American Journal of Mathematics and Statistics

p-ISSN: 2162-948X    e-ISSN: 2162-8475

2018;  8(2): 27-35

doi:10.5923/j.ajms.20180802.01

 

Improved Estimation of Population Mean Using Information on Size of the Sample

Rajesh Kumar Gupta1, S. K. Yadav2

1Tata Consultancy Services, 755 W Big Beaver Road STE 800, 8th Floor, Troy MI, USA

2Department of Statistics, Babasaheb Bhimrao Ambedkar University, Lucknow, India

Correspondence to: Rajesh Kumar Gupta, Tata Consultancy Services, 755 W Big Beaver Road STE 800, 8th Floor, Troy MI, USA.

Email:

Copyright © 2018 Scientific & Academic Publishing. All Rights Reserved.

This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/

Abstract

In the present paper, the sample size has been used as information for improved estimation of population mean of the main variable under study. A generalized ratio type estimator of population mean has been proposed. The large sample properties the bias and the mean squared error of the proposed estimator have been obtained up to first order of approximation. The optimum value of the characterizing scalar which minimizes the mean squared error has been obtained and the minimum value of the mean squared error of the proposed estimator for this optimum value has also been obtained. A comparison of the proposed estimator has been made with mean per unit estimator and other existing estimators of population mean. A numerical study is also carried out to judge the performances of the proposed and existing estimators of the population mean.

Keywords: Study variable, Auxiliary variable, Bias, Mean squared error, Efficiency

Cite this paper: Rajesh Kumar Gupta, S. K. Yadav, Improved Estimation of Population Mean Using Information on Size of the Sample, American Journal of Mathematics and Statistics, Vol. 8 No. 2, 2018, pp. 27-35. doi: 10.5923/j.ajms.20180802.01.

1. Introduction

Estimation of the population parameters is necessary when the size of the population is very large and we wish to get the result in very less time and with minimum cost, labor etc. To estimate any parameter the best estimator is the corresponding statistic. Thus for estimating population mean, sample mean is the most suitable estimator but it has a reasonably large sampling variance. Our aim is to search for the estimator with higher efficiency that is minimum variance or mean squared error. This aim is achieved through the use of auxiliary information supplied by the auxiliary variable. Auxiliary variable is highly positively or negatively correlated with the main variable under study. When main variable under study is positively correlated with the auxiliary variable and the line of regression passes through origin, ratio type estimators are used for improved estimation of population mean. Product type estimators are used when main and auxiliary variables are negatively correlated otherwise regression type estimators are used for the estimation of population mean.
Let the finite population under consideration consist of N distinct and identifiable units and let be a bivariate sample of size n taken from (X, Y) using a SRSWOR scheme. Let and respectively be the population means of the auxiliary and the study variables, and let and be the corresponding sample means. It is well known and has been seen in practice that in simple random sampling scheme, sample means and are unbiased estimators of population means of and respectively. Population mean is one of the very important measures of central tendency in almost all fields of society including field of Medical sciences, Biological sciences, Agriculture, Industry, social sciences, humanities etc. Thus the estimation of population mean is of great significance in above fields. In the present manuscript a modified ratio type estimator of population mean of the study variable using information on size of the sample has been proposed and its large sample properties have been studied up to the first order of approximation.

2. Review of Existing Estimators

The usual and the most suitable estimator for estimating population mean is the corresponding sample mean given by,
(1)
It is unbiased for population mean and its variance up to the first order of approximation is given by,
(2)
Where, .
Cochran (1940) used the positively correlated auxiliary variable with the study variable and proposed the following usual ratio estimator of population mean as,
(3)
The above estimator is a biased estimator of population mean and its bias and mean squared error, up to the first order of approximation respectively are,
(4)
Where, ,
.
In literature various modified estimators of population mean of study variable using auxiliary variables have been given by various authors. For detailed study of the modified ratio type estimators, latest references can be made of Kadilar and Cingi (2004, 2006a, 2006b, 2009), Singh (2003), Singh and Tailor (2003, 2005), Koyuncu and Kadilar (2009), Subramani (2013), Subramani and Kumarapandiyan (2012a,b,c, 2013a,b), Tailor and Sharma (2009), Yan and Tian (2010), Yadav and Pandey (2011), Yadav and Adewara (2013), Yadav et al. (2014, 2015), Yadav et al. (2016a, 2016b, 2016c, 2016d), Abid et al. (2016).
Following table-1 represents some modified estimators their constants, biases and mean squared errors.
Table 1. Various estimators, their constants, biases and mean squared errors
Thus biases and mean squared errors of above estimators may be written as,
(5)

