1. Introduction

Nonresponse always exists when surveying human populations as people hesitate to respond in surveys. Nonresponse as an aspect in almost every type of sample survey creates problems for estimation which cannot simply be eliminated by increasing sample size.

This phenomenon of nonresponse in a sample survey reduces the precision of parameters estimates and increases bias in estimates resulting in larger mean square error, thus ultimately reducing their efficiency.

An important technique to address these problems is by calibration. Calibration as a tool for reweighting for nonresponse was first introduced by[4] for the estimation of finite population characteristics like means ratios and totals.

Deville and Sarndal calibration estimation procedure provides a valuable class of techniques for combining data sources. The basic idea is to use estimates from one set of sources, which may be treated as sufficient accurate to act as benchmark. Estimates based on data from further sample source are then adjusted so as to agree with these benchmarks. The process of adjustment is called Calibration. The constraints that the estimates of the benchmarks based on this source should agree with the benchmarks are called Calibration Constraints.

The problem of calibration of design weights is well known in the literature of survey sampling.[4] used the method of calibration of estimators using auxiliary information. Their calibration method provides a class of estimates. Some of the well known estimators such as classical ratio estimator belong to this class. Several authors including[5-6, 9-13] among others considered the[4] method and derived important calibrated estimators. But so far derivation of calibrated estimators from the class of calibrated estimators derived by[4] method in the context of domain estimation is not well known in the literature. Our objective therefore is to extend calibration estimation to domain estimation.

Consider the finite population under study of size divided into domains; of sizes respectively. Domain membership of any population unit is unknown before sampling. It is assumed that domains are quite large[7].

The technique of estimation by calibration is based on the idea to use auxiliary information to obtain a better estimate of a population statistic. Consider a finite population of size with units labels Let be the study variable and be the -dimensional vector of auxiliary variables associated with unit .

Suppose we are interested in estimating the domain total . We draw a sample using a probability sampling design , with probability , where the first and second order inclusion probabilities are and respectively.

An estimate of is the Horvitz-Thompson (HT) estimator

(1)

where is the sampling weight defined as the inverse of the inclusion probability for unit .

An attractive property of the HT-estimator is that it is guaranteed to be unbiased regardless of the sampling design [8]. It variance under is given as:

(2)

Suppose there are auxiliary variables at unit and may or may not be known a priori. is the domain total for , and is known a priori. Ideally, we would like

(3)

but often times this is not true.

The idea behind calibration estimation is to find weights close to based on a distance function such that

(4)

Equation (4) is the calibration constraint. We wish to find weights similar to so as to preserve the unbiased property of the HT-estimator. Once is found, then our propose calibration estimator for is:

(5)

where .

Thus

(6)

This can be written in regression form as in equation (13).

In section 2 we discuss how to find the design weights for a given sample given a distance function. The expectation, variance, and variance estimation of the domain calibration estimator as well as the relationship of the domain calibration estimator to the generalized regression (GREG) estimator is also discussed. In section 3, we discuss the approach for finding the best sample design for the domain calibration estimator using appropriate design weights. Section 4 discusses the approach for the derivation of optimum stratum sample sizes that would minimize the variance of the domain calibration estimator and reduce the objective function under six different criteria. In section 5 we present data analysis and discussion. Section 6 presents the conclusions for the paper.

2. Derivation of Calibration Estimators for Domain

Given a sample , we want to find close to based on a distance function subject to the constraint in equation (4). This is an optimization problem where we wish to minimize

(7)

using the method of Lagrange Multipliers.

We will derive our calibration weights using the chi-squared distance where is a tuning parameter that can be manipulated to achieve the optimal minimum of equation (7).

However, in practice, it should be noted that the choice of distance function depends on the statistician and the problem considered.

(8)

Thus equation (7) becomes

(9)

Differentiating equation (9) with respect to and equating to zero we have

(10)

substituting equation (10) into (4) and solving for we have

(11)

where .

substituting (11) into (10) we have

(12)

Where

Following from equation (1): That is

Thus

(13)

Equation (13) is our proposed calibration estimator. It is a version of the Generalized Regression Estimator (GREG - estimator). This implies that the GREG-estimator is a special case of the calibration estimator in equation (5). Our result in (13) conforms to the GREG-estimator proposed by[2]. In fact, the GREG-estimator is a special case of the calibration estimator when the chosen distance function is the chi-square distance (see[4]).

