Improved estimator of population mean under compromise method of imputation

Introduction

Nonresponses or missing values are common occurrences in sample surveys. Item nonresponse and unit nonresponse are the two types of nonresponse. Item nonresponse is deal with the imputation method, while unit nonresponse is deal with the weight method. Similarly, the most common method for solving missing values is by imputation, which replaces missing values using the existing data as a source. Additionally, many researchers have studied the auxiliary information available in order to increase the accuracy of population mean estimation simple random sampling without replacement (SRSWOR). For example, Cochran applied the auxiliary information at the estimation stage and proposed an estimator to estimate the population mean. Bahl and Tuteja (1991) [2] first proposed new ratio-type exponential method for estimating the mean of population using information on auxiliary variable and they are more efficient than the existing estimators. Let the mean of the population

A sample s of size n is drawn from the population by simple random sampling without replacement (SRSWOR) to estimate population mean (𝑌 ̅ ). Let the number of responding units be r out of sampled units. The set of responding units be R and the remaining non responding units be R c . For every units i ∈ R, the value of yi is observed and the remaining units i ∈ R c are missed and have to be imputed using the imputation techniques. Here assume that the imputation has carried using the available auxillary information x such that xi, the values x for unit i. is known and it is positive for all i ∈ s. The following notations used are are given by Lee et al. (1994) for the case of single value imputation. If the i th unit requires the imputation, the value of 𝑏 ̂𝑥𝑖 is imputed where

. The data after the imputation becomes

The above imputation method used is known as Ratio method of imputation. The point estimator of population mean under this method of imputation is given by

Another method of imputation is mean method of imputation. In this method the data after imputation becomes

The point estimator for the above method of imputation is given by

Singh and Horn (2000) have another imputation procedure called compromise imputation which is giving more efficient result than the other existing imputation procedures. The data after imputation becomes

The point estimator of the population mean is given by

The study explores the derivation and compotation procedure of an alternate estimator of population mean using estimator given by Khoshnevisan (2007) [5] under compromised imputation. The article is orientated so that it first discusses the estimators' theoretical foundation. In the following section, the proposed estimator and its characteristics are described. The subsequent part demonstrated how the appropriate estimator performed in comparison to the current estimators using a case study. It ends later with some general remarks.

Theoretical background

Rao and Sitter (1995) gave the concept of two-phase sampling and the mechanism of that for given r and n we have

The properties (Bias and MSE) of the above discussed imputation methods are as follows,

The bias and MSE of Ratio method of imputation is,

The bias and MSE of the compromised method of imputation is,

(suitable constant to reduce the variance)

Proposed imputation method for missing data By using the estimator given by Khoshnevisan (2007) [5] , here we proposed a new estimator for missing data. The data after imputation takes as, The bias of the proposed point estimator is given as

Theorem 3.3

The mean squared error (MSE) of the proposed estimator 𝑦 ̅ 𝑝 is given by

Case Study

In this section, the proposed estimator is compared with existing estimators. For this study the two population data was used which are given by Murthy (1967), the population constants are as follows. The above table 2 is explaining about the bias of the proposed estimator and for the previously existing estimators (Ratio estimator and compromise imputation estimator) for both the populations. According to the above population it clearly tells us that for both populations the proposed estimator is showing less bias compared to the other estimators. Hence the proposed estimator is giving better results than other existing estimators. The table 3 showing the MSE of Proposed estimator and the existing estimators (Ratio estimator and compromise imputation estimator). Form the above table for both the populations MSE of proposed estimator is less than the existing estimators by this we can conclude that the proposed estimator is preferable over the existing estimators by obtaining efficient point estimator. Figure 1 indicates that the proposed estimator has the lowest bias compared to other two estimators. Similarly, in case of MSE, the proposed estimator outperforms than the other estimators. In the present study, the alternative estimator of population mean under compromised imputation method has been proposed. Its bias and MSE also been proposed. For the proposed estimator the efficiency has been checked and prove to be more efficient than the existing estimators like ratio method of imputation, mean method of imputation and compromised imputation.

Acknowledgement

AcknowledgmentsWe also thank two anonymous referees and the editor for their helpful comments and suggestions that have substantially improved the presentation of this article.

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