Keywords

Imputation; Bias; Mean Square Error (MSE); Missing Data; Large Sample Approximation; Simple Random Sampling without Replacement (SRSWOR).

Introduction

The sampling unit refuses to participate in the sample survey, cannot respond, cannot be contacted, or accidentally loses some of the information collected due to unexpected factors, resulting in incomplete survey responses. To deal with missing data effectively Kalton et al. (1981) and Sande (1979) suggested imputation methods that make an incomplete data set structurally complete and its analysis simple. Hyunshik Lee & Särndal (1994); and Lee et al. (1995) used the information on an auxiliary variable for the purpose of imputation. Later Singh and Horn (2000) introduced a compromised method of imputation based on auxiliary variables. Ahmed et al. (2006) discussed several new imputation based estimators that used the information on an auxiliary variate and compared their performance with the mean method of imputation.

Singh and Horn (2000); Wright & Capps (2011), Singh & Gogoi (2017); Singh & Nath (2018b; 2019) and Joyce et al. (2021) discussed designing mixed sampling plan based on IPD and some imputation methods of missing data for estimating the population mean using two-phase sampling scheme.

The objective of the present research work is to provide more efficient alternative estimators than the existing ones, when population parameter of auxiliary information is missing or unknown.

Notations

Let be a finite population of size N and Y is the study variable and X, Z are the auxiliary variable where Y, X and Z are the population mean of the variable Y, X and Z respectively.

Consider a first phase sample of size drawn from the population by using SRSWOR method and a second sample S of size drawn from or S₁

Case-I: when second sample S is drawn from S₁ i.e. second sample S is depends on first sample S₁ (denoted by design I) as in Figure 1.

Case-II: when second sample S is drawn from i.e. second sample S is independent of first sample S₁ (denoted by design II) as in Figure 1.

Let the second sample S contains responding units forming a sub space R and non-responding units with sub space R^c , such that For every unit is observed available. For every unit values are missing and imputed values are observed available. For every unit values are missing and imputed values are computed. The of auxiliary variables are used as a source of imputation for missing data when assuming that in S and S₁ the data are known.

population correlation coefficient between X and Y , Y and Z & Z and X respectively.

the coefficient of variation of X, Y and Z respectively.

Now using the concept of two-phase sampling and denoting E₁and E₂ as the expectation over first phase and second phase respectively we have the following expected values.

Case I: when S is drawn from S₁

Similarly,

Similarly,

Similarly,

Case II: when S is drawn from

Similarly

Similarly,

Similarly,

Some Existing Imputation Techniques

Mean Method of Imputation

Under Mean method of imputation

Using above the point estimator of population mean

The bias and Variance are given by

Ratio Method of Imputation (Hyunshik Lee & Särndal, 1994)

Under Ratio Method of Imputation

Using above the point estimator of population mean Y is

The bias and MSE are given by

Compromised Method of Imputation (Singh & Horn, 2000)

Under this method of imputation

Using the above the point estimator of population mean Y is

Where β is a constant to be determined such that MSE of y_comp is minimum.

The optimum Bias and MSE are-

Exponential Ratio Method of Imputation in two-phase sampling (Pandey et al., 2015)

Under this method of imputation

Using above the point estimator of population mean Y is

Where α is a drawn constant such that MSE of is minimum.

The optimum Bias and MSE are given by

Dual to Ratio Method of Imputation in two-phase sampling (Singh & Nath, 2018a)

Under this method of imputation

Using above the point estimator of population mean is

Where β is a drawn constant such that MSE of is minimum.

The optimum Bias and MSE are given by

Proposed Imputation Strategies

Motivating the above imputation methods of population mean, we have proposed the following Multivariate Ratio type imputation methods of population mean in two-phase sampling.

Imputation Method

The imputation scheme is as follows:

Imputation Method

The imputation scheme is as follows

Imputation Method

The imputation scheme is as follows

Imputation Method

The imputation scheme is as follows

Point estimators for population mean under the proposed four types of imputation methods can easily be deduced. We have the point estimators-

In general the above four imputation method can be defined as

The imputation scheme is a follows

Point estimator for population mean

Where α₁ and α₂ are suitable chosen constants to be determined such that MSE of the point estimator has minimum and

Expanding in terms of retaining the terms upto first order approximate we have

Where,

Properties of Proposed Estimator

The bias, MSE and min MSE of the proposed point estimators have been derived in the following theorems.

Theorem 1

Bias of the estimators under design I and design II upto first order of approximation are as:

Proof: Taking expectation on both sides of equation (1) we have

Putting the expected values under design I we have

Putting the expected values under design II we have

Theorem 2

MSE of the estimators under design I and design II upto first order approximation are as-

Proof: Taking expectation after squaring the both sides of (3) we have

Putting the expected values under design I we have

Putting the expected values under design II we have

The optimum value of is obtained by minimizing given in equation (6) and (7) by using the method of maxima and minima we have-

Putting the optimum values of under the design I and design II in equation (2) and (3) and solving for we have

Putting the optimum values of under design I and II in equation (4) & (5) we have

Theorem 3

The estimator is unbiased for optimum values of under design I and II.

