Almost Unbiased Ratio Cum Product Estimators for Finite ... · Jambulingam Subramani and Master...

International Journal of Statistics and Systems

ISSN 0973-2675 Volume 12, Number 4 (2017), pp. 645-663

© Research India Publications

http://www.ripublication.com

Almost Unbiased Ratio Cum Product Estimators for

Finite Population Mean with Known Ranges and

It’s Functions

Jambulingam Subramani and Master Ajith S

Department of Statistics, Ramanujan School of Mathematical sciences

Pondicherry University, Puducherry, 605014, India.

Abstract

This manuscript deals with a new class of almost unbiased modified ratio cum

product estimators for the estimation of population mean of the study variable

by using the known ranges and its functions of the auxiliary variable. The bias

and mean squared error of proposed estimators are obtained. Empirical

conditions are developed.Numerical comparisons are done by using some

natural populations and the simulation study is conducted by using

morgnstern’s bivariate exponential population. It shows that the proposed

estimators are almost unbiased and more efiicient than the existing estimators.

Keywords: Auxiliary variable, Bias, Mean squared error, Natural populations,

Ratio and Product estimators, Simple random sampling.

1. INTRODUCTION

In literature, several estimators exist with auxiliary variables. When the auxiliary

information is to be used in the estimation stage ratio, product and regression methods

are widely used.The commonly used the population parameters of the auxiliary

variables are mean, median, the coefficient of variation, the coefficient of skewness,

the coefficient of kurtosis etc.If the correlation between study variable and auxiliary

variables are positive the ratio method of estimation and its modifications are used and

the correlation between study and auxiliary variables are negative the product method

of estimation and its modifications are used. Several researchers have directed their

mailto:[email protected]

mailto:[email protected]

646 Jambulingam Subramani and Master Ajith S

efforts towards to get efficient estimators of the population mean. These estimators are

biased but the percentage relative efficiency is better than that of simple random

sampling, ratio and product estimators. In this reason, we consider the problem of

estimation of population mean of study variable using ranges and their functions of the

auxiliary variable. So we have suggested new modified ratio cum product estimators

for estimating the population means of the study variable. To know more about

historical developments of the estimation of population mean, the readen are referred

to Cochran[1,2], J.Subramani and Master Ajith [3,4], Murthy[7], Sisodia and Dwivedi

[9], Subramani[10], Subramani and kumarapandiyan [11,12], Upadhyaya and

Singh[13], Yan and Tian [14] and the references cited there in.etc.

2. NOTATIONS AND LITERATURE REVIEW

Let 𝑈 = {𝑈1, 𝑈2, 𝑈3 … . 𝑈𝑁} be a finite population having N units. Each Ui=(Xi,Yi ),i

=1,2,3....N has a pair of values. Here Y be the study variable and X be the auxiliary

variable which is correlated with Y. Let 𝑦 = {𝑦1, 𝑦2, 𝑦3. . . . 𝑦𝑛}, and 𝑥 =

{𝑥1, 𝑥2, 𝑥3. . . . 𝑥𝑛} be n sample values. Then �� and �� be the sample means of the study

and auxiliary variables,𝑆𝑦2 =

1

𝑁−1∑ (𝑌𝑖 − ��)2𝑁

𝑖=1 ,𝑆𝑥2 =

1

𝑁−1∑ (𝑋𝑖 − ��)2𝑁

𝑖=1 and 𝑆𝑥𝑦 =1

𝑁−1∑ (𝑌𝑖 − ��)(𝑋𝑖 − ��)𝑁

𝑖=1 be the population variance and co-variance of the study

variable and auxiliary variable. Similarly the coefficient of variations and coefficient of

co-variance of these variables are defined as 𝐶𝑦 =𝑆𝑦

��, 𝐶𝑥 =

𝑆𝑥

�� and 𝐶𝑥𝑦 =

𝑆𝑥𝑦

��=

𝜌𝐶𝑥𝐶𝑦where 𝜌 is the correlation coefficient. In this study we consider the minimum

value ( R1) maximum value (RN) , range 𝑅 = (RN − R1), Rd = (RN−R1

2) and Ra =

(RN+R1

2) of the auxiliary variable.

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

without replacement, the estimator of population mean ��𝑠𝑟𝑠 is an unbiased estimator for

the population mean Y and its variance is

V(��𝑠𝑟𝑠) = δ𝑆𝑦2= 𝛿��2𝐶𝑦

2 (1)

where δ = (1−f

n) is the finite population correction

The ratio estimator ( Cochran [1]) is given

��𝑅 = ��

��X = ��

the bias and mean squared error of ratio estimator up to first order approximations are

B(��𝑅) = δ�� [𝐶𝑥2 − 𝜌𝐶𝑥𝐶𝑦]

Almost Unbiased Ratio Cum Product Estimators for Finite Population Mean… 647

MSE(��𝑅) = δ��2 [𝐶𝑦2 + 𝐶𝑥

2− 2𝜌𝐶𝑥𝐶𝑦] (2)

Based on Mohanty and Sahoo [5] estimators, modified ratio estimators with minimum

value (R1) and maximum value (RN) of the auxiliary variable is given by,

Y𝑀𝑅1 = y [X + R1

x + R1]

YMR2 = y [X + R𝑁

x + RN]

Modified ratio estimators using range and its functions of the auxiliary variable as given

by,

YMR3 = y [X + 𝑅

x + R]

YMR4 = y [X + Rd

x + Rd]

YMR5 = y [X + Ra

x + Ra]

The biases and mean squared errors of these estimators are given by

B(��𝑀𝑅𝑖) = δ��[𝜃𝑖2𝐶𝑥

2 − 𝜃𝑖ρCxCy]

MSE(��𝑀𝑅𝑖) = δ��2[𝐶𝑦2 + 𝜃𝑖

2𝐶𝑥

2 − 2𝜃𝑖ρCxCy]

Where 𝜃𝑖 = ��

��+𝑇𝑖, i= 1,2,3 ,4,5 𝑇1 = R1,𝑇2 = RN, 𝑇3 = 𝑅, 𝑇4 = R𝑑 , and 𝑇5 = Ra

The auxiliary variable and study variable are negatively correlated, the product

estimator and its modifications are used. The product estimator ( Murthy [6] ) is given

by

��𝑝 = ��

��.

