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Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme



Krishnajyothi Nath and B.K. Singh
 
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ABSTRACT

Background and Objective: In literature there has been a study on ratio cum product estimator of a finite population mean in two-phase sampling in sample surveys, but it lacks study when there is non-response on sample observations. So the main objective of this paper was to propose three generalized classes of ratio cum product compromised imputation techniques in presence of missing values in two-phase sampling design and its properties have been studied. Materials and Methods: The estimators were compared with other existing estimators in two different designs. The bias and M.S.E. of suggested estimators were derived in the form of population parameters using the concept of large sample approximations. Results: The results showed the superiority of the proposed estimators over the existing methods. Numerical studies are performed over two population data sets using the expressions of bias and M.S.E. and their efficiencies are compared with other existing estimators. Conclusion: It was observed that the proposed estimators were performing better than the estimators taken for comparisons in the presence of missing data.

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  How to cite this article:

Krishnajyothi Nath and B.K. Singh, 2018. Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme. Asian Journal of Mathematics & Statistics, 11: 27-39.

DOI: 10.3923/ajms.2018.27.39

URL: https://scialert.net/abstract/?doi=ajms.2018.27.39
 
Received: July 11, 2018; Accepted: November 06, 2018; Published: January 05, 2019


Copyright: © 2018. This is an open access article distributed under the terms of the creative commons attribution License, which permits unrestricted use, distribution and reproduction in any medium, provided the original author and source are credited.

INTRODUCTION

Missing data is a problem encountered in almost every data collection activity but particularly in sample survey. To overcome the problem of missing observations or non-response in sample surveys, the technique of imputation is frequently used to replace the missing data. In literature, several imputation techniques are described, some of them are better over others. To deal with missing values effectively Kalton et al.1 and Sande2 suggested imputation that make an incomplete data set structurally complete and its analysis simple. Lee et al.3,4 used the information on an auxiliary variate if it is available. Later Singh and Horn5 suggested a compromised method of imputation. Shukla6, Singh and Deo7, Ahmed et al.8, Arnab and Singh9, Rueda and Gonzalez10, Gonzalez et al.11, Bouza12, Kadilar and Cingi13, Shukla and Thakur14, Shukla et al.15,16, Baraldi and Enders17, Diana and Perri18, Singh et al.19, Shukla et al.20, Thakur et al.21, Shukla et al.22, Thakur et al.23, Singh et al.24,25, Thakur et al.26, Singh and Gogoi27, Nath and Singh28 and Singh and Espejo29 suggested several new imputation based estimators that use information on an auxiliary variate under single and double sampling scheme. The objective of the present research work was to derive some imputation methods for mean estimation in case population parameter of auxiliary information is missing or unknown.

NOTATIONS

Let W = {1, 2,..., N} be a finite population with Yi as a variable of main interest and Xi (i = 1, 2,..., N) an auxiliary variable. As usual:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

are population means,Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme is unknown and Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme under investigation.

Consider a preliminary large sample S’ of size n’ is drawn from population Ω by Simple Random Sampling Without Replacement (SRSWOR) and a secondary sample S of size n(n<n’) is drawn in either of the following manners:

Case-I: as a sub-sample from sample S’ (denoted by design I) as in Fig. 1a
Case-II: Independent to sample S’ (denoted by design II) as in Fig. 1b, without replacing S’

Let sample size S of n units contains r responding units (r<n) forming a sub-space R and (n-r) non-responding with sub-space RC in S = R∪RC. For every i∈R, yi is observed available. For i∈RC, the yi values are missing and imputed values are computed. The ith value xi of auxiliary variate is used as a source of imputation for missing data when i∈RC. Assume for S, the data xs = {xi: i∈S} and for I’∈S’, the data {xi: I’∈S’} are known with mean:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

respectively. The following symbols are used hereafter:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme, Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme = The population mean of X and Y
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme = The sample mean of X and Y
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme = Sample mean of X and Y for corresponding responding units
ρxy = The correlation co-efficient between X and Y
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme = The population mean squares of X and Y
Cx, Cy = The co-efficient of variation of X and Y

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

LARGE SAMPLE APPROXIMATION

Let, Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme and Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme which implies the results Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme and Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme Now by using the concept of two-phase sampling, following Rao and Sitter30 and the mechanism of missing completely at random (MCAR), for given r, n and n’, we have:

Under design F1 [Case I]: Under design F1 is shown in

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Fig. 1(a-b): (a) Design I, F1 and (b) Design II, F2

Under design F2 [Case II]: Under design F2 is shown in Fig.1b:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

