝑌 ̅ 2 𝐶 𝑦 2 , 𝐸(𝑒 1 2 ) = ( 1-𝑓 𝑛 ) 𝑋 ̅ 2 𝐶 𝑥 2 , 𝐸(𝑒 0 𝑒 1 ) = ( 1-𝑓 𝑛 ) 𝜌𝐶 𝑥 𝐶 𝑦
Substitute the values of 0 and 1 in equation ( 6) and neglecting the high order expressions, we get 
𝐵(𝑌 ̅ ̂𝑃𝑟 ) = 𝐸(𝑌 ̅ ̂𝑃𝑟 -𝑌 ̅ ) 𝐵(𝑌 ̅ ̂𝑃𝑟 ) = 𝑌 ̅ (𝛼𝜆 1 + (1 -𝛼)𝜆 2 -1) + ( 1 -𝑓 𝑛 ) 𝑌 ̅ {𝛼𝜆 1 𝜃 2 𝐶 𝑥 2 -𝜃𝜌𝐶 𝑥 𝐶 𝑦 (𝛼𝜆 1 -(1 -𝛼)𝜆 2 )}
The detailed derivation of the mean squared error is given in the appendix and the final expression is obtained with only first order approximation in the Taylor series expansion as 
𝑀𝑆𝐸(𝑌 ̅ ̂𝑃𝑟 ) = 𝐸(𝑌 ̅ ̂𝑃𝑟 -𝑌 ̅ ) 𝑀𝑆𝐸(𝑌 ̅ ̂𝑃𝑟 ) = 𝑌 ̅ 2 (𝛼𝜆 1 + (1 -𝛼)𝜆 2 -1) 2 + ( 1 -𝑓 𝑛 ) 𝑌 ̅ 2 {𝐶 𝑦 2 (𝛼𝜆 1 + (1 -𝛼)𝜆 2 ) 2 + 𝜃 2 𝐶 𝑥 2 (3𝛼 2 𝜆 1 2 + (1 -𝛼) 2 𝜆 2 2 -2𝛼𝜆 1 ) + 2𝜃𝜌𝐶 𝑥 𝐶 𝑦 (𝛼𝜆 1 -(1 -𝛼)𝜆 2 ) -2(𝛼 2 𝜆 1 2 -(1 -𝛼) 2 𝜆 2 2 )}
where 1 and 2 are as defined above. If we assume that 1 =0, 2 = 0 and = 1 then the proposed estimator are exactly equal to the existing modified ratio estimator and if 1 = 0 , 2 = 0 and = 0 then the proposed estimator is exactly equal to the existing modified product estimator we discussed above. If we assume that 1 = ( ̅ ̂ ) , 2 = ( ̅ ̂ ) then the proposed estimators are almost unbiased ratio cum product estimator under optimum values of . The optimal value of is determined by minimizing the MSE ( ̅ ̂ ) with respect to . For this differentiate MSE with respect to and equate to zero. 
 = 0, and we get the value of , as 
𝛼 = (𝜆 2 -1)(𝜆 2 -𝜆 1 ) + ( 1-𝑓 𝑛 ) {𝐶 𝑦 2 𝜆 2 (𝜆 2 -𝜆 1 ) + 𝜃 2 𝐶 𝑥 2 (𝜆 1 + 𝜆 2 2 ) -𝜃𝜌𝐶 𝑥 𝐶 𝑦 (𝜆 1 + 𝜆 2 -4𝜆 2 2 )} (𝜆 1 -𝜆 2 ) 2 + ( 1-𝑓 𝑛 ) {(𝜆 1 -𝜆 2 ) 2 𝐶 𝑦 2 + 𝜃 2 𝐶 𝑥 2 (3𝜆 1 2 + 𝜆 2 2 ) + 4𝜃𝜌𝐶 𝑥 𝐶 𝑦 (𝜆 2 2 -𝜆 1 2 )}