The bias and mean squared error of the product estimator are given by

B (��𝑝) = δ�� [ρCxCy]

MSE (��𝑝) = δ��2 [𝐶𝑦2 + 𝐶𝑥

2+ 2ρCxCy] (3)

The modified product estimators for the population mean 𝑌 with known minimum,

maximum values, ranges and it’s functions are given by


��𝑀𝑃1 = �� (�� + 𝑅1

�� + 𝑅1

)

��𝑀𝑃2 = �� (�� + 𝑅𝑁

�� + 𝑅𝑁

)

��𝑀𝑃3 = �� (�� + 𝑅

�� + 𝑅)

��𝑀𝑃4 = �� (�� + 𝑅𝑑

�� + 𝑅𝑑

)

��𝑀𝑃4 = �� (�� + 𝑅𝑎

�� + 𝑅𝑎

)

The biases and mean squared errors of the modified product estimators are given by

B(��𝑀𝑃𝑖) = δ��[𝜃𝑖ρCxCy]

MSE(��𝑀𝑃𝑖) = δ��2[𝐶𝑦2 + 𝜃𝑖

2𝐶𝑥

2 + 2𝜃𝑖ρCxCy] (4)

Where 𝜃𝑖 = ��

��+𝑇𝑖, i= 1,2,3 ,4,5 𝑇1 = R1,𝑇2 = RN, 𝑇3 = 𝑅, 𝑇4 = R𝑑 , and 𝑇5 = Ra

3. PROPOSED CLASS OF ESTIMATORS

We have suggested class of ratio cum product estimators for the population mean by

using the known population Range and their functions of the auxiliary variable X

The proposed class of ratio cum product estimators with known Ranges and their

functions are given as

��𝑃1 = �� [𝛼1𝜆1 (��+𝑅1

��+𝑅1) + (1 − 𝛼1)𝛾1 (

��+𝑅1

��+𝑅1)]

��𝑃2 = �� [𝛼2𝜆2 (��+𝑅𝑁

��+𝑅𝑁) + (1 − 𝛼2)𝛾2 (

��+𝑅𝑁

��+𝑅𝑁)]

��𝑃3 = �� [𝛼3𝜆3 (��+𝑅

��+𝑅) + (1 − 𝛼3)𝛾3 (

��+𝑅

��+𝑅)]

��𝑃4 = �� [𝛼4𝜆4 (��+𝑅𝑎

��+𝑅𝑑) + (1 − 𝛼4)𝛾4 (

��+𝑅𝑑

��+𝑅𝑑)]

��𝑃5 = �� [𝛼5𝜆5 (��+𝑅𝑎

��+𝑅𝑎) + (1 − 𝛼5)𝛾5 (

��+𝑅𝑎

��+𝑅𝑎)]

Where 𝜆𝑖 = 𝑆𝑦

𝑆𝑦+𝑎𝑖𝐶𝑦, 𝛾𝑖 =

𝑆𝑦

𝑆𝑦+𝑏𝑖𝐶𝑦,i = 1,2,3,4,5 . Here 𝑎𝑖’s and 𝑏𝑖’s are constants


3.1 The Bias and Mean Squared error of the Proposed Estimators

To obtain the bias and mean squared error of the proposed estimators,

��𝑃𝑖 = �� [𝛼𝑖𝜆𝑖 (��+𝑇𝑖

��+𝑇𝑖) + (1 − 𝛼𝑖)𝛾𝑖 (

��+𝑇𝑖

��+𝑇𝑖)]

The expected values of the proposed class of estimators and neglecting the high order

expressions are given

𝐵(��𝑃𝑖) = 𝐸(��𝑃𝑖 − �� )

𝐵(��𝑃𝑖) = ��(𝛼𝑖𝜆𝑖 + (1 − 𝛼𝑖)𝛾𝑖 − 1) + δ��{𝛼𝑖𝜆𝑖𝜃𝑖2𝐶𝑥

2 − (𝛼𝑖𝜆𝑖 + (1 −

𝛼𝑖)𝛾𝑖)𝜃𝑖ρCxCy}

𝐵(��𝑃𝑖) = ��(𝑃𝑖 − 1) + δ��{(𝑃𝑖 + 𝑄𝑖

2) 𝜃𝑖

2𝐶𝑥2 + 𝑄𝑖𝜃𝑖ρCxCy}

Where δ = (1 − f

n) , 𝑃𝑖 = 𝛼𝑖𝜆𝑖 + (1 − 𝛼𝑖)𝛾𝑖 , 𝑄𝑖 = 𝛼𝑖𝜆𝑖 − (1 − 𝛼𝑖)𝛾𝑖

𝜃𝑖 = ��

�� + 𝑇𝑖

, i = 1,2,3,4,5 𝑇1 = R1, 𝑇2 = RN, 𝑇3 = 𝑅, 𝑇4 = R𝑑, and 𝑇5 = Ra

Where 𝜆𝑖 and 𝛾𝑖 are as defined above. If we assume that 𝑎𝑖 =0 , 𝑏𝑖 = 0 and 𝛼𝑖 = 1

then the proposed estimators are exactly equal to existing modified ratio estimator. If

𝑎𝑖 =0 , 𝑏𝑖 = 0 and 𝛼𝑖 = 0 then the proposed estimators are exactly equal to existing

modified product estimator. If we assume that 𝑎𝑖 = 𝐵(��𝑀𝑅𝑖) , 𝑏𝑖 = 𝐵(��𝑀𝑃𝑖) and 𝛼𝑖 =

1 𝑜𝑟 𝛼𝑖 = 0 , then the proposed estimators are almost unbiased ratio or product

estimators corresponding to the existing estimators. The detailed derivation of the

biases mean squared errors are given in the appendix and the final expression is

obtained with only first-order approximation in the Taylor series expansion as,

𝑀𝑆𝐸(��𝑃𝑖) = 𝐸(��𝑃𝑖 − ��)2

𝑀𝑆𝐸(��𝑃𝑖) = ��2(𝑃𝑖 − 1)2 + (1 − 𝑓

𝑛) ��2{𝐶𝑦

2 𝑃𝑖2 + 𝜃𝑖

2𝐶𝑥2( 𝑃𝑖

2 + (𝑃𝑖 + 𝑄𝑖)(𝑄𝑖 − 1)

− 2𝜃𝑖ρCxCy(𝑄𝑖(2𝑃𝑖 − 1))}

Where δ = (1 − f


𝜃𝑖 = ��

�� + 𝑇𝑖

, i = 1,2,3,4,5 𝑇1 = R1, 𝑇2 = RN, 𝑇3 = 𝑅, 𝑇4 = R𝑑 , and 𝑇5 = Ra


If 𝑎𝑖 = 𝐵(��𝑀𝑅𝑖) , 𝑏𝑖 = 𝐵(��𝑀𝑃𝑖) and 𝛼𝑖 is optimum, then the proposed estimators are

less bias (almost unbiased) ratio cum product estimators. The optimal value of 𝛼𝑖 can

be determined by minimizing the MSE (��𝑝𝑖) with respect to 𝛼𝑖. For this differentiate

MSE with respect to 𝛼𝑖 and equate to zero.