3. Proposed Estimator

Motivated by Jerajuddin and Kishun (2016) estimator of population mean, we propose the following generalized estimator of the population mean using information on size of the sample as,
(6)
where is a suitably chosen constant to be defined such that the mean squared error of the proposed estimator is minimum.
To study the large sample properties of the proposed estimator, we define, and such that for and , , .
Expressing (6) in terms of , we have
(7)
Where
We assume that , so that may be expanded. Now expanding the right-hand side of (7), we have,
(8)
(9)
Retaining the terms up to the first order of approximation, we have
(10)
Subtracting from both sides of (10), we get
(11)
Taking expectations on both sides of (11) and putting the values of different expectations, we get the bias of as
(12)
Squaring both sides of (11) and retaining the terms up to the first order of approximation, we have,
(13)
Taking expectation both sides of (13) and putting the values of different expectations, we get the mean square error of , up to the first order of approximation, as
(14)
which is minimum for,
Thus the minimum MSE of is,
(15)

4. Theoretical Efficiency Comparison

From equation (15) and equation (2), proposed estimator is better than the mean per unit estimator if,
(16)
From equation (15) and equation (2), proposed estimator is better than the usual ratio estimator by Cochran (1940) if,
(17)
From equation (15) and equation (2), proposed estimator is better than the usual ratio estimator by Cochran (1940) if,
or,
(18)

5. Empirical Study

To judge the performances of the proposed estimator and the existing estimators of population mean using auxiliary variable, we have considered four natural populations from two sources. First two populations, population-1 and population-2 are from Murthy (1967) and rest two populations, population-3 and population-4 are from Mukhopadhyay (2009).
Murthy (1967)
Population 1: Y = Output for 80 factories in a region and X= Number of workers
Population 2: Y = Output for 80 factories in a region and X = Fixed Capital
Mukhopadhyay (2009)
Population 3: Y = Output for 40 factories in a region and X = Number of workers
Population 4: Y = Output for 40 factories in a region and X = Fixed Capital
Following Table-2 and Table-3 represents the biases, mean squared errors of proposed and existing estimators of population mean. Table-4 shows the efficiencies of the proposed estimator over other existing estimators of population mean.
Table 2. Biases of the Existing and proposed modified ratio estimators for four natural populations
     
Table 3. Variance/Mean squared errors of the Existing and proposed modified ratio estimators for four natural populations
     
Figure 1
Table 4. Percentage relative efficiency of the proposed estimator over other existing modified ratio estimators compare to estimator
      for four natural populations
     
Figure 2

6. Results and Conclusions

In the present manuscript we have proposed a generalized ratio type estimator of the study variable by making use of information on the size of the sample. The expressions for the bias and mean squared errors of the proposed estimator have been derived up to the first order of approximation. The optimum value of the characterizing scalar which minimizes the mean squared of the proposed estimator has been obtained and the minimum value of the mean squared error for this optimum value of the characterizing scalar has also been obtained. Table-2 and Table-3 represents the biases and the mean squared errors of the proposed and the existing estimators. Table-4 represents the percentage relative efficiencies of the proposed estimator over other estimators. From Table-2 and Table-3 we see that the proposed estimator has minimum biases and mean squared errors in all four natural populations. The relative efficiency of the proposed estimator has been given in Table-4. It can be seen from Table-4, the proposed estimator has least mean squared error for all four populations. Thus it is best for all populations among all other estimator of population mean. Thus the proposed estimator should be used for improved estimation of population mean.

