An Efficient Compromised Imputation Method for Estimating Population Mean
1. INTRODUCTION
Auxiliary
information is important for survey practitioner as it is utilized to improve
the performance of the methods. It may be utilized at the design stage or the
estimation stage of the survey to get the more efficient estimator. At
estimation stage ratio, product and regression methods are traditionally used. Bhal
and Tuteja (1991) introduced
exponential ratio and product estimator for estimation of population mean. Many
modifications have been proposed using these methods till date. For handling
missing data on the study variable several extensions and developments were
proposed in the literature. Singh (2003) suggested
product estimation for imputation. Shakti Prasad (2018) adapts exponential
product type estimator given by Bahal and Tuteja (1991) and
proposed exponential estimators for imputation. Kadilar and Cingi (2008) investigated some ratio-type imputation
methods and proposed three new estimators to overcome the problem of the
missing data. Diana and Perri (2010)
proposed three regression type estimators which were
more efficient than the Kadilar and Cingi (2008). The
present article suggests a general ratio product exponential type method of
imputation and accordingly proposed three estimators using the different amount
of available auxiliary information as utilized by Ahmad
et al. (2006), Kadilar and Cingi (2008), and Diana and Perri (2010). The proposed methods are
than compared by traditional procedure of imputation. The proposed estimators
come out to be more efficient than the usual ratio, product, regression, and
exponential method for handling missing observations to estimate the population
mean. Given a finite population ,
the objective is to estimate the
population mean . A
simple random sample wor, ,
of size is drawn from the population .
Let the responding units be from the sampled units. Let us denote as the set of responding units and the set of non-responding units, i.e., is observed for but for units in the values are not available and hence imputed
values are derived by some method. In this paper we shall use the following
notations: : Population
Size; Sample size; : Number of
responding units; : Population
means of study variate and auxiliary variate respectively; : Standard Deviation of study variate and auxiliary variate respectively; : Coefficient
of variation of study variate and auxiliary variate respectively; : Correlation
coefficient between and ; . 2. Some
existing methods of imputation 1)
The Then the estimator of the population mean is given by
and its MSE is given by
(1.1) 2)
The ratio method of imputation uses information on one auxiliary
variable and calculates the missing values by
Where This gives the resulting estimator by The MSE of is given by (1.2) It is noted that, in the presence of missing data, the
availability of information on auxiliary variable in the data set supports
suggesting efficient estimators. 3) Diana
and Perri (2010) proposed three estimators as by using different regression-type
method of imputation such that the imputed data is given by For these
methods the resulting estimators are
(1.2)
(1.3)
(1.4) They proved
that the suggested estimators are more efficient than the Kadilar and Cingi (2008) estimators. is always more efficient
than both and , whereas perform better than if the condition 3. The proposed Estimator With the above
imputation method, the resulting estimator of the population mean is obtained as (2.1) and are constant chosen
suitably so that their choice minimizes the mean square error of the resultant
estimator and is a real constant. Our
goal in this paper is to discuss the suggested estimators for different values
of and have a comparative
study of the suggested estimator for these values of in order
to get the minimum MSE. 4. First
Degree Approximation to the Bias To derive the Bias and MSE expressions of the proposed estimator upto , we define Thus, we have The expectation of these are And under simple random sampling without replacement, where , . Now representing (2.1) in terms of , we have We assume that the sample is large enough to make and so small that contributions from powers of
degree higher than two are negligible. By retaining powers up to and , we get
(2.2)
Where and
From (2.2) we have
(2.3) Taking expectation on both side we obtain the bias of to order as |