𝜕𝑀𝑆𝐸

𝜕𝛼𝑖= 0, and we get the value of 𝛼𝑖, as

𝛼𝑖 =(𝜆𝑖−1)(𝜆𝑖−𝛾𝑖)+(

1−𝑓

𝑛){𝐶𝑦

2𝛾𝑖(𝛾𝑖−𝜆𝑖)+𝜃𝑖2𝐶𝑥

2(𝜆𝑖+𝛾𝑖2)−𝜃𝑖𝜌𝐶𝑥𝐶𝑦(𝜆𝑖+𝛾𝑖−4𝛾𝑖

2)}

(𝜆𝑖−𝛾𝑖)2+(1−𝑓

𝑛){(𝜆𝑖−𝛾𝑖)2𝐶𝑦

2+𝜃𝑖

2𝐶𝑥2(3𝜆𝑖

2+𝛾𝑖2)+4𝜃𝑖𝜌𝐶𝑥𝐶𝑦(𝛾𝑖

2−𝜆𝑖2)}

4. EFFICIENCY COMPARISON

In this section, compare the efficiencies proposed estimators with that of the existing

estimators such as simple random sampling without replacement sample mean, ratio

estimator, product estimator, modified ratio estimator and modified product estimator

The detailed derivation of the comparisons of the estimators are given in appendix.

4.1 Comparison with Simple random sampling without replacement sample

mean

By comparing the proposed estimators with that of simple random sampling without

replacement sample mean, we arrive Ypi is more efficient than ��𝑠𝑟𝑠 only if

𝑉(��𝑠𝑟𝑠) ≥ 𝑀𝑆𝐸(��𝑃𝑖)

𝜌 ≥(𝑃𝑖 − 1)2 + δ[𝐶𝑦

2(𝑃𝑖2 − 1) + 𝜃𝑖

2𝐶𝑥2 (𝑃𝑖

2 + (𝑃𝑖 + 𝑄𝑖)(𝑄𝑖 − 1))]

2δ𝜃𝑖𝐶𝑥𝐶𝑦𝑄𝑖(2𝑃𝑖 − 1)

4.2 Comparison with Ratio estimator

By comparing the proposed estimators with that of ratio estimator, we arrive ��𝑃𝑖 more

efficient than ��𝑅 only if

𝑉(��𝑅) ≥ 𝑀𝑆𝐸(��𝑃𝑖)

𝜌 ≥(𝑃𝑖 − 1)2 + δ[𝐶𝑦

2(𝑃𝑖2 − 1) + 𝐶𝑥

2(𝜃𝑖2(𝑃𝑖

2 + (𝑃𝑖 + 𝑄𝑖)(𝑄𝑖 − 1)) − 1)]

2δ𝐶𝑥𝐶𝑦(𝜃𝑖𝑄𝑖(2𝑃𝑖 − 1) − 1)


4.3 Comparison with Product estimator

By comparing the proposed estimators with that of product estimator, we arrive Ypi is

more efficient than ��𝑃 only if

𝑉(��𝑃) ≥ 𝑀𝑆𝐸(��𝑃𝑖)

𝜌 ≥(𝑃𝑖 − 1)2 + δ[𝐶𝑦

2(𝑃𝑖2 − 1) + 𝐶𝑥


2 + (𝑃𝑖 + 𝑄𝑖)(𝑄𝑖 − 1)) − 1)]

2δ𝐶𝑥𝐶𝑦(1 + 𝜃𝑖𝑄𝑖(2𝑃𝑖 − 1))

4.4 Comparison with Modified Ratio estimator

By comparing the proposed estimators with that of ratio estimator, we arrive ��𝑃𝑖 is more

efficient than ��𝑀𝑅𝑖 only if

𝑉(��𝑀𝑅𝑖) ≥ 𝑀𝑆𝐸(��𝑃𝑖)

𝜌 ≥(𝑃𝑖 − 1)2 + δ[𝐶𝑦

2(𝑃𝑖2 − 1) + 𝜃𝑖

2𝐶𝑥2(𝑃𝑖

2 + (𝑃𝑖 + 𝑄𝑖)(𝑄𝑖 − 1) − 1)]

2δ𝜃𝑖𝐶𝑥𝐶𝑦(𝑄𝑖(2𝑃𝑖 − 1) − 1)

4.5 Comparison with Modified Product estimator

By comparing the proposed estimators with that of product estimator, we arrive ��𝑃𝑖 is

more efficient than ��𝑀𝑃𝑖 only if

𝑉(��𝑀𝑃𝑖) ≥ 𝑀𝑆𝐸(��𝑃𝑖)

𝜌 ≥(𝑃𝑖 − 1)2 + δ[𝐶𝑦

2(𝑃𝑖2 − 1) + 𝜃𝑖

2𝐶𝑥2(𝑃𝑖

2 + (𝑃𝑖 + 𝑄𝑖)(𝑄𝑖 − 1) − 1)]

2δ𝜃𝑖𝐶𝑥𝐶𝑦(1 + 𝑄𝑖(2𝑃𝑖 − 1))

5. NUMERICAL STUDY

In this section, we consider two natural populations, for accessing the performance of

the proposed estimators. The computed values of constants and parameters of these

populations are given below:

Population 1: (Cochran [2] page 325)

N = 10 n=3 ��= 101.1 ��=58.8 ρ= 0.6515 Sx=7.9414 Sy=15.4448

Cx = 0.1350 Cy=0.1527 β1=0.2363 β2=2.2388 R1=48 RN= 72


R =24 Rd =12 Ra= 60

Population 2: (Singh and Chaudhary [8] , page 177)

N = 34 n=5 ��= 856.4117 ��=199.4412 ρ = 0.4453 Sx=733.1407

Sy=150.2150 Cx = 0.8561 Cy=0.7532 β1=7.9550 β2=13.3667 R1 = 96

RN = 4170 R = 4074 Rd = 2037 Ra = 2133

Table 1: Constants and parameters of the population

Constants Population 1 Population 2

𝜆i 1 1 1 1 0.999 0.946 1.005 1.005 1.004 1.004

𝛾i 0.999 0.998 0.998 0.999 0.997 0.958 0.992 0.992 0.986 0.986

𝛼𝑖 1.319 1.169 1.018 1.244 0.943 0.724 1.603 1.582 1.135 1.156

𝜃𝑖 0.551 0.449 0.71 0.495 0.831 0.899 0.17 0.174 0.296 0.286

These values are obtained when 𝑎𝑖= 𝐵(��𝑀𝑅𝑖),𝑏𝑖 = 𝐵(��𝑀𝑃𝑖) and 𝛼𝑖 ( i = 1,2,3,4,5) is

optimum and these parameters are used to obtained the biases and mean squared errors

of the proposed and existing estimators, and also compare the percentage relative

efficiency of proposed estimators with that of the existing estimators such as simple

random sampling sample mean, ratio, product, modified ratio and modified product

estimators with quartiles and their functions.