2.1. Variance and Variance Estimation

We will follow the procedure proposed by[4] to derive an approximate variance for our proposed calibration estimator .

To find the expectation and variance of , we use the linearization technique to find an approximation of and with respect to a probability design . Let be the population level version of . Then a linear approximation of is:

(14)

Where the first term is of order , the second term is of order and the last term is of order as shown by[4]. Consequently, the last term can be omitted since it is of order . Thus, we can rewrite (14) as:

(15)

Using equation (15), the design-based expectation of is:

(16)

Thus is an approximately design-unbiased estimator of the domain total . Note that

since

Again using equation (15), the design-based asymptotic variance of is

(17)

where

The variance estimator is;

(18)

Note that the , a consistent and approximate unbiased estimator of variance (18) is:

where .

It should be noted that, it is acceptable to use the design weights in the variance estimation as in equation (18), but[4] suggest that the calibration weights be used in equation (18) as this makes the variance estimator both design-consistent and nearly model unbiased. Moreover, since the calibration estimator is asymptotically equivalent to the GREG-estimator, it can be inferred that calibration estimators are more efficient compared to HT-estimator if there is a strong correlation between and [2].

3. Sample Design for the Calibration Estimator

Consider a stratified random sampling design with strata and such that elements are considered from in stratum . Then, the design weights needed for the point estimation are for all in stratum . However, the design weights needed for the variance estimation if and both and are in different strata, say stratum and stratum is:

(19)

Using equation (18): .

Then, we have;

(20)

It should be noted that in calibration, it is assumed that elements respond independently so that . Consequently, from the following Theorem in stratified sampling design according to[3];

Theorem

If the samples are drawn independently in different strata,

where is the variance of over repeated samples from stratum .

Since

is a linear function of the with fixed weights . Hence we may quote the result in statistics for the variance of a linear function

But since samples are drawn independently in different strata, all covariance terms varnish[3 p. 92].

We have variance estimator of (18) as:

(21)

4. Optimal Sample Allocations

We shall now deduce the optimum , that minimize the variances of the proposed calibration estimators for a specified cost, or that minimize the cost for a specified variance.

Let us consider the simple linear sampling cost function of the form:

(22)

where is the overhead cost and is the cost per unit of obtaining the necessary information in -th stratum. In this paper, we shall consider the following allocation methods: optimum allocation, Neyman allocation, optimal power allocation, Neyman power allocation, square root allocation and Neyman square root allocation.

(i) Optimum allocation

The problem of optimum allocation consists in minimizing the sampling variance for a given overall sampling cost of the survey or minimizing the overall sampling cost for specified sampling variance.

Let us consider the simple linear sampling cost function of equation (22), the corresponding Lagragian is:

(23)

Differentiating (23) with respect to and and equating to zero we have respectively

and

Thus

(24)

and

Finally to obtain a solution for , we substitute for into (24) as follows:

(25)

(ii) Neyman allocation

If the cost per unit is the same in all strata, (that is, ) , then (25) reduces to

(26)

The type of allocation of (26) where (that is, where the cost per unit is the same in all strata) is called the Neyman allocation.

(iii) Optimal power allocation

The power allocation was first considered by[1]. He considered a compromise allocation between equal allocation and Neyman allocation in which the within stratum sample size is proportional to and called it power allocation. Suppose that a stratified random sampling is to be selected. Let be some measure of size or importance for the th stratum. It is desired to determine stratum sample sizes , that the loss function

(27)

is minimized subject to the constraint where and is a constant in the range and is called the power of the allocation.

Following from our sample design, the loss function is

The corresponding lagragian is

(28)

Differentiating (28) with respect to and and equating to zero we have

and

Thus,

(29)

and

Finally to obtain a solution for , we substitute for into (29) as follows:

(30)

The exponent is called the power of the allocation. The choice of results in significantly different allocations, for example, if the cost per unit is the same across strata (that is, ) and setting , we obtain the Neyman allocation of (26). Choosing a value of between 0 and 1 can be viewed as a compromise allocation between Neyman allocation and the equal allocation.

