Estimation of general parameters using auxiliary information in simple random sampling without replacement
⁎Corresponding author. irfii.st@amu.ac.in (Irfan Ali)
-
Received: ,
Accepted: ,
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.
Abstract
Estimation of population parameters plays a vital role in the area of sampling. Many authors have proposed several estimators for estimating population parameter(s) using auxiliary information. This paper has attempted to suggest a new estimator for estimating the general parameter
Keywords
Auxiliary variable
Conventional estimator
MSE
Efficiency
Simulation study
1 Introduction
Auxiliary information, when suitably used, improves the efficiency of the population parameters estimators. There are various techniques for applying auxiliary information, which helps improve the estimator’s performance. These techniques include product methods, ratio, and regression, among others. Auxiliary information can be available in various forms, such as in the form of variables or attributes. Many researchers have widely used it to formulate various estimators for estimating population parameters in different sampling schemes. Srivastava (1971), Kadilar and Cingi (2004) and Gupta and Shabbir (2008) have proposed estimators for estimating unknown population values. Another estimator’s class that uses correlation coefficient value and minimum MSE have been used in population mean estimation as documented in Srivastava and Jhajj (1980), Srivastava and Jhajj (1981), Srivastava and Jhajj (1986). Shabbir and Gupta (2007) proposed an estimator using the information in the form of attributes. Tracy et al. (1996) and Gupta and Shabbir (2007) used information on two auxiliary variables and proposed estimators to estimate the population constants. Some other related work can be found in Singh and Singh (2001), Tripathi et al. (2002), Khoshnevisan et al. (2007), Singh et al. (2008a), Singh et al. (2008b), Singh and Kumar (2011), Singh and Solanki (2011), Singh and Malik (2014), Sharma et al. (2017), Adichwal et al. (2016), Adichwal et al. (2019), Adichwal et al. (2017), Singh et al. (2018) and Mishra and Singh (2017).
Motivated by the work of these authors, we have proposed an improved estimator in the SRSWOR scheme for the general parameter of the population.
2 Notations
Let us consider a sample of size n is drawn from a population
In general, consider the following population parameters
Note that
Let us define,
Subject to the condition
Ignoring fpc, we have
3 Conventional estimator
The general form of the parameter under consideration can be stated as
In Eq. (3.1), a and b are scalars, suitably chosen and
For choices of a, b the general parameter
-
If
, reduces to -
If
, reduces to -
If
, reduces to -
If
, reduces to
The general parameter
Expressing Eq. (3.2) in terms of
Eq. (3.3) can be written as
or equivalently,
Squaring Eq. (3.4) and neglecting the terms of
Take the expectations on both sides of Eq. (3.5), we have MSE’s estimators of
4 Proposed estimator
We propose a class of difference-cum exponential ratio type estimators to estimate the parameter of the population
We express Eq. (4.1) in terms of
Multiplying out and neglecting the higher-order terms of
or
Squaring Eq. (4.4) both sides and neglect the higher order of
or
From Eq. (4.5), we have
Take the expectation of Eq. (4.7), we have
Differentiate partially Eq. (4.9) w.r.to
After simplifying, we obtain the optimum value of
Substituting
Remark 4.1: The optimum values of constants A1 and A2 at (4.10) involve unknown population parameters. The values of these quantities can be guessed accurately through a pilot sample survey or sample data at hand or experience gathered in due course of time, see Srivastava and Jhajj (1980), Singh et al. (2003) and Singh and Solanki (2013).
When
MSE of the estimator
Table 4.1 presents the existing estimators obtained from Eq. (4.14) on taking appropriate values of a, b, k,
Subset of the proposed estimator | a | b | K |
|
|
---|---|---|---|---|---|
|
a | b | 0 | 1 |
|
|
1 | 0 | 0 | 1 |
|
|
1 | 0 | 0 | 1 | 2 |
|
1 | 0 | 0 | 0 |
|
Using known
Table 4.2 presents a set of existing estimators obtained from (4.17) by suitable a, b,
Subset of the proposed estimator | A | b |
|
|
---|---|---|---|---|
|
a | b | 1 |
|
|
0 | 2 | 1 |
|
|
0 | 2 | 1 | 2 |
5 Efficiency comparison
We note that
It can be observed from (5.1), (5.2) and (5.3), the proposed difference-cum exponential ratio type conventional estimator t performs efficiently than the estimators
6 Estimation of the population mean of the study variable Y
For
MSE’s expressions of the estimator
MSE of
Putting the value of
Subset of the proposed estimator | k |
|
w2 |
|
|
---|---|---|---|---|---|
|
0 | 1 | 0 |
|
|
|
0 | 1 | 0 | 2 | |
|
0 | 0 | 0 |
|
The minimum MSE’s of the estimator
7 Efficiency comparison
8 Empirical study
In this section, we compare the performance of the proposed estimators using a known population data set.