Table 2: Bias and MSE of Proposed Class of Estimators

Proposed Estimator Population 1 Population 2

𝐁𝐢𝐚𝐬 MSE 𝐁𝐢𝐚𝐬 MSE

��𝑃1 -9.76E-15 32.079 -5.32E-15 2678.769

��𝑃2 -7.64E-15 32.0095 5.15E-14 3159.852

��𝑃3 1.00E-14 32.042 -3.15E-14 3159.052

��𝑃4 -1.67e-14 32.003 1.31E-14 3122.974

��𝑃5 -1.23E-14 32.088 -3.53E-14 3126.263


Table 3: Bias and MSE of Existing Estimators

Estimators Population 1 Population 2

𝐁𝐢𝐚𝐬 MSE 𝐁𝐢𝐚𝐬 MSE

��𝑠𝑟𝑠 - 55.6603 - 3849.248

��𝑅 0.1132 35.0447 15.165 4925.325

��𝑝 0.3171 163.283 9.768 12718.48

��𝑀𝑅1 -0.5559 34.7135 11.376 4366.155

��𝑀𝑅2 -0.0415 33.5455 -0.941 3329.697

��𝑀𝑅3 -0.0081 32.0655 -0.945 3322.446

��𝑀𝑅4 0.0334 32.4152 -0.707 3131.558

��𝑀𝑅5 -0.0515 34.5819 -0.752 3141.065

��𝑀𝑃1 0.1745 93.2761 8.784 11373.28

��𝑀𝑃2 0.1425 104.1486 1.664 4657.512

��𝑀𝑃3 0.2252 123.1333 1.697 4676.116

��𝑀𝑃4 0.2634 138.9182 2.891 5438.228

��𝑀𝑃5 0.1569 98.0534 2.799 5373.66

Table 4: Percentage Relative Efficiency of Proposed Estimators

Proposed Estimators

Population 1 Population 2

��𝑠𝑟𝑠 ��𝑅 ��𝑝 ��𝑠𝑟𝑠 ��𝑅 ��𝑝

��𝑃1 173.42 109.19 508.73 143.69 183.87 474.78

��𝑃2 173.51 109.24 509 121.82 155.87 402.5

��𝑃3 173.71 109.37 509.58 121.84 155.91 402.6

��𝑃4 173.92 109.51 510.22 123.26 157.91 407.26

��𝑃5 173.45 109.21 508.85 123.13 157.55 406.83


Table 5: Percentage Relative Efficiency of Proposed Estimators

Proposed Estimators Modified Ratio Modified Product

Population 1 Population 2 Population 1 Population 2

��𝑃1 111.004 162.99 290.61 424.59

��𝑃2 104.57 105.38 324.66 147.39

��𝑃3 100.07 105.17 384.29 148.03

��𝑃4 101.29 100,27 434.08 174.14

��𝑃5 107.76 100.47 305.56 171.88

6. SIMULATION STUDY

In this section a finite population of size N= (100,100) is generated from morgnstern’s

bivariate exponential distribution ( MTBED ) with parameters alpha= 0.8 and theta 1

= 1.5 and theta 2 = 0.9. The simulation process is repeated under 10000 times. The

average value of the biases, mean squared errors and percentage relative efficiencies

with respect to the existing and proposed estimators are obtained for a random sample

of size n = 25 and n= 50 are drawn by SRSWOR. The parameter values, biases, mean

squared errors and the percentage relative efficiencies are given in the following tables.

