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.

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 (s_i), 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 z_i, z_j/(1-z_i), and z_k/(1-z_i-z_j).

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 i^th unit in group c and an be expressed as

where = population size in group c, = number of units with complete data

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

(1)

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

(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

(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

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, ……… , U_k 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 U_c, S_c, and are N_c, n_c, and m_c respectively, then by letting m_c> 1, we have

Consider any class c (c = 1, 2, …..k), m_c is used to represent n_c. 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 N_c,

(6)

Equation (6) implies that a sample of size m_c is used to represent a population of size N_c. The overall adjusted estimator can thus be written as

(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

Define a vector so that

Now,

Hence the estimator is unbiased.

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 m_c to estimate parameters in a population of size N_c, 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,

(8)

Now,

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

Therefore, overall variance of the estimator is

(9)

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.

From equation (8), we assume that N_c () is known. Suppose that N_c is not known, we need to estimate N_c and consequently a new Suppose the classification is such that the subpopulation ratio is equal to That is, sampling distribution of is centered on

(10)

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

(11)

And consequently becomes

(12)

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

Bias since is constant. (Cochran, 1977)

(13)

But from previous workings,

(14)

From equations (6) and (7)

(15)

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

(16)

Clearly, Bias () vanishes if

Where,

References

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