ACKNOWLEDGEMENTS

The authors are very much thankful to the editor and anonymous referee for critically examining the manuscript and giving valuable comments for improvement in the earlier draft.

Appendix

R Code to calculate bias and mse’s.

References

[1]  ABID, M., ABBAS, N. SHERWANI, R.A.K. AND NAZIR, H. Z. (2016). Improved Ratio Estimators for the population mean using non-conventional measure of dispersion. Pakistan Journal of Statistics and Operations Research, XII (2), 353-367.
[2]  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 Agric. Science, 30, 262-275.
[3]  JERAJUDDIN, M., KISHUN, J. (2016). Modified Ratio Estimators for Population Mean Using Size of the Sample, Selected From Population, IJSRSET, 2, 2, 10-16.
[4]  KADILAR, C., CINGI, H. (2004). Ratio estimators in simple random sampling, Applied Mathematics and Computation, 151, 893–902. sss.
[5]  KADILAR, C., CINGI, H. (2006a). An improvement in estimating the population mean by using the correlation co-efficient, Hacettepe Journal of Mathematics and Statistics, 35 (1), 103–109.
[6]  Gupta, R.K. and Misra, S. (2006). Estimation of Population Variance Using Ratio Type Estimator” in Indian Journal of Mathematics and mathematical Sciences. Vol.2, No.2, 169-176.
[7]  Misra, S. and Gupta, R.K. (2008). Almost Unbiased Jacknifed Ratio type estimator of Population variance” in International Journal of Agricultural and Statistical Sciences (IJASS), Vol.4, No.2, 345-350.
[8]  Misra, S., Gupta, R.K. and Shukla, A.K. (2012). Generalized Class of Estimators for Estimation of finite Population Variance” in International Journal of Agricultural and Statistical Sciences, Vol. 8, No. 2, pp. 447-458.
[9]  KADILAR, C., CINGI, H. (2006b). Improvement in estimating the population mean in simple random sampling, Applied Mathematics Letters, 19, 75–79.
[10]  KADILAR, C., CINGI, H. (2009). Advances in sampling theory - Ratio method of estimation, Bentham Science Publishers.
[11]  KOYUNCU, N., KADILAR, C. (2009). Efficient estimators for the population mean, Hacettepe Journal of Mathematics and Statistics, 38(2), 217–225.
[12]  MUKHOPADHYAY, P. (2009). Theory and methods of survey sampling, PHI Learning, 2nd edition, New Delhi.
[13]  MURTHY, M.N. (1967). Sampling Theory and Methods; Statistical Publishing Society, Calcutta.
[14]  SINGH, G.N. (2003). On the improvement of product method of estimation in sample surveys, Journal of Indian Society of Agricultural Statistics, 56 (3), 267–265.
[15]  SINGH, H. P., TAILOR, R. (2003). Use of known correlation coefficient in estimating the finite population means, Statistics in Transition, 6 (4), 555–560.
[16]  SINGH, D., CHAUDHARY, F. S. (1986). Theory and analysis of sample survey designs, New Age International Publisher, New Delhi.
[17]  SINGH, H. P., TAILOR, R. (2003). Use of known correlation coefficient in estimating the finite population means, Statistics in Transition, 6 (4), 555–560.
[18]  SINGH, H. P., TAILOR, R. (2005). Estimation of finite population mean with known co-efficient of variation of an auxiliary variable, Statistica, anno LXV, 3, 301–313.
[19]  SISODIA, B.V.S. and DWIVEDI, V.K. (1981). A Modified Ratio Estimator using Coefficient of Variation of Auxiliary Variable; Jour. of Indian. Soc. of Agri. Stat., Vol. 33(1). Pp. 13–18.
[20]  SUBRAMANI, J. (2013). Generalized modified ratio estimator of finite population mean, Journal of Modern Applied Statistical Methods, 12 (2), 121–155.