Table 5: Population Parameters Simulated data

N = 100 n = 25 ��=1.0877 ��=0.5707 Cx =1.1321 Cy=1.1051

ρ = 0.2509 R1= 0.0039 RN = 2.7722 R= 2.7681 R𝑑 = 1.3841 R𝑎= 1.3881

N = 100 n = 50 ��= 1.0568 ��=0.7717 Cx = 1.1368 Cy=0.8787

ρ = 0.0919 R1= 0.0047 RN = 4.8238 R= 4.8191 R𝑑 = 2.4095 R𝑎= 2.4143


Table 6: Population Parameters Simulated data

Parameters

/constants When sample size n =25

𝜃𝑖 0.9903 0.1669 0.1672 0.2854 0.2846

𝜆𝑖 0.9774 1.0002 1.0002 0.9993 0.9993

𝛾𝑖 0.9941 0.9990 0.9990 0.9983 0.9983

𝛼𝑖 0.6166 1.1146 1.1136 0.8598 0.8608

Parameters

/constants When sample size n=50

𝜃𝑖 0.9903 0.1667 0.1670 0.2851 0.2843

𝜆𝑖 0.9923 1.0001 1.0001 0.9998 0.9998

𝛾𝑖 0.9981 0.9997 0.9997 0.9994 0.9994

𝛼𝑖 0.6066 1.1127 1.1117 0.8570 0.8580

Table 7: Biases and MSE of simulated Data

Estimators n=25 n=50

bias MSE bias MSE

Existing

estimators

��𝑠𝑟𝑠 0.0371 0.0123

��𝑅 0.0088 0.0198 0.0088 0.0198

��𝑝 0.0022 0.0296 0.0022 0.0296

��𝑀𝑅1 0.0257 0.0196 8.63E-03 0.0196

��𝑀𝑅2 -0.0002 0.0119 -6.17E-05 0.0119

��𝑀𝑅3 -0.0002 0.0119 -6.13E-05 0.0119

��𝑀𝑅4 0.0007 0.0119 2.57E-04 0.0119

��𝑀𝑅5 0.0007 0.0119 2.54E-04 0.0119


��𝑀𝑃1 0.0022 0.0293 0.0022 0.0293

��𝑀𝑃2 0.0004 0.0135 0.0004 0.0135

��𝑀𝑃3 0.0004 0.0135 0.0004 0.0135

��𝑀𝑃4 0.0006 0.0147 0.0006 0.0147

��𝑀𝑃5 0.0006 0.0147 0.0006 0.0147

Proposed

Estimators

��𝑃1 3.43E-19 0.0338 9.85E-19 0.0116

��𝑃2 -1.12E-18 0.0352 -2.31E-18 0.0117

��𝑃3 -1.41E-18 0.0352 8.83E-19 0.0117

��𝑃4 9.92E-20 0.0351 -7.19E-20 0.0117

��𝑃5 1.68E-18 0.0351 2.29E-18 0.0117

Table 8: Percentage Relative Efiiciency of Simulated data

Proposed estimators

Sample size n=5 Sample size n=50

��𝑠𝑟𝑠 ��𝑅 ��𝑝 ��𝑠𝑟𝑠 ��𝑅 ��𝑝

��𝑃1 109.8612 174.9820 263.2365 106.7319 170.9454 255.4924

��𝑃2 105.3091 167.7317 252.3294 105.2325 168.5439 251.9031

��𝑃3 105.3096 167.7324 252.3305 105.2326 168.5441 251.9034

��𝑃4 105.5446 168.1068 252.8936 105.3107 168.6692 252.0904

��𝑃5 105.5427 168.1037 252.8890 105.3101 168.6682 252.0889


Table 9: Percentage Relative Efficiency of Simulated data

Proposed Estimators Modified Ratio Estmator Modified Product Estimator

n=25 n=50 n=25 n=50

��𝑃1 173.3341 260.7518 169.3316 253.0694

��𝑃2 101.1478 115.2022 101.1968 114.9955

��𝑃3 101.1463 115.2241 101.1951 115.0170

��𝑃4 101.8834 125.9913 101.8648 125.5087

��𝑃5 101.8676 125.9074 101.8498 125.4265

7. CONCLUSIONS

In this paper, we have proposed a class of modified ratio cum product estimators for

the estimation of finite population mean of the study variable Y with known ranges and

their functions of the auxiliary variable. The biases and mean squared errors of the

proposed estimators are obtained and compared with that of simple random sampling,

ratio, product, modified ratio and modified product estimators. The performances of the

proposed estimators for some known natural populations are also observed and a

simulation study is carried out by morgenstern’s bivariate exponential distribution (

MTBED ),and is repeated under 10000 times. The proposed class of estimators have

less bias (almost unbiased) and mean squared error than all the existing estimators.

Tables 4,5,8, and 9 shows that the proposed class of estimators have heighest

percentage relative efficiency compared to the existing estimators and these estimators

are performing better than all the existing estimators.Hence the proposed class of

estimators are recommended for practical applications.

REFERENCES

[1] Cochran W.G. (1940): 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.

[2] Cochran, W. G. (1977). Sampling Techniques. Third Edition, Wiley Eastern

Limited.

[3] Jambulingam Subramani and Master Ajith (2016): Improved Ratio cum Product

Estimator with Known Coefficient of Variation in Simple Random Sampling J.


Adv. Res. Appl. Math. Stat.; 1(2), 60-70.

[4] Jambulingam Subramani and Master Ajith (2016): Modified Ratio cum Product

Estimators for Estimation of Finite Population Mean with Known Correlation

Coefficient Biom. Biostat. Int J , 4(6): 00113.

[5] Mohanty, S., Sahoo, J.(1995). A note on improving the ratio method of

estimation through linear transformation using certain known population

parameters.Sankhyā, Series B . 57:93-102.

[6] Murthy, M.N. (1964): Product method of estimation. Sankhya A, 26, 69-74.

[7] Murthy, M.N. (1967). Sampling theory and methods. Statistical Publishing

Society, Calcutta, India.

[8] Singh, D. and Chaudhary, F.S. (1986), Theory and analysis of sample survey

designs. New Age International Publisher.

[9] Sisodia B.V.S. and V.K. Dwivedi (1981)., “A modified ratio estimator using

coefficient of variation of auxiliary variable”, Jour. Ind. Soc. Agri. Stat., Vol.

33(1), Pp. 13-18.

[10] Subramani J (2013): "Generalized modified ratio estimator for estimation of

finite population mean," Journal of Modern Applied Statistical Methods,

vol.12, pp.121-155.

[11] Subramani J .and G. Kumarapandiyan (2012) Modified Ratio Estimators for

Population Mean Using Function of Quartiles of Auxiliary Variable, Bonfring

International Journal of Industrial Engineering and Management Science, Vol.

2, No. 2,

[12] Subramani, J., and Kumarapandiyan, (2012) G. Variance estimation using

quartiles and their functions of an auxiliary variable, International Journal of

Statistics and Applications 2 (5),67-72.

[13] Upadhyaya, L.N. and Singh, H.P(1999).. Use of transformed auxiliary variable

in estimating the finite population mean, Biometrical Journal 41 (5), 627-636.

[14] Yan, Z. and Tian, B. (2010). Ratio Method to the Mean Estimation Using Co-

efficient of Skewness of Auxiliary Variable, ICICA 2010, Part II, CCIS 106,

pp. 103–110.

https://www.statindex.org/authors/7172

https://www.statindex.org/authors/87274

https://www.statindex.org/articles/128785



https://www.statindex.org/journals/1533

https://www.statindex.org/journals/1533/57


APPENDIX

Bias and MSE of Proposed Estimators

The proposed class of ratio cum product estimators with known Ranges and their functions are

given as

��𝑃1 = �� [𝛼1𝜆1 (��+𝑅1

��+𝑅1) + (1 − 𝛼1)𝛾1 (

��+𝑅1

��+𝑅1)]

��𝑃2 = �� [𝛼2𝜆2 (��+𝑅𝑁

��+𝑅𝑁) + (1 − 𝛼2)𝛾2 (

��+𝑅𝑁

��+𝑅𝑁)]

��𝑃3 = �� [𝛼3𝜆3 (��+𝑅

��+𝑅) + (1 − 𝛼3)𝛾3 (

��+𝑅

��+𝑅)]

��𝑃4 = �� [𝛼4𝜆4 (��+𝑅𝑎

��+𝑅𝑑) + (1 − 𝛼4)𝛾4 (

��+𝑅𝑑

��+𝑅𝑑)]

��𝑃5 = �� [𝛼5𝜆5 (��+𝑅𝑎

��+𝑅𝑎) + (1 − 𝛼5)𝛾5 (

��+𝑅𝑎

��+𝑅𝑎)]