The description of population data sets is as follows.
Population
Y = Number of person per block
X = Number of rooms per block
The values of the parameters are
n = 10,
Tables 8.1 is concluded that the performance of the proposed estimator
Estimators | PRE |
---|---|
|
100.0000 |
|
195.1564 |
|
173.1602 |
|
72.5100 |
|
173.1602 |
9 Simulation study
This section shows the procedure for comparing estimators
Step 1: Simulate
Step 2: Let
Step 3: Gives the pair (Y, X).
Step-4: Let define the parameters
Step-5: Similarly, use parameters
Step-6: Use SRS to draw 500 samples
Step-7: Calculate
and PRE of an estimator t with respect to the usual estimator
Tables 9.1 and 9.2 shows simulation results that have been obtained following the stepwise procedures.
|
N | Average PRE’s | ||||
---|---|---|---|---|---|---|
|
|
|
|
|
||
0.5 | 40 | 100.0000 | 140.2242 | 138.7793 | 80.36167 | 138.7793 |
50 | 100.0000 | 139.6806 | 138.6224 | 80.1948 | 138.6224 | |
60 | 100.0000 | 139.5516 | 138.5826 | 80.29106 | 138.5826 | |
0.6 | 40 | 100.0000 | 165.8203 | 164.1184 | 72.5535 | 164.1184 |
50 | 100.0000 | 165.1557 | 163.9098 | 72.47781 | 163.9098 | |
60 | 100.0000 | 164.9832 | 163.841 | 72.51843 | 163.841 | |
0.8 | 40 | 100.0000 | 299.7129 | 296.6709 | 60.68814 | 296.6709 |
50 | 100.0000 | 298.4635 | 296.2357 | 60.68819 | 296.2357 | |
60 | 100.0000 | 298.0521 | 296.0032 | 60.66814 | 296.0032 |
|
N | Average PRE’s | ||||
|
|
|
|
|
||
0.5 | 40 | 100.0000 | 119.2466 | 118.0128 | 97.4543 | 118.0128 |
50 | 100.0000 | 118.8099 | 117.9053 | 96.43723 | 117.9053 | |
60 | 100.0000 | 118.7146 | 117.8876 | 96.7558 | 117.8876 | |
0.6 | 40 | 100.0000 | 130.8356 | 129.4851 | 87.2013 | 129.4851 |
50 | 100.0000 | 130.3386 | 129.3491 | 86.7238 | 129.3491 | |
60 | 100.0000 | 130.2249 | 129.3196 | 86.9226 | 129.3196 | |
0.8 | 40 | 100.0000 | 191.0631 | 189.1082 | 68.4768 | 189.1082 |
50 | 100.0000 | 190.2839 | 188.8529 | 68.3965 | 188.8529 | |
60 | 100.0000 | 190.0674 | 188.7543 | 68.4277 | 188.7543 |
From Tables 9.1 and 9.2, we observe that from the various values of correlation coefficient
10 Conclusion
The present study proposed an improved estimator for a general parameter estimation using the information on “additional value X”. Differential cum exponential ratio type estimator ‘t’ of a general parameter is suggested in SRSWOR. The estimator ‘t’ proposed in Eq. (4.1) can be used to estimate different population parameters. We have verified the performance of the proposed estimator using simulated data set for the case of estimation of population mean only. The result of this study shows that the usual mean estimator, Singh and Pal (2017), Upadhyaya et al. (2011), Bahl and Tuteja (1991), Yadav and Kadilar (2013) and Singh et al. (2011) can be shown as the members of the proposed class. An empirical study has been carried out using real data set (presented in Table 8.1) and simulated data sets (presented in Table 9.1 and Table 9.2 by taking two different simulated population data sets) to illustrate the efficiency and effectiveness of the proposed estimator. The result of simulation studies shows that the average PRE of the suggested estimator
Acknowledgement
The authors wish to acknowledge the editor in chief and the anonymous reviewers for their constructive comments, which improved the quality of the work.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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