SOME EXISTING IMPUTATION TECHNIQUES

Let, Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme be the mean of the finite population under consideration. A simple random sampling without replacement (SRSWOR) S of size n is drawn from Ω = {1, 2,..., N} to estimate the population mean Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme. Let the number of responding units out of sampled n units be denoted by r(r<n), the set of responding units by R and that of non-responding units by RC. For every unit i∈R the value yi is observed, but for the units i∈RC, the observations yi are missing and instead imputed values are derived. The ith value xi of auxiliary variate is used as a source of imputation for missing data when i∈RC. Assume for S, the data xs = {xi:i∈S} are known with mean Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme Under this setup, some well known imputation methods are given below:

Mean methods of imputation: The mean imputation method is to replace each missing datum with the mean of the observed value. The data after imputation becomes:

For yi define yoi as:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Using above, the imputation-based estimators of population mean Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme is Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

The bias and mean square error is given by:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Ratio method of imputation: Following the notations of Lee et al.3 in the case of single imputation method, if the ith unit requires imputation, the value is imputed.

For yi and xi, define yoi as:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Using above, the imputation-based estimator is:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Where:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

The bias and mean square error of Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme is given by:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Where:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Compromised method of imputation: Singh and Horn5 suggested a compromised method of imputation. It based on using information from imputed values for the responding units in addition to non-responding units. In case of compromised imputation procedures, the data take the form:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

where, α is a suitably chosen constant, such that the resultant variance of the estimator is optimum. The imputation-based estimator, for this case, is:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

The bias, mean square error and minimum mean square error at Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme of are given by:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Ahmed methods: For the case where, yji denotes the ith available observation for the jth imputation method, the three imputation methods y1i, y2i and y3i are given as follows:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

where, β1 is a suitably chosen constant, such that the variance of the resultant estimator is minimum. Under this method, point estimator of y1i is:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

The bias, mean square error and minimum mean square error at Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme of t1 are given by:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

where, β2 is a suitably chosen constant, such that the variance of the resultant estimator is minimum. Under this method, the point estimator of y2i is:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

The bias, mean square error and minimum mean square error at Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme of t2 are given by:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

where, β3 is a suitably chosen constant, such that the variance of the resultant estimator is minimum. Under this method, the point estimator of y3i is:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

The bias, mean square error and minimum mean square error at Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme of t3 are given by:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Singh and Gogoi method of imputation: For the case where, yji denotes the ith available observation for the jth imputation method, the imputation method y1i, y2i, y3i is given as follows:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

The point estimator of the population meanImage for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Schemeunder proposed method of imputation is:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

The bias, mean square error and minimum mean square error of Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme respectively are given by:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

The point estimator of the population meanImage for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Schemeunder proposed method of imputation is:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

The bias, mean square error and minimum mean square error of Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme respectively are given by:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

The point estimator of the population meanImage for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Schemeunder proposed method of imputation is:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

The bias, mean square error and minimum mean square error of Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme respectively are given by:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

MATERIALS AND METHODS

For estimating the population mean of the study variate y, Singh and Espejo29 considered an estimator of the ratio-product type in two phase sampling given by:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

where, Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme and Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme are the sample means of y and x, respectively based on a sample size n out of the population of N units. Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme‘ is the sample mean of sample size n’ and k is suitably chosen constant.

Motivated by Singh and Espejo29 ratio-product estimator of a finite population mean and for the case where yji denotes the ith available observation for the jth imputation method, we here proposed the following three ratio cum product type methods of imputation in two phase sampling:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

where, α is a suitably chosen constant, such that the resultant variance of the estimator is minimum.

Under this strategy, the point estimator of Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme is:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
(1)

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

where, α is a suitably chosen constant, such that the resultant variance of the estimator is minimum.

Under this strategy, the point estimator of Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme is:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
(2)

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

where, α is a suitably chosen constant, such that the resultant variance of the estimator is minimum.

Under this strategy, the point estimator of Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme is:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
(3)

PROPERTIES OF PROPOSED ESTIMATORS

Let B(.)t and M(.)t denote the bias and Mean Square Error (MSE) of an estimator under the given sampling design t = I, II. The properties of Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme and Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme are derived in the following theorems respectively.

Theorem 1: Estimators Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme and Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme in terms of ei; I = 1, 2, 3 and can be expressed upto first order of approximation as:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

by ignoring the termsImage for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme for r+s>2, where, r, s = 0.1, 2,... and i = 1, 2, 3; j = 2, 3 which is first approximation.