Where 𝜆𝑖 = 𝑆𝑦

𝑆𝑦+𝑎𝑖𝐶𝑦, 𝛾𝑖 =

𝑆𝑦

𝑆𝑦+𝑏𝑖𝐶𝑦,i = 1,2,3,4,5 . Here 𝑎𝑖’s and 𝑏𝑖’s are constants

Consider 𝑒0 =��−��

��, 𝑒1 =

��−��

��, 𝜃𝑖 =

��

��+𝑇𝑖, T1 = Xmax,T2 = Xmin , T3 = R, T4 = Rd, T5 = Ra

𝐸(𝑒0) = 𝐸(𝑒1) = 0, 𝐸(𝑒02) = (

1−𝑓

𝑛) ��2𝐶𝑦

2 , 𝐸(𝑒12) = (

1−𝑓

𝑛) ��2𝐶𝑥

2, 𝐸(𝑒0𝑒1) =

(1−𝑓

𝑛) 𝜌𝐶𝑥𝐶𝑦

Substitute the values of 𝑒0and 𝑒1 in ��𝑝𝑖 and neglecting the higher order expressions, we get

��𝑝𝑖 = 𝛼𝑖𝜆𝑖�� (�� + Ti

�� + Ti) + (1 − 𝛼𝑖)𝛾𝑖�� (

�� + Ti

�� + Ti

)

= 𝛼𝑖𝜆𝑖��(1 + 𝑒0)(1 + 𝜃𝑖𝑒1)−1 + (1 − 𝛼𝑖)𝛾𝑖��(1 + 𝑒0)(1 + 𝜃𝑖𝑒1)

= ��{𝛼𝑖𝜆𝑖(1 + 𝑒0)(1 − 𝜃𝑖𝑒1 + 𝜃𝑖2𝑒1

2) + (1 − 𝛼𝑖)𝛾𝑖(1 + 𝑒0)(1 + 𝜃𝑖𝑒1)}

= ��{𝛼𝑖𝜆𝑖(1 − 𝜃𝑖𝑒1 + 𝜃𝑖2𝑒1

2 + 𝑒0 − 𝜃𝑖𝑒1𝑒0) + (1

− 𝛼𝑖)𝛾𝑖(1 + 𝜃𝑖𝑒1 + 𝑒0 + 𝜃𝑖𝑒0𝑒1)}

��𝑝𝑖 − �� = ��{𝛼𝑖𝜆𝑖(1 − 𝜃𝑖𝑒1 + 𝜃𝑖2𝑒1

2 + 𝑒0 − 𝜃𝑖𝑒1𝑒0)

+ (1 − 𝛼𝑖)𝛾𝑖(1 + 𝜃𝑖𝑒1 + 𝑒0 + 𝜃𝑖𝑒0𝑒1)}

𝐵(��𝑃𝑖) = 𝐸(��𝑃𝑖 − �� )

= 𝐸{��(𝛼𝑖𝜆𝑖(1 − 𝜃𝑖𝑒1 + 𝜃𝑖2𝑒1

2 + 𝑒0 − 𝜃𝑖𝑒1𝑒0) + (1 − 𝛼𝑖)𝛾𝑖(1 + 𝜃𝑖𝑒1 + 𝑒0 +

𝜃𝑖𝑒0𝑒1) − 1)}


= ��{𝛼𝑖𝜆𝑖(1 + δ𝜃𝑖2𝐶𝑥

2 − δ𝜃𝑖𝜌𝐶𝑥𝐶𝑦) + (1 − 𝛼𝑖)𝛾𝑖(1 + δ𝜃𝑖𝜌𝐶𝑥𝐶𝑦) − 1}

𝐵(��𝑃𝑖) = ��(𝛼𝑖𝜆𝑖 + (1 − 𝛼𝑖)𝛾𝑖 − 1) + δ��{𝛼𝑖𝜆𝑖𝜃𝑖2𝐶𝑥

2 − (𝛼𝑖𝜆𝑖 + (1 − 𝛼𝑖)𝛾𝑖)𝜃𝑖ρCxCy}

𝐵(��𝑃𝑖) = ��(𝑃𝑖 − 1) + δ��{(𝑃𝑖 + 𝑄𝑖

2) 𝜃𝑖

2𝐶𝑥2 + 𝑄𝑖𝜃𝑖ρCxCy}

Where δ = (1 − f


The mean squared error of the proposed estimator is

𝑀𝑆𝐸(��𝑃𝑖) = 𝐸(��𝑃𝑖 − ��)2

= 𝐸{��(𝛼𝑖𝜆𝑖(1 − 𝜃𝑖𝑒1 + 𝜃2𝑒12 + 𝑒0 − 𝜃𝑖𝑒1𝑒0) + (1 − 𝛼𝑖)𝛾𝑖(1 + 𝜃𝑖𝑒1 + 𝑒0 + 𝜃𝑖𝑒0𝑒1)

− 1)2}

= 𝐸 {��2(𝛼𝑖2𝜆𝑖

2(1 − 𝜃𝑖𝑒1 + 𝜃𝑖2𝑒1

2 + 𝑒0 − 𝜃𝑖𝑒1𝑒0)2

+ (1 − 𝛼𝑖)2𝛾𝑖2(1 + 𝜃𝑖𝑒1 + 𝜃𝑖𝑒0𝑒1)2 +

1 + 2𝛼𝑖𝜆𝑖(1 − 𝛼𝑖)𝛾𝑖(1 − 𝜃𝑖𝑒1 + 𝜃𝑖2𝑒1

2 + 𝑒0 − 𝜃𝑖𝑒1𝑒0)(1 + 𝜃𝑖𝑒1 + 𝑒0 + 𝜃𝑖𝑒0𝑒1) −

2𝛼𝑖𝜆𝑖(1 − 𝜃𝑒1 + 𝜃𝑖2𝑒1

2 + 𝑒0 − 𝜃𝑖𝑒1𝑒0) − 2(1 − 𝛼𝑖)𝛾𝑖(1 + 𝜃𝑖𝑒1 + 𝑒0 + 𝜃𝑖𝑒0𝑒1)}

= δ��2{𝛼𝑖2𝜆𝑖

2(1 + 3δ𝜃𝑖2𝐶𝑥

2 + δ𝐶𝑦2 − 4δ𝜃𝑖𝜌𝐶𝑥𝐶𝑦) + (1 − 𝛼𝑖)2𝛾𝑖

2(1 + δ𝜃𝑖2𝐶𝑥

2 + δ𝐶𝑦2 +

4δ𝜃𝑖𝜌𝐶𝑥𝐶𝑦) + 2𝛼𝑖𝜆𝑖(1 − 𝛼𝑖)𝛾𝑖(1 + δ𝐶𝑦2) − 2𝛼𝑖𝜆𝑖(1 + δ𝜃𝑖

2𝐶𝑥2 − δ𝜃𝑖𝜌𝐶𝑥𝐶𝑦) − 2(1 −

𝛼𝑖)𝛾𝑖(1 + δ𝜃𝑖𝜌𝐶𝑥𝐶𝑦)}

𝑀𝑆𝐸(��𝑃𝑖) = ��2(𝛼𝑖𝜆𝑖 + (1 − 𝛼𝑖)𝛾𝑖 − 1)2 + δ��2{𝐶𝑦2( 𝛼𝑖𝜆𝑖 + (1 − 𝛼𝑖)𝛾𝑖 )