Efficiency comparison

To assess the efficiencies of the proposed estimator with that of the estimators we discussed above, the mean squared error under optimality condition of these estimators are used and it satisfies the following cases. The proposed estimator under optimality condition is more efficient than of existing estimators such as ( ̅ ), ( ̅ ̂), ( ̅ ̂ ), ( ̅ ̂), ( ̅ ̂ ), and it satisfying the following conditions. 
Case ( 1) These values are used to obtain the bias and mean squared error of proposed estimator and existing estimators under optimality conditions. The mean square error of the proposed estimator is used to compare the percentage relative efficiency with that of the existing estimators such as the simple random sample mean, ratio, product, modified ratio estimator and the modified product estimator. 
𝑉(𝑦 ̅ 𝑠𝑟𝑠 ) ≥ 𝑀𝑆𝐸(𝑌 ̅ ̂𝑃𝑟 ) 𝜌 ≥ (𝐴 -1) 2 + ( 1-𝑓 𝑛 ) {(𝐴 2 -1)𝐶 𝑦 2 + 𝜃 2 𝐶 𝑥 2 (𝐴 2 + (𝐴 + 𝐵)(𝐵 -1))} ( 1-𝑓 𝑛 ) 2𝜃𝐶 𝑥 𝐶 𝑦 (𝐵(2𝐴 -1) Case (2) MSE(𝑌 ̅ ̂𝑅) ≥ 𝑀𝑆𝐸(𝑌 ̅ ̂𝑃𝑟 ) 𝜌 ≥ (𝐴 -1) 2 + ( 1-𝑓 𝑛 ) {(𝐴 2 -

CONCLUSION

The performance of the proposed estimator for some known natural populations is also observed. 
when 1 = ( ̅ ̂ ) , 2 = ( ̅ ̂ ) and α is optimum the proposed estimator is less bias (almost unbiased ) than the existing estimators and it is more efficient than all these existing estimators. In the fact the percentage relative efficiency is ranging from  In the case of simple random sampling, the percentage relative efficiency ranged from 173.79 to 668.76. 