Proof: Putting the optimum values of under design I in equation (6) we have

Similarly put ting the optimum values of under design II in equation (7) we have

Comparison

In this section we divide the conditions under which the suggested estimator is superior to the existing estimators in design I and design II. To compare the different estimators we use the following theorem of multiple correlation coefficients.

Comparison with Mean Method of Imputation

is always efficient than in design I and design II.

Comparison with Ratio Method of Imputation

is always efficient than in design I and design II.

Comparison with Compromised Method of Imputation

Comparison with Exponential Ratio Method of Imputation

Comparison with dual to Ratio Method of Imputation

is always efficient than in design I and design II.

Empirical Study

To examine the performance of the proposed estimator of the population mean in twophase sampling, we have considered the following three populations (Tables 1-10).

Table 1 Mse of The Different Estimators Under Design I
Point Estimator	Population I	Population II	Population III
	2.026693	2559.906609	10.369737
	1.875253	1631.420886	.955755
	1.757726	1630.937426	8.752709
	1.324985	1084.832174	7.845193
	1.152552	448.613009	6.731424
	1.090778	446.556316	6.273775

 

Table 2 Mse of The Different Estimators Under Design II
Point Estimator	Population I	Population II	Population III
	2.026693	2559.906609	10.369737
	1.875253	1631.420886	8.955755
	1.757726	1630.937426	8.752709
	1.117529	828.239053	6.4115601
	1.117496	405.529871	6.356754
	1.053244	403.431209	5.851978

 

Table 3 Pre of The Different Estimators With Respect To  Under Design I
Point Estimator	Population I	Population II	Population III
	100.000	100.000	100.000
	106.686	100.029	102.319
	141.530	150.385	114.156
	162.704	363.659	133.044
	171.918	365.334	142.749

 

Table 4 Pre of The Different Estimators With Respect To  Under Design Ii
Point Estimator	Population I	Population II	Population III
	100.000	100.000	100.000
	106.686	100.029	102.319
	167.803	196.975	139.681
	167.808	402.294	140.886
	178.045	404.386	153.038

 

Table 5 Pre of The Different Estimators With Respect To  Under Design I
Point Estimator	Population I	Population II	Population III
	100.000	100.000	100.000
	132.660	150.340	111.568
	152.507	363.551	130.027
	161.144	365.225	139.513

 

Table 6 Pre of The Different Estimators With Respect To  Under Design Ii
Point Estimator	Population I	Population II	Population III
	100.000	100.000	100.000
	157.287	196.916	136.514
	157.292	402.174	137.691
	166.887	404.267	149.568

 

Table 7 Pre of The Different Estimators With Respect To  Under Design I
Point Estimator	Population I	Population II	Population III
	100.000	100.000	100.000
	114.961	241.819	116.546
	121.472	242.933	125.047

 

Table 8 Pre of The Different Estimators With Respect To  Under Design Ii
Point Estimator	Population I	Population II	Population III
	100.000	100.000	100.000
	100.003	204.236	100.086
	106.103	205.299	109.562

 

Table 9 Pre of The Different Estimators With Respect To  Under Design I
Point Estimator	Population I	Population II	Population III
	100.000	100.000	100.000
	105.666	100.460	107.295

 

Table 10 Pre of The Different Estimators With Respect To  Under Design Ii
Point Estimator	Population I	Population II	Population III
	100.000	100.000	100.000
	106.100	100.495	108.626

Population I (Cochran, 1977)

Y : Number of placebo children

X : Number of paralytic polio cases in the placebo group

Z :Number of paralytic polio cases in the ‘not inoculated group

Population II (Murthy, 1967)

Y : Area under wheat in 1964

X : Area under wheat in 1963

Z :Cultivated area in 1961

Population III (Anderson, 2003)

Y : Head length of second son

X : Head length of first son

Z :Head breathe of first son

Conclusion

From the above tables, it is obvious that the suggested have smaller Mean Square Error (MSE) than the MSE’s of the other existing estimators both theoretically as well as empirically under dependent and independent cases. Also the Bias of the proposed estimator vanishes at the optimum values of α^₁ and α^₂ . . Therefore it is concluded that the proposed estimator is preferable to use over other existing estimator.

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Received: 10-Feb-2022, Manuscript No. JMIDS-22-11218; Editor assigned: 15-Feb-2022, PreQC No. JMIDS-22-11218(PQ); Reviewed: 07-Mar-2022, QC No. JMIDS-22-11218; Revised: 29-Mar-2022, Manuscript No. JMIDS-22-11218 (R); Published: 05-Apr-2022