2

+ 𝜃𝑖2𝐶𝑥

2 ( 3𝛼𝑖2𝜆𝑖

2+ (1 − 𝛼𝑖)2𝛾𝑖

2 − 2𝛼𝑖𝜆𝑖 ) + 2𝜃𝑖ρCxCy(𝛼𝑖𝜆𝑖

− (1 − 𝛼𝑖)𝛾𝑖 − 2(𝛼𝑖2𝜆𝑖 − (1 − 𝛼𝑖)2𝛾𝑖

2))}

𝑀𝑆𝐸(��𝑃𝑖) = ��2(𝑃𝑖 − 1)2 + δ��2{𝐶𝑦2 𝑃𝑖

2 + 𝜃𝑖2𝐶𝑥

2( 𝑃𝑖2 + (𝑃𝑖 + 𝑄𝑖)(𝑄𝑖 − 1)

+ 2𝜃𝑖ρCxCy(𝑄𝑖(2𝑃𝑖 − 1))}

Where δ = (1 − f


𝜃𝑖 = ��

�� + 𝑇𝑖

, i = 1,2,3,4,5 𝑇1 = R1, 𝑇2 = RN, 𝑇3 = 𝑅, 𝑇4 = R𝑑 , and 𝑇5 = Ra

The optimal value of 𝛼𝑖 is determined by minimize the MSE (��𝑝𝑖) with respect to 𝛼𝑖. For this

differentiate MSE with respect to 𝛼𝑖 and equate to zero.

ie, 𝜕𝑀𝑆𝐸

𝜕𝛼𝑖= 0, and we get the value of 𝛼𝑖, as


2(𝛼𝑖𝜆𝑖 + (1 − 𝛼𝑖)𝛾𝑖 − 1)(λi − 𝛾𝑖 ) + δ {Cy22(𝛼𝑖𝜆𝑖 + (1 − 𝛼𝑖)𝛾𝑖 )(λi − 𝛾𝑖 )

+ θ2Cx2(6𝛼𝑖λi

2 − 2(1 − 𝛼𝑖)𝛾𝑖 2 − 2λi) + 2θ𝜌𝐶𝑥𝐶𝑦(λi + 𝛾𝑖

− 2(2𝛼𝑖λi2 + 2(1 − 𝛼𝑖)𝛾𝑖

2)} = 0

(𝛼𝑖λ𝑖 − 𝛼𝑖𝛾𝑖 )(λi − 𝛾𝑖 ) + δ {Cy2(𝛼𝑖λi − 𝛼𝑖𝛾𝑖 )(λi − 𝛾𝑖 ) + θ2Cx

2(3𝛼𝑖λi2 + 𝛼𝑖𝛾𝑖

2) +

θ𝜌𝐶𝑥𝐶𝑦(−4𝛼𝑖λi2 + 4𝛼𝑖𝛾𝑖

2)} = (𝛾𝑖 − 1)(𝛾𝑖 − λi) + δ{Cy2𝛾𝑖 (𝛾𝑖 − λi) + θ2Cx

2(λi +

𝛾𝑖 2) − θ𝜌𝐶𝑥𝐶𝑦(λi + 𝛾𝑖 − 4𝛾𝑖

2)}

𝛼𝑖 =(𝜆𝑖 − 1)(𝜆𝑖 − 𝛾𝑖) + δ{𝐶𝑦

2𝛾𝑖(𝛾𝑖 − 𝜆𝑖) + 𝜃𝑖2𝐶𝑥

2(𝜆𝑖 + 𝛾𝑖2) − 𝜃𝑖𝜌𝐶𝑥𝐶𝑦(𝜆𝑖 + 𝛾𝑖 − 4𝛾𝑖

2)}

(𝜆𝑖 − 𝛾𝑖)2 + δ{(𝜆𝑖 − 𝛾𝑖)2𝐶𝑦2

+ 𝜃𝑖2𝐶𝑥

2(3𝜆𝑖2 + 𝛾𝑖

2) + 4𝜃𝑖𝜌𝐶𝑥𝐶𝑦(𝛾𝑖2 − 𝜆𝑖

2)}

Where δ = (1−f

n),𝜃𝑖 =

��

��+Ti, and T1 = Q1,T2 = Q3 T3 = Qd,T4 = Qa,T5 = Qr, i = 1,2,3,4,5

Comparison with Simple random sampling without replacement sample mean

By comparing the proposed estimators with that of simple random sampling without

replacement sample mean, we arrive Ypi is more efficient than ��𝑠𝑟𝑠 only if

𝑉(��𝑠𝑟𝑠) ≥ 𝑀𝑆𝐸(��𝑃𝑖)

(1−𝑓

𝑛) ��2𝐶𝑦

2 ≥ ��2(𝑃𝑖 − 1)2 + δ��2{𝐶𝑦2 𝑃𝑖


2( 𝑃𝑖2 + (𝑃𝑖 + 𝑄𝑖)(𝑄𝑖 − 1) +

2𝜃𝑖ρCxCy(𝑄𝑖(2𝑃𝑖 − 1))}

2δ𝜌𝜃𝑖𝐶𝑥𝐶𝑦(𝑄𝑖(2𝑃𝑖 − 1)) ≥ (𝑃𝑖 − 1)2 + δ{𝐶𝑦2 (𝑃𝑖

2 − 1) + 𝜃𝑖2𝐶𝑥

2(( 𝑃𝑖2 + (𝑃𝑖 + 𝑄𝑖)(𝑄𝑖 −

1))}

𝜌 ≥(𝑃𝑖 − 1)2 + δ[𝐶𝑦

2(𝑃𝑖2 − 1) + 𝜃𝑖

2𝐶𝑥2 (𝑃𝑖

2 + (𝑃𝑖 + 𝑄𝑖)(𝑄𝑖 − 1))]

2δ𝜃𝑖𝐶𝑥𝐶𝑦𝑄𝑖(2𝑃𝑖 − 1)

