Estimation of Population Total in the Presence of Missing Values Using a Modified Murthy's Estimator and the Weight Adjustment Technique
1Department of Statistics and Actuarial Science, Kenyatta University-Kenya
Use of Murthy’s method in estimation of population parameters, such as population totals, population means, and population variances has been limited to surveys where survey data values are complete. This study applies weight adjustment technique to estimate a population total under simple random sampling without replacement. The asymptotic properties show that the estimated population total is sufficient for the true population total. The proposed estimator is obtained by symmetrizing Murthy’s estimator.
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Keywords: Murthy’s estimator, missing values, weight adjustment
American Journal of Applied Mathematics and Statistics, 2014 2 (3),
Received March 27, 2014; Revised May 08, 2014; Accepted May 22, 2014Copyright © 2014 Science and Education Publishing. All Rights Reserved.
Cite this article:
- Odhiambo, Oyoo David, and Christopher Ouma Onyango. "Estimation of Population Total in the Presence of Missing Values Using a Modified Murthy's Estimator and the Weight Adjustment Technique." American Journal of Applied Mathematics and Statistics 2.3 (2014): 163-167.
- Odhiambo, O. D. , & Onyango, C. O. (2014). Estimation of Population Total in the Presence of Missing Values Using a Modified Murthy's Estimator and the Weight Adjustment Technique. American Journal of Applied Mathematics and Statistics, 2(3), 163-167.
- Odhiambo, Oyoo David, and Christopher Ouma Onyango. "Estimation of Population Total in the Presence of Missing Values Using a Modified Murthy's Estimator and the Weight Adjustment Technique." American Journal of Applied Mathematics and Statistics 2, no. 3 (2014): 163-167.
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In sample surveys, modeling an optimal estimator that best estimates finite population total has been of interest to modern statisticians (Ouma et al., 2010). Estimation methods for some population parameters include, among others, ratio estimation, Horvitzand Thompson estimation, and Yates and Grundy estimation. From these studies, various estimators of population have been obtained. In this paper, we have obtained a new estimator by symmetrizing Murthy’s estimator. We have then estimated finite population total in the presence of missing data using the derived estimator. As a way of correcting the ‘missingness’ of data, weight adjustment method has been used.1.1 Background of the Problem
In sample surveys, completeness of observed datais one factor that influences inferences made on results of a study. Daroga and Chaudhary (2002) explained that missing data distort validity and reliability of a study. Consequently, variousmethods of correcting missing data have been proposed in sample surveys. Some of the methods include: imputation techniques,partial deletion andresampling (Brewer, 2002, Broemeling,2009). Singh and Solanki (2012) later not only supported Broemeling’sproposal (2009), but also observed that previous studies have not extensively used samples with missing data. This research has, therefore, filled this gap by using a sample with missing values. Singh and Solanki (2012) further observed that previous studies have only focused on ordered sampling procedures. However, not all sets of data are ordered. In filling this gap, this study utilizes Murthy’s estimation method, which involves unordered sampling procedures (Murthy, 1957).
2. Murthy’s Estimation
Murthy's estimator has been used for constructing unbiased estimators of population totals and/or mean from a sample of fixed size. Let be an estimator of population parameter based on the ordered sample (si), Murthy’s estimator for population total is given by
P(s/i) = conditional probability of getting the set of units that was drawn, given that the i-th unit was drawn first.
P(s) = unconditional probability of getting the set of units that was drawn
Consider a random selection of three population units i, j, and k are randomly selected from a population of size N with the corresponding selection probabilities be zi, zj/(1-zi), and zk/(1-zi-zj).
Then we can show that Murthy’s estimator, is unbiased for the population total Y and its variance for n = 2 is given by
Which can be rearranged as follows
3. Proposed Estimator
The proposed estimator is given by
Where weight adjustment of ith unit in group c and an be expressed as
where = population size in group c, = number of units with complete data3.1. Derivation of the Proposed Estimator
By assuming any two population units and and the corresponding selection probabilities and Shahbaz (2004) modified Murthy’s estimator as
And Shahbaz and Ayesha (2008) symmetrized the partitioned estimator as and given by
Suppose the symmetrization is such that then define as
Equation (1) is only for selecting 2 units. Suppose we consider n units, we get given by
Since the study involves estimating finite population total in the presence of missing data, we apply weighting adjustment to correct the “missingness” of responses. We proceed as follows;
For any population of size N, as , then and That is, for large n, the inclusion probabilities are asymptotically equal and (Cochran 1977)
Using the results for large n and asymptotic value of , equation (2) reduces to
Equations (3) and the proposed estimator are similar if the weighting constant and Our task is therefore to determine the value of
Consider the set and be a set chosen from U.
Define a population of size N as and a sample of size n as .
Let the respective population and sample totals be
And the corresponding population and sample means are given by
Since is unbiased for it follows that N and hence the estimator of population is
Suppose the population can be classified to form k groups based on auxiliary information Using the definition of S above, let us partition S as
Using the k classes, there exists partitions U1, U2, ……… , Uk such that ,
Let be the set containing identified numbers of responding units in class c (i.e with no missing information).
Let the sizes of Uc, Sc, and are Nc, nc, and mc respectively, then by letting mc> 1, we have
Consider any class c (c = 1, 2, …..k), mc is used to represent nc. This implies that each of the mc units has a weight of
Let be a study observation with an identification number i in class c. If we define
And from equation (4), can be estimated by That is,
Then, for known Nc,
Equation (6) implies that a sample of size mc is used to represent a population of size Nc. The overall adjusted estimator can thus be written as
Where And can be expressed as = w1.w2, where w1 = is the base weight in class c and w2 = is the non-response adjusted weight in class c.
4. Properties of the Proposed Estimator4.1. Unbiasedness
Define a vector so that
Hence the estimator is unbiased.4.2. Variance of the Proposed Estimator
Since the nature of sampling makes the entire sampling procedure analogous to Simple Random Sampling (SRS). Suppose we consider one of the classes and use a sample of size mc to estimate parameters in a population of size Nc, we can apply the procedures in SRS to derive this variance.
Since is unbiased for it follows that
Recall: (Cochran, 1977)
which on simplification gives,
and this simplifies to,
Hence, where, Thus,
But in SRS, sample variance () is unbiased for population variance (). Where
Therefore, overall variance of the estimator is
Consider the proposed estimator and finite population total . A sequence of point estimators is said to be weakly consistent for if converges in probability to
Proof: By Chebychev’s inequality, for every .
Taking limits as the right hand side .
Hence, which is the necessary and sufficient condition for consistency.4.4. Bias of the Proposed Estimator
From equation (8), we assume that Nc () is known. Suppose that Nc is not known, we need to estimate Nc and consequently a new Suppose the classification is such that the subpopulation ratio is equal to That is, sampling distribution of is centered on
Replacing equation (10) in equation (7), we have
And consequently becomes
We can thus obtain Bias () instead of Bias ()
Bias since is constant. (Cochran, 1977)
But from previous workings,
From equations (6) and (7)
Substituting (14) and (15) in equation (13) and simplifying, we obtain
Clearly, Bias () vanishes if4.5. Expected Mean Squared Error (MSE) of the Proposed Estimator
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