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Estimation of Population Total in the Presence of Missing Values Using a Modified Murthy's Estimator and the Weight Adjustment Technique

Oyoo David Odhiambo, Christopher Ouma Onyango*

Department of Statistics and Actuarial Science, Kenyatta University-Kenya

Abstract: 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.

Keywords: Murthy’s estimator, missing values, weight adjustment

Received March 27, 2014; Revised May 08, 2014; Accepted May 22, 2014

Citation: Oyoo David Odhiambo, 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). 163-167. January 2014. doi:10.12691/ajams-2-3-12

1. Introduction

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

Where,

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 data

3.1. 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

and

where

Suppose the symmetrization is such that then define as

[picture] | (1)

Equation (1) is only for selecting 2 units. Suppose we consider n units, we get given by

[picture] | (2)

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

[picture] | (3)

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

3.2. Weighting Adjustment

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,

[picture] | (6)

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

[picture] | (7)

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 Estimator

4.1. Unbiasedness

Define a vector so that

Now,

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)

Now

Define

In SRS,

And

Hence

which on simplification gives,

and this simplifies to,

Hence, where, Thus,

[picture] | (8)

Now,

But in SRS, sample variance () is unbiased for population variance (). Where

Therefore, overall variance of the estimator is

[picture] | (9)

4.3. Consistency of the Proposed Estimator

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

That is,

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

[picture] | (10)

Replacing equation (10) in equation (7), we have

[picture] | (11)

And consequently becomes

[picture] | (12)

We can thus obtain Bias () instead of Bias ()

Bias since is constant. (Cochran, 1977)

[picture] | (13)

But from previous workings,

[picture] | (14)

From equations (6) and (7)

[picture] | (15)

Substituting (14) and (15) in equation (13) and simplifying, we obtain

[picture] | (16)

Clearly, Bias () vanishes if

4.5. Expected Mean Squared Error (MSE) of the Proposed Estimator

Where,

References

[1]   | Brick, J.M. and Kalton, G. (1996) Handling missing data in survey research. Statistical Methods in Medical Research, 5, 215-238.
[2]   | Broemeling, D. L. (2009). Bayesian Methods for Measures of Agreement (Chapman & Hall/CRC Biostatistics Series). Chapman and Hall/CRC Press.
[3]   | Cochran, W. G. (1977). Sampling Techniques. 3rd Edition. New York, John Wiley.
[4]   | Chang, C. and Ferry, B. (2012). Weighting Methods in Survey Sampling. Section on Survey Research Methods-JSM, 4768-4782.
[5]   | Daroga, S. and Chaudhary, F. (2002). Theory and Analysis of Sample Survey Designs. New Delhi: New Age International (P) Limited Publishers.
[6]   | Murthy, M. N. (1957). Ordered and unordered estimators in sampling without replacement. Sankhya, 18, 379-390.
[7]   | Ouma, C., Odhiambo, R. and Orwa, G. (2010). Bootstrapping in Model-Based Estimation of a Finite Population Total Under Two-Stage Cluster Sampling With Unequal Cluster Sizes. Annals of Statistics, July Issue, 171-184.
[8]   | Salehi, M. and Seber, G. A. (2002). Theory & Methods: A New Proof of Murthy's Estimator which Applies to Sequential Sampling. Australian & AMP New Zealand Journal of Statistics, 43(3), 281-286.
[9]   | Shahbaz, Q. M., and Ayesha, S. (2008). A new symmetrized estimator of population total in unequal probability sampling. Journal of Statistics, 13(1), 20-25.
[10]   | Singh, H. P. and Solanki, R. S. (2012). An alternative procedure for estimating the population mean in simple random sampling. Pakistan Journal of Statistics and Operation Research, 8(2), N 1816-2711.