APPENDIX


Bias and MSE of Proposed Estimators

The proposed ratio cum product estimator is given by 
𝑌 ̅ ̂𝑝𝑟 = 𝛼𝜆 1 𝑦 ̅ (
𝑌 ̅ ̂𝑝𝑟 = 𝛼𝜆 1 𝑦 ̅ ( 𝑋 ̅ + M d 𝑥̅ + M d ) + (1 -𝛼)𝜆 2 𝑦 ̅ ( 𝑥̅ + M d 𝑋 ̅ + M d ) = 𝛼𝜆 1 𝑌 ̅ (1 + 𝑒 0 )(1 + 𝜃𝑒 1 ) -1 + (1 -𝛼)𝜆 2 𝑌 ̅ (1 + 𝑒 0 )(1 + 𝜃𝑒 1 ) = 𝑌 ̅ {𝛼𝜆 1 (1 + 𝑒 0 )(1 -𝜃𝑒 1 + 𝜃 2 𝑒 1 2 ) + (1 -𝛼)𝜆 2 (1 + 𝑒 0 )(1 + 𝜃𝑒 1 )} = 𝑌 ̅ {𝛼𝜆 1 (1 -𝜃𝑒 1 + 𝜃 2 𝑒 1 2 + 𝑒 0 -𝜃𝑒 1 𝑒 0 ) + (1 -𝛼)𝜆 2 (1 + 𝜃𝑒 1 + 𝑒 0 + 𝜃𝑒 0 𝑒 1 )} 𝑌 ̅ ̂𝑝𝑟 -𝑌 ̅ = 𝑌 ̅ {𝛼𝜆 1 (1 -𝜃𝑒 1 + 𝜃 2 𝑒 1 2 + 𝑒 0 -𝜃𝑒 1 𝑒 0 ) + (1 -𝛼)𝜆 2 (1 + 𝜃𝑒 1 + 𝑒 0 + 𝜃𝑒 0 𝑒 1 )} 𝐵(𝑌 ̅ ̂𝑃𝑟 ) = 𝐸(𝑌 ̅ ̂𝑃𝑟 -𝑌 ̅ ) = 𝐸{𝑌 ̅ (𝛼𝜆 1 (1 -𝜃𝑒 1 + 𝜃 2 𝑒 1 2 + 𝑒 0 -𝜃𝑒 1 𝑒 0 ) + (1 -𝛼)𝜆 2 (1 + 𝜃𝑒 1 + 𝑒 0 + 𝜃𝑒 0 𝑒 1 ) -1)} = 𝑌 ̅ {𝛼𝜆 1 (1 + ( 1-𝑓 𝑛 ) 𝜃 2 𝐶 𝑥 2 -( 1-𝑓 𝑛 ) 𝜃C xy ) + (1 -𝛼)𝜆 2 (1 + ( 1-𝑓 𝑛 ) 𝜃C xy ) -1} 𝐵(𝑌 ̅ ̂𝑃𝑟 ) = 𝑌 ̅ {(𝛼𝜆 1 + (1 -𝛼)𝜆 2 -1) + ( 1 -𝑓 𝑛 ) {𝛼𝜆 1 𝜃 2 𝐶 𝑥 2 -𝜃C xy (𝛼𝜆 1 -(1 -𝛼)𝜆 2 )}}
The mean squared error of the proposed estimator is 
𝑀𝑆𝐸(𝑌 ̅ ̂𝑃𝑟 ) = 𝐸(𝑌 ̅ ̂𝑃𝑟 -𝑌 ̅ ) 2 = 𝐸{𝑌 ̅ (𝛼𝜆 1 (1 -𝜃𝑒 1 + 𝜃 2 𝑒 1 2 + 𝑒 0 -𝜃𝑒 1 𝑒 0 ) + (1 -𝛼)𝜆 2 (1 + 𝜃𝑒 1 + 𝑒 0 + 𝜃𝑒 0 𝑒 1 ) -1) 2 } = 𝐸{𝑌 ̅ 2 (𝛼 2 𝜆 1 2 (1 -𝜃𝑒 1 + 𝜃 2 𝑒 1 2 + 𝑒 0 -𝜃𝑒 1 𝑒 0 ) 2 + (1 -𝛼) 2 𝜆 2 2 (1 + 𝜃𝑒 1 + 𝑒 0 + 𝜃𝑒 0 𝑒 1 ) 2 + 1 + 2𝛼𝜆 1 (1 -𝛼)𝜆 2 (1 -𝜃𝑒 1 + 𝜃 2 𝑒 1 2 + 𝑒 0 -𝜃𝑒 1 𝑒 0 )(1 + 𝜃𝑒 1 + 𝑒 0 + 𝜃𝑒 0 𝑒 1 ) -2𝛼𝜆 1 (1 -𝜃𝑒 1 + = 𝑌 ̅ 2 {𝛼 2 𝜆 1 2 (1 + 3 ( 1-𝑓 𝑛 ) 𝜃 2 𝐶 𝑥 2 + ( 1-𝑓 𝑛 ) 𝐶 𝑦 2 -4 ( 1-𝑓 𝑛 ) 𝜃C xy ) + (1 -𝛼) 2 𝜆 2 2 (1 + ( 1-𝑓 𝑛 ) 𝜃 2 𝐶 𝑥 2 + ( 1-𝑓 𝑛 ) 𝐶 𝑦 2 + ( 1-𝑓 𝑛 ) 4𝜃C xy ) + 2𝛼𝜆 1 (1 -𝛼)𝜆 2 (1 + ( 1-𝑓 𝑛 ) 𝐶 𝑦 2 ) -2𝛼𝜆 1 (1 + ( 1-𝑓 𝑛 ) 𝜃 2 𝐶 𝑥 2 - ( 1-𝑓 𝑛 ) 𝜃C xy ) -2(1 -𝛼)𝜆 2 (1 + ( 1-𝑓 𝑛 ) 𝜃C xy )} 𝑀𝑆𝐸(𝑌 ̅ ̂𝑃𝑟 ) = 𝑌 ̅ 2 {(𝛼𝜆 1 + (1 -𝛼)𝜆 2 -1) 2 + ( 1 -𝑓 𝑛 ) {𝐶 𝑦 2 (𝛼𝜆 1 + (1 -𝛼)𝜆 2 ) 2 + 𝜃 2 𝐶 𝑥 2 (3𝛼 2 𝜆 1 2 + (1 -𝛼) 2 𝜆 2 2 -2𝛼𝜆 1 ) + 2𝜃C xy (𝛼𝜆 1 -(1 -𝛼)𝜆 2 ) -2(𝛼 2 𝜆 1 2 -(1 -𝛼) 2 𝜆 2 2 )}} Where 𝜃 = 𝑋 ̅ 𝑋 ̅ +M d
The optimal value of is determined by minimize the MSE ( ̅ ̂ ) with respect to . 

Theorem:

The proposed estimator under optimality condition is more efficient than of existing 

,

   1 , 2 are constants To obtain the bias and mean squared error of the proposed estimator, Substitute these values in ̅ ̂ and neglecting the high order expressions, we get 




  estimators such as ( ̅ ), ( ̅ ̂), ( ̅ ̂ ), ( ̅ ̂), ( ̅ ̂ ) : The proposed estimator is more efficient than simple random sampling without replacement estimator if ( ̅ ) -( ̅ ̂ ) ≥ 0 


Table 1 .

 1 Bias and MSE of proposed and existing estimators from different populations 



Table 2 .