[21]  SUBRAMANI, J., KUMARAPANDIYAN, G. (2012a). Estimation of population mean using coefficient of variation and median of an auxiliary variable, International Journal of Probability and Statistics, 1 (4), 111–118.
[22]  SUBRAMANI, J., KUMARAPANDIYAN, G. (2012b). Modified ratio estimators using known median and coefficient of kurtosis, American Journal of Mathematics and Statistics, 2 (4), 95–100.
[23]  SUBRAMANI, J., KUMARAPANDIYAN, G. (2012c). Estimation of population mean using known median and coefficient of skewness, American Journal of Mathematics and Statistics, 2 (5), 101–107.
[24]  SUBRAMANI, J., KUMARAPANDIYAN, G. (2013a). Estimation of population mean using deciles of an auxiliary variable, Statistics in Transition-New Series, 14 (1), 75–88.
[25]  SUBRAMANI, J., KUMARAPANDIYAN, G. (2013b). A new modified ratio estimator of population mean when median of the auxiliary variable is known, Pakistan Journal of Statistics and Operation Research, Vol. 9 (2), 137–145.
[26]  TAILOR, R., SHARMA, B. (2009). A modified ratio-cum-product estimator of finite population mean using known coefficient of variation and coefficient of kurtosis, Statistics in Transition-New Series, 10 (1), 15–24.
[27]  UPADHYAYA, L.N. & SINGH, H.P. (1999). Use of transformed auxiliary variable in estimating the finite population means, Biometrical Journal 41 (5), 627–636.
[28]  YADAV, S.K. AND PANDEY, H. (2011). Improved Exponential Estimators of Population Mean Using Qualitative Auxiliary Information under Two Phase Sampling, Investigations in Mathematical Sciences, Vol. 1, 85-94.
[29]  YADAV, S. K. AND ADEWARA, A.A. (2013). On Improved Estimation of Population Mean using Qualitative Auxiliary Information, Mathematical Theory and Modeling, 3, 11, 42-50.
[30]  YADAV, S.K; MISHRA, S.S. AND SHUKLA, A.K. (2014). Improved Ratio Estimators for Population Mean Based on Median Using Linear Combination of Population Mean and Median of an Auxiliary Variable. American Journal of Operational Research, 4, 2, 21-27.
[31]  YADAV, S.K; MISHRA, S.S. AND SHUKLA, A.K. (2015). Estimation Approach to Ratio of Two Inventory Population Means in Stratified Random Sampling, American Journal of Operational Research, 5, 4, 96-101.
[32]  YADAV, S. K; MISHRA, S. S; SHUKLA, A.K; KUMAR, S. AND SINGH, R. S. (2016a). Use of Non-Conventional Measures of Dispersion for Improved Estimation of Population Mean, American Journal of Operational Research, 6, 3, 69-75.
[33]  YADAV, S. K; GUPTA, SAT; MISHRA, S. S. AND SHUKLA, A. K. (2016b). Modified Ratio and Product Estimators for Estimating Population Mean in Two-Phase Sampling, American Journal of Operational Research, 6, 3, 61-68.
[34]  YADAV, S.K; SUBRAMANI, J; MISHRA, S.S. AND SHUKLA, A .K. (2016c). Improved Ratio-Cum-Product Estimators of Population Mean Using Known Population Parameters of Auxiliary Variables, American Journal of Operational Research, 6, 2, 48-54.
[35]  YADAV, S.K; MISRA, S; MISHRA, S.S. AND CHUTIMAN, N. (2016d). Improved ratio estimators of population mean in Adaptive Cluster Sampling, Journal of Statistics Applications and Probability Letter, 3, 1, 1-6.
[36]  YAN, Z. & TIAN, B. (2010). Ratio Method to the Mean Estimation using Coefficient of Skewness of Auxiliary Variable; ICICA, Part-II, CCIS 106, pp. 103 – 110.
[37]  Gupta, R.K. and Yadav, S.K. (2017). New Efficient Estimators of Population Mean Using Non-Traditional Measures of Dispersion. Open Journal of Statistics, 7, 394-404. https://doi.org/10.4236/ojs.2017.73028.