Estimation of mean with imputation of missing data using factor-type estimator under adaptive cluster sampling in sample surveys

Introduction

Adaptive cluster sampling is a sampling approach where all the units that satisfy a given criterion are included in the sample coupled with a selected sample unit. Yadav et al. (2016) [20] provides an explanation of adaptive cluster sampling (ACS). For example in epidemic time, all those units were included in the sample, which are in relation with the Corona patient for last 14 days. In the event that a Corona patient is discovered during testing, everyone who has had contact with the patient during the previous 14 days is also examined. Let 𝛺 = {1, 2,3, … , 𝑁} be a population and the ordered pair (𝑌 𝑖 , 𝑋 𝑖 ); 𝑖 = 1,2,3, … , 𝑁 be the observations of 𝑖 𝑡ℎ individual. Assume the population mean 𝑌 ̅ = is the population mean of auxiliary variable and known in advance by past data or previous experience. Let a preliminary sample of size 𝑛 is drawn by SRSWOR from 𝛺. Also, let 𝐴 𝑖 ; 𝑖 = 1,2,3, … … , 𝑛 be the network of size 𝑚 𝑖 of unit i, that is selected in the preliminary sample. Let the means of 𝑌 and 𝑋 in the 𝑖 𝑡ℎ respectively. Also,

are the unbiased estimator of

The unbiased estimate of population mean is 𝑌 ̅ 𝑎𝑐 = 𝜔 ̅ 𝑦 = (1.1)

Review of Literature

The authors Pochai (2008) [8] , Saheli (2003) [10] , Turk and Borkowski (2005) [19] , Singh and Yadav (2019) [14] , Yadav et al. (2016) [20] , Thompson (1990) [18] and others have all done https://www.mathsjournal.com work based on "Adaptive cluster sampling to estimate the population mean". "Auxiliary variable-based estimators of population mean under adaptive cluster sampling" have been proposed by Chao (2004) [2] , Dryver and Chao (2007) [6] , Chutiman and Kumphon (2008) [5] , Chutiman (2013) [4] and Chaudhury and Hanif (2015) [3] . All of the techniques described in the literature rely on data from the sample, but they are ineffective if certain observations of the primary study variable are unintentionally or for other reasons absent. In this publication, a methodology for estimating missing data and population mean estimates was established.

In this earlier discussion of the concept of missing observation, Rubin (1976) [9] presented three examples of missing data patterns. The first is the missing at random (MR), the second is the observed at random (OR) and the third is parametric distribution (PD).

The missing completely at random (MCR) distribution pattern, first described by Heitjan and Basu (1996) [7] is examined in this work. Thakur et al. (2011) [15] used two-phase sampling to produce missing data. The theory of modern missing data analyses is proposed by Baraldi and Enders (2010) [1] . Further recommendations for "Missing data in stratified random sampling" were made by Thakur, Yadav, and Pathak (2014) [17] .

Proposed Imputation Methods

Shukla and Thakur (2008) [11] used an F-T (factor-type) estimator as a tool of imputation and then derived an estimator for population mean under SRSWOR. Shukla, et al. ( 2009) [12] -discussed factor-type estimator in two-phase sampling under imputation of missing data for estimating population mean. Some other contributions of "Factor-type estimators to estimate the population mean" are due to Thakur et al. ( 2016) [16] , Thakur and Shukla (2022) [13] etc. he following are some suggested imputation techniques for missing data under adaptive cluster sampling based on the inspiration of these papers:

According to sections 3.1, 3.2 and 3.3 the point estimators for Y̅ are:

Properties of Proposed Imputation Methods in Adaptive Cluster Sampling

for the specified 𝑤 ̅ 𝑦𝑟 , 𝑤 ̅ 𝑥𝑟 and 𝑤 ̅ 𝑥𝑛 respectively. Then, The estimator 𝑇 𝐹𝑇𝐶1 with ε, δ and 𝜂 up to first order of approximation, could be written as:

Use of large sample approximation with section 4.0

[𝟏. 𝟐]: Bias of 𝑇 𝐹𝑇𝐶1 up to first order approximation is:

[𝟏. 𝟑]: Mean squared error of 𝑇 𝐹𝑇𝐶1 up to first order is:

[𝟏. 𝟒]: The minimum m.s.e. of (T FTC1 ) when 𝜽= -V and expression is:

Proof: Minimum m.s.e. occurs when

The minimum m.s.e. of (𝑇 𝐹𝑇𝐶2 ) at 𝜽 = -V and expression is:

Proof: Minimum m.s.e., when

[𝟏. 𝟏]:

The estimator 𝑇 𝐹𝑇𝐶3 with ε, δ and 𝜂 up to first order of approximation, could be written as:

[𝟏. 𝟐]: Bias of 𝑇 𝐹𝑇𝐶3 up to first order approximation is:

[𝟏. 𝟑]: Mean squared error of 𝑇 𝐹𝑇𝐶3 up to first order is:

[𝟏. 𝟒]: The minimum m.s.e. of (𝑇 𝐹𝑇𝐶3 )when 𝜽 = -V and expression is:

Proof: Minimum m.s.e. occurs when

Values of k with optimality condition

The optimality condition 𝜃 = -𝑉 provides the equation

This is a cubic equation in terms of k, and for which M.S.E. is optimal, we obtain a maximum of three k values, such as 𝑘 1 , 𝑘 2 and 𝑘 3 . The best criterion for determining k's value is:

Step 1: For i, j = 1, 2, 3, compute |𝐵(𝑇 𝐹𝑇𝐶 ) 𝑘 𝑗 |

Step 2: For given values of i, choose 𝑘 𝑗 as

In the end, this provides bias control at the ideal m.s.e. Since 𝐷 3 > 0, therefore 𝑦 ̅ 𝐹𝑇𝑐3 is always better than 𝑦 ̅ 𝐹𝑇𝑐2 .

Comparison

Empirical Study

An artificial population in Table 7.1(a) and 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 24 5 0 0 0 0 0 0 0 0 0 0 0 0 0 1 22 5 4 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 4 8 0 0 0 0 0 33 0 0 0 27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 6 7 1 0 5 0 0 0 0 0 0 0 0 0 21 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 5 7 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 5 7 0 7 7 6 3 0 0 0 0 0 0 0 0 0 5 4 3 0 5 8 4 5 1 0 5 0 0 0 0 0 0 0 7 65 0 4 5 0 9 0 0 0 0 0 3 1 0 0 0 0 0 0 1 4 5 0 7 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 1 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 27 0 0 21 0 0 0 0 0 0 0 0 0 0 0 0 0 29 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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