 2 Percentage relative efficiency of the proposed estimator 





  In the case of ratio and modified ratio estimators, the PRE is ranged from 109.76 to 2858.75 and 109.48 and 2452.84  In case of product and modified product estimators, the percentage relative efficiencies are ranged from 127.04 to574.21 and 103.65 to 568.48 


		Population 1	Population 2	Population 3
	Estimator						
		Bias	MSE	Bias	MSE	Bias	MSE
	Proposed (opt)	-4.62e-13	66429.21	8.88e-16	3013.05	1.06e-14	6.83
	𝑦 ̅ 𝑠𝑟𝑠	-	631805.10	-	3849.91	-	2722.40
	𝑌 ̅ ̂𝑅	304.37	948789.10	15.16	4925.33	1.54	111.43
	𝑌 ̅ ̂𝑝	243.21	5990618.00	9.77	12718.48	21.97	11343.93
	𝑌 ̅ ̂𝑀𝑅	50.35	139073.90	1.78	3177.31	-4.60	401.08
	𝑌 ̅ ̂𝑀𝑃	145.42	3153694.00	5.15	7287.23	13.56	7332.58
	Estimators	Population 1	Population 2	Population 3
	𝑦 ̅ 𝑠𝑟𝑠		668.76	174.33	173.79
	𝑌 ̅ ̂𝑅		2858.75	109.76	222.37
	𝑌 ̅ ̂𝑝		127.04	511.41	574.21
	𝑌 ̅ ̂𝑀𝑅		2452.84	109.48	219.87
	𝑌 ̅ ̂𝑀𝑃		103.65	509.51	568.	

48 www.tucds.edu.np 


  For this differentiate MSE with respect to and equate to zero. 


		ie,	𝜕𝑀𝑆𝐸 𝜕𝛼	= 0, and we get the value of 𝛼, as
	2(αλ 1 + (1 -α)λ 2 -1)(λ 1 -λ 2 ) + (	1-𝑓 𝑛	) {C y	2 2(αλ 1 + (1 -α)λ 2 )(λ 1 -λ 2 ) + θ 2 C x	2 (6αλ 1	2 -
	2(1 -α)λ 2	2 -2λ 1 ) + 2θC xy (λ 1 + λ 2 -2(2αλ 1	2 + 2(1 -α)λ 2	2 )} = 0
		(αλ 1 -αλ 2 )(λ 1 -λ 2 ) + (	1-𝑓 𝑛	) {C y	2 (αλ 1 -αλ 2 )(λ 1 -λ 2 ) + θ 2 C x	2 (3αλ 1	2 + αλ 2	2 ) +
	θC xy (-4αλ 1	2 + 4αλ 2	2 )} = (λ 2 -1)(λ 2 -λ 1 ) + (	1-𝑓 𝑛	) {C y	2 λ 2 (λ 2 -λ 1 ) + θ 2 C x	2 (λ 1 + λ 2	2 ) -
	θC xy (λ 1 + λ 2 -4λ 2	2 )}
	𝛼 =	(𝜆 2 -1)(𝜆 2 -𝜆 1 ) + ( (𝜆 1 -𝜆 2 ) 2 + ( 1-𝑓 1-𝑓 𝑛 𝑛 ) {(𝜆 1 -𝜆 2 ) 2 𝐶 𝑦 ) {𝐶 𝑦 2 𝜆 2 (𝜆 2 -𝜆 1 ) + 𝜃 2 𝐶 𝑥 2 + 𝜃 2 𝐶 𝑥 2 (3𝜆 1 2 (𝜆 1 + 𝜆 2 2 + 𝜆 2 2 ) + 4𝜃C xy (𝜆 2 2 ) -𝜃C xy (𝜆 1 + 𝜆 2 -4𝜆 2 2 -𝜆 1 2 )}	2 )}

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www.tucds.edu.np 

www.tucds.edu.np

It shows that proposed estimator performing better than all these existing estimators. 


CONFLICT OF INTEREST

The authors declared that there is no conflict of interest. 
Nepalese Journal of Statistics, Vol. 1, 1-14 J. Subramani 

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Almost unbiased ratio cum product estimator 12 www.tucds.edu.np 

Case (2):

The proposed estimator is more efficient than ratio estimator if 
) 2 ((2 -1) -1) 

Case (3):

The proposed estimator is more efficient than modified ratio estimator if, 
Case (4): The proposed estimator is more efficient than product estimator if 
Nepalese Journal of Statistics, Vol. 1, 1-14 
Copyright @ Central Department of Statistics, TU, 2017 13 

Case (5):

The proposed estimator is more efficient than modified product estimator if
Jambulingam Subramani
Almost Unbiased Ratio cum Product Estimator for Finite Population Mean with Known Median in Simple Random Sampling