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Some Improved Ratio Estimators for Estimating Mean of Finite Population

Muhammad Ijaz1*, Hameed Ali2

1Department of Statistics, University Of Peshawar, Pakistan

2Department of Statistics, Islamia College Peshawar, Pakistan

Corresponding Author:
Muhammad Ijaz
Department of Statistics, University of Peshawar, Pakistan
E-mail: ijaz.statisticstics@gmail.com

Received date: 27/03/2018 Accepted date: 21/06/2018 Published date: 25/06/2018.

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Abstract

In this paper we have proposed an efficient estimator for estimating the mean of finite population under simple random sampling schemes. We have proposed a modified ratio estimator whose efficiency is the same as of regression estimator. It is a well establish fact that linear regression estimator is more efficient than most of the ratio estimators. We have found the Bias and MSE up to first order of approximation. The conditions under which the proposed estimators perform well as compared to other estimators. These properties are supported by real data sets.

Keywords

Auxiliary information, Study variable, Bias, MSE, Percentage relative efficiency

Introduction

The efficiency of an estimator can be increased largely, if we incorporate the auxiliary/benchmark variable(s) correlated with the study variable. Utilizing auxiliary information in a way so that the results become highly efficient. The use of auxiliary information is a challenging problem. Many statisticians use auxiliary information in their own way. It was Cochran who first uses auxiliary information in estimating the mean of finite population. Many other Statisticians make use of auxiliary information at estimation stages [1-6].

The Classical estimator of the mean of the finite population equation This estimator is an unbiasedestimator of population mean and its variance is given by

equation

Cochran [1] introduces the traditional ratio type estimator and is given by

equation

The Bias and MSE are

equation

And

equation

Where equation the coefficient of variation of the study variable Y is,equation is the coefficient ofvariation of the auxiliary variableequation is the coefficient of covariance between the studyvariable and the auxiliary variable andequation is the coefficient of correlation between Y and X.Sisodia and Dwivedi introduces the ratio type estimator for estimating the mean of finitepopulation and is follows as,

equation

The mean square error is

equation

An exponential ratio type estimator due to Bhal and Tuteja[7] is given by

equation

The Bias and MSE of (1.6) is given by

equation

&

equation

Sing and tailor [8] proposed another estimator for estimating the mean of finite population for theknown value of correlation coefficient between the study variable and auxiliary variable. The estimatoris given by

equation (1.9)

The MSE is written as

equation

Proposed Estimators

We suggest the following estimators

equation (2.0)

equation

Where equation are constants or some functions of auxiliary information which is to be determined, sothat to get minimum MSE for the proposed estimator.

Properties of the First Proposed Estimator

We will come across through the following terms and notations to compute the Bias and MSE for theproposed estimator,

equation

equation

Then we can write (2.0) as follows

equation

By neglecting the higher power terms, we have

equation

The Bias corresponding to (2.0) is given by

equation

Or

equation

For MSE, Squaring and taking expectation of equation (2.3), we have

equation (2.5)

Since,

equation

equation (2.6)

We can find the optimum value of equation by minimizing the MSE of equation with respect to ω1

Differentiating (2.6) w.r.to equation and equating to zeroequation we get

equation

By substituting equation , in (2.4) and (2.6) we get

equation

equation

Properties of the Second Proposed Estimator

equation (2.9)

Terms with power higher than two is ignored, we have

equation

We can write

equation

or

equation

For MSE, Squaring and taking expectation of equation (3.3), we have

equation (2.12)

Since,

equation

equation

The optimum value of ω2 can be find out by minimizing (3.6) with respect to the

Differentiating (3.6) w.r.to ω2 and equating to zeroequation we get

equation

Substituting (3.4) (3.6) for equation we get

equation (2.14)

equation (2.15)

Theoretical Comparison of Proposed Estimators

Following are the conditions under which the suggested estimator performs well than the existingestimators considered here.

equation

Which is always true if and only if ρ ≠ 0

equation

If

equation

This is always true,

equation

If,

equation

equation

If

equation

equation

If

equation

equation

If

equation

Obviously the above conditions will always true when we apply it to real data sets.

Applications in SRS

Here in this section we will apply our proposed estimator to different real data sets taken from variousfield of life. The table 2 below shows that our proposed estimator is best as compared to the existingestimators, discussed in the literature. The following data sets have been considered for the comparisonpurpose.

Parameters data set 1 Source: Murthy (1967), Data set 2 Source: Murthy (1967), Data set 3 Source: US Agriculture Statistics(2010) Data set 4 Source: Koyuncu and Kadilar (2009) Data set 5 Source : Pakistan MFA (2004)
N 108 80 69 923 97
n 16 20 17 180 25
equation 461.3981 11.2664 4505.16 11440.498 3050.28
equation 172.704 51.8264 4514.9 436.43 3135.62
ρ  0.828315 0.3542 1.3756 1.718299 2.302173
ρ 0.6903 0.7507 1.18324 1.8645 2.327893
ρ 0.7896 0.9513 0.902327 0.9543 0.9871
β1  1.3612 1.05 5.141563 3.9365 28.345
β2 1.6307 -0.06339 29.77932 18.7208 50.32

Table 1: Differeent data sets with their parameteres values.

Estimators Population 1 Population 2 Population 3 Population 4 Population 5
equation 100 100 100 100 100
equation 263.83 66.28 439.899 939.7 3818.46
equation 263.92 82.5 440.15 940.11 3823.8
equation 168.81 200.13 448.38 817.15 895.53
equation 263.93 87.067 440.05 939.91 3820.9
equation 264.03 65.05 448.34 943.8 3895.86
equation 266.15 877.54 538.19 1119.7 3994.77

Table 2: Percentage relative efficiency of the proposed estimators against some existing estimators

Conclusion

It is clear from above table that efficiency of our proposed estimators is optimum than all estimatorsconsidered in the literature, for all data sets. The conditions mentioned above also supported by the real data. Both estimators are equally efficient and give best results tan all others considered above. So we can modify some basic ratio estimators by assigning some suitable constants to them and hence their efficiency can be increases considerably.

References