Proof:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Theorem 2: Biases Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme and Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme of and under design I and II, upto first order of approximation are:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
(4)

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
(5)

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
(6)

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
(7)

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
(8)

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
(9)

Proof:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Theorem 3: Mean squared errors of Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme and Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme under the design I and II, upto first order of approximation can be written as:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
(10)

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
(11)

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
(12)

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
(13)

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
(14)

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
(15)

Proof:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Theorem 4: Minimum mean squared error of Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme and Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme under the design I and II are given as:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Proof:

Differentiating (Eq. 10) with respect to α and equating it to zero, we get:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Putting the value of α in Eq. 10, we obtain:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Similarly proceeding for Eq. 11-15, we have:
 
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

COMPARISON

It derived the conditions under which the suggested estimators are superior to Ahmed et al.8 estimators in design I and II:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme is better than t1 when:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme is better than t1 when:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Where:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

which is always true.

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

which is always true.

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme is better than t3 when:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme is better than t3 when:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Where:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

NUMERICAL ILLUSTRATIONS

It consider two populations A and B, first one is artificial population of size15,16 N = 200 and another one is from Ahmed et al.8 with the following parameters as given in Table 1.

Let there be n’ = 60, n = 40, r = 5 for population A and n’ = 2000, n = 500, r = 450 for population B, then the bias and the MSE of the suggested estimators under design I and II and that of Ahmed et al.8 methods for population A and B respectively are given in Table 2-4. The sampling efficiency of suggested estimators under design I and II over Ahmed et al.8 is defined as:

Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

RESULTS

The present work proposed some imputed estimators for estimating population mean in two phase sampling under two different designs. The study of the estimators had been made taking two different population data sets (Table 1) for their efficiencies. The results of the proposed estimators certainly showed the superiority of it over the traditional estimators viz., the mean method of imputation, ratio method of imputation, compromised method of imputation and feel better or little bit inferior depending on sampling design and population data sets over Singh and Gogoi and three Ahmed methods of imputation (Table 2-4) as far as MSEs are concerned.

Table 1: Parameters of population A and B
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Table 2: Bias and MSE of suggested estimators for population A
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Table 3: Bias and MSE of suggested estimators for population B
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme

Table 4: Bias and MSE for population A and B for estimators m, RAT, COMP
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Source: Ahmed et al.8 and Singh and Gogoi27

Table 5: Efficiency of suggested estimators for population A and B
Image for - Population Mean Estimation Using Ratio-cum Product Compromised-method of Imputation in Two-phase Sampling Scheme
Source: Ahmed et al.8 and Singh and Gogoi27

The Table 5 gives the relative efficiency of the proposed estimators. All the three proposed estimators have been found to be better in their performances over the estimators proposed by Singh and Gogoi and Ahmed’s estimators in design I of population B. The second proposed estimator is showing better efficiency in both the designs of the population A and population B and other proposed estimators are either little bit inferior or having close proximity with estimators under comparisons.

DISCUSSION

The ratio method of estimation (or the product method of estimation31,32, yields a more efficient estimator than the simple unbiased estimator provided the correlation coefficient between study variate y and auxiliary variate x has high positive value (or high negative value). Further, the ratio estimator is most effective and is as efficient as the regression estimator, when the relationship between the study variate, y and the auxiliary variate, x, is linear through the origin and the variate of y is proportional to x. However, in many practical situations, the line does not pass through the vicinity of the origin. Keeping this deed in view, an attempt has been made to improve these estimators in compromised imputation taking three sets of linear combination of ratio and product estimators when there is non-response either on study variable or auxiliary variable or both in two-phase sampling. Kalton et al.1 and Sande2 were the first to discussed imputation technique that make an incomplete data set structurally complete and its analysis simple. Lee et al.3,4 used the information on an auxiliary variate for the purpose of imputation. Later Singh and Horn 5 suggested a compromised method of imputation. Ahmed et al.8 suggested several new imputation based estimators that use the information on an auxiliary variate.

CONCLUSION

The present work proposed some imputed estimators for estimating population mean in two phase sampling scheme in sample survey. The results demonstrated the superiority of these proposed estimators over the traditional estimators’ viz., the mean method of imputation, ratio method of imputation, compromised method of imputation and the second proposed estimator is showing better efficiency in both the designs of the population A and population B.

SIGNIFICANCE STATEMENT

This study discovers that the estimators under considerations are better in comparison to traditional estimators viz., the mean method of imputation, ratio method of imputation, compromised method of imputation and better than Singh and Gogoi and three Ahmed methods of imputation depending on sampling design in estimating the population mean. This study will help the researcher to uncover the area of formulation of imputation methods experimenting different sampling designs.

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