Comparison with Ratio estimator

By comparing the proposed estimators with that of ratio estimator, we arrive ��𝑃𝑖 more efficient

than ��𝑅 only if

𝑉(��𝑅) ≥ 𝑀𝑆𝐸(��𝑃𝑖)

δ��2(𝐶𝑦2 + 𝐶𝑥

2 − 2Cxy) ≥ ��2(𝑃𝑖 − 1)2 + δ��2{𝐶𝑦2 𝑃𝑖


2( 𝑃𝑖2 + (𝑃𝑖 + 𝑄𝑖)(𝑄𝑖 −

1) − 2𝜃𝑖ρCxCy(𝑄𝑖(2𝑃𝑖 − 1))}

2δ𝜌𝐶𝑥𝐶𝑦(𝜃𝑖𝑄𝑖(2𝑃𝑖 − 1) − 1) ≥ (𝑃𝑖 − 1)2 + δ{𝐶𝑦2 (𝑃𝑖

2 − 1) + 𝐶𝑥2(𝜃𝑖

2( 𝑃𝑖2 + (𝑃𝑖 +

𝑄𝑖)(𝑄𝑖 − 1) − 1)}


𝜌 ≥(𝑃𝑖 − 1)2 + δ[𝐶𝑦

2(𝑃𝑖2 − 1) + 𝐶𝑥


2 + (𝑃𝑖 + 𝑄𝑖)(𝑄𝑖 − 1)) − 1)]

2δ𝐶𝑥𝐶𝑦(𝜃𝑖𝑄𝑖(2𝑃𝑖 − 1) − 1)

Comparison with Product estimator

By comparing the proposed estimators with that of product estimator, we arrive Ypi is more

efficient than ��𝑃 only if

𝑉(��𝑃) ≥ 𝑀𝑆𝐸(��𝑃𝑖)

δ��2(𝐶𝑦2 + 𝐶𝑥

2 + 2ρCxCy) ≥ ��2(𝑃𝑖 − 1)2 + δ��2{𝐶𝑦2 𝑃𝑖


2( 𝑃𝑖

2 + (𝑃𝑖 + 𝑄𝑖)(𝑄𝑖 −

1) − 2𝜃𝑖ρCxCy(𝑄𝑖(2𝑃𝑖 − 1))}

2δ𝜌𝐶𝑥𝐶𝑦(𝜃𝑖𝑄𝑖(2𝑃𝑖 − 1) + 1) ≥ (𝑃𝑖 − 1)2 + δ{𝐶𝑦2 (𝑃𝑖

2 − 1) + 𝐶𝑥2(𝜃𝑖

2( 𝑃𝑖2 + (𝑃𝑖 +

𝑄𝑖)(𝑄𝑖 − 1) − 1)}

𝜌 ≥(𝑃𝑖 − 1)2 + δ[𝐶𝑦

2(𝑃𝑖2 − 1) + 𝐶𝑥


2 + (𝑃𝑖 + 𝑄𝑖)(𝑄𝑖 − 1)) − 1)]

2δ𝐶𝑥𝐶𝑦(1 + 𝜃𝑖𝑄𝑖(2𝑃𝑖 − 1))

Comparison with Modified Ratio estimator

By comparing the proposed estimators with that of ratio estimator, we arrive ��𝑃𝑖 is more

efficient than ��𝑀𝑅𝑖 only if

𝑉(��𝑀𝑅𝑖) ≥ 𝑀𝑆𝐸(��𝑃𝑖)

δ��2(𝐶𝑦2 + 𝜃𝑖

2𝐶𝑥2 − 2𝜃𝑖ρCxCy) ≥ ��2(𝑃𝑖 − 1)2 + δ��2{𝐶𝑦



2 + (𝑃𝑖 +

𝑄𝑖)(𝑄𝑖 − 1) − 2𝜃𝑖ρCxCy(𝑄𝑖(2𝑃𝑖 − 1))}

2δ𝜌𝜃𝑖𝐶𝑥𝐶𝑦(𝑄𝑖(2𝑃𝑖 − 1) − 1) ≥ (𝑃𝑖 − 1)2 + δ{𝐶𝑦2 (𝑃𝑖

2 − 1) + 𝜃𝑖2𝐶𝑥

2(( 𝑃𝑖2 + (𝑃𝑖 +

𝑄𝑖)(𝑄𝑖 − 1) − 1)}

𝜌 ≥(𝑃𝑖 − 1)2 + δ[𝐶𝑦

2(𝑃𝑖2 − 1) + 𝜃𝑖

2𝐶𝑥2(𝑃𝑖

2 + (𝑃𝑖 + 𝑄𝑖)(𝑄𝑖 − 1) − 1)]

2δ𝜃𝑖𝐶𝑥𝐶𝑦(𝑄𝑖(2𝑃𝑖 − 1) − 1)

Comparison with Modified Product estimator

By comparing the proposed estimators with that of product estimator, we arrive ��𝑃𝑖 is more

efficient than ��𝑀𝑃𝑖 only if

𝑉(��𝑀𝑃𝑖) ≥ 𝑀𝑆𝐸(��𝑃𝑖)

δ��2(𝐶𝑦2 + 𝜃𝑖

2𝐶𝑥2 + 2𝜃𝑖ρCxCy) ≥ ��2(𝑃𝑖 − 1)2 + δ��2{𝐶𝑦



2 + (𝑃𝑖 +


𝑄𝑖)(𝑄𝑖 − 1) − 2𝜃𝑖ρCxCy(𝑄𝑖(2𝑃𝑖 − 1))}

2δ𝜌𝜃𝑖𝐶𝑥𝐶𝑦(1 + 𝑄𝑖(2𝑃𝑖 − 1)) ≥ (𝑃𝑖 − 1)2 + δ{𝐶𝑦2 (𝑃𝑖

2 − 1) + 𝜃𝑖2𝐶𝑥

2(( 𝑃𝑖2 + (𝑃𝑖 +

𝑄𝑖)(𝑄𝑖 − 1) − 1)}

𝜌 ≥(𝑃𝑖 − 1)2 + δ[𝐶𝑦

2(𝑃𝑖2 − 1) + 𝜃𝑖

2𝐶𝑥2(𝑃𝑖

2 + (𝑃𝑖 + 𝑄𝑖)(𝑄𝑖 − 1) − 1)]

2δ𝜃𝑖𝐶𝑥𝐶𝑦(1 + 𝑄𝑖(2𝑃𝑖 − 1))

Almost Unbiased Ratio Cum Product Estimators for Finite ... · Jambulingam Subramani and Master...

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Transcript of Almost Unbiased Ratio Cum Product Estimators for Finite ... · Jambulingam Subramani and Master...