(iv) Neyman power allocation

If the cost per unit is the same across strata, then;

(31)

(v) Square root allocation

The square root allocation is a special case of the power allocation. When the power of the allocation is set to one-half (that is, setting ), we obtain

(32)

(vi) Neyman square root allocation

Again, if the cost per unit is the same across strata (that is, ), and the power of the allocation is set to one-half (that is, setting ), then, we obtain what may be called the Neyman square root allocation as:

(33)

5. Data Analysis and Discussion

5.1. Background and Analytical Set-up

The data used is obtained from the 2005 socio-economic household survey of Akwa Ibom State conducted by the ministry of economic development, Uyo, Akwa Ibom State, Nigeria.

The study variable, , represents the household expenditure on food and auxiliary variable, , represents the household income. The statistic of interest is the total cost of food for household and its corresponding estimator for male and female heads of household.

The population of household heads was stratified into two strata that constitute the domains; as the male household heads and the female household heads respectively. For the population of individual household heads, we want a separate estimates for male and female household heads defined as two domains of the population. The number of the male household heads and female household heads in the survey are known. We used the calibration estimator for the domain total and the following formulation is specified: The number of male household heads, and female household heads, are known and the auxiliary vector has two possible values; namely, for all male household heads and for all female household heads. The population total of the auxiliary vector is which is also known and for all .

An assisting model of the form was designed for the calibration estimators, where is the number of strata (domains) and are independently generated by the standard normal distribution.

5.2. The Sampling Design Variance Estimation

To obtain an optimum value of that minimizes the design variance , a population was generated with the following parameters: . Table 1 shows the summary of values of for the six allocation criteria. The variance for the domain calibration estimator using the optimum values of from the six different allocation criteria are presented in table 2.

Table 1. Optimum Value of

Table 2. Optimum Variance

The variance estimator from the stratified random sampling design is;

where and and is the stratum variance of the residuals where . The optimum value of for the Neyman allocation gave the minimum variance. The results of the design variance estimation are presented in table 3.

Table 3. Variance Estimation

5.3. Comparisons with Greg-estimator

To compare the performance of each estimator we use the following criteria; bias (B), relative bias (RB), mean square error (MSE), average length of confidence interval (AL) and the coverage probability (CP) of . Let be the estimate of in the simulation run; we define

where

where and are the upper and lower confidence limit of the corresponding confidence interval.

Coverage probability of 95% confidence interval is the ratio of the number of times the true domain total is included in the interval to the total number of runs or the number of replicates.

For each estimator of , a 95% confidence interval is constructed, where

where is the lower confidence limit, is the upper confidence limit and .

The analytical study was conducted using the R-statistical package. There were runs in total. For the run , a Bernoulli sample is drawn where each unit is selected into the sample independently, with inclusion probability where . Following the results of analysis for optimum stratum sample sizes, we fixed and and the corresponding calibration estimators of the domain totals were computed. For simplicity, the tuning parameter was set to unity .

For each estimator of , a 95% confidence interval is constructed, where is the lower confidence limit, and is the upper confidence limit.The results of the analysis are given in table 4.

Table 4. Comparison of estimators from analytical study

5.4. Discussion

An assisting model of the form was designed where is the number of strata (domains) and The results of the residual diagnostics showed the value as 0.588 indicating that the model is significant and that the calibration estimators are unbiased with respect to the sampling design. The correlation between the study variable and the auxiliary variable is is strong and sufficient implying that the calibration estimators would provide better estimates of the domain totals.

The Neyman allocation criterion provides the optimum stratum sample sizes and that minimized the variance of the calibration estimators as reflected in table 2.

The design strata estimates are 19,753.1468 and 4,678.6777 for stratum 1 and stratum 2 respectively. Similarly, the variance estimate is 24,431.8245. Following from the above estimates, we deduced that the design strata estimates sum up to the finite population estimates.

Analysis for the comparison of performance of estimators showed that the biases of 0.82 percent and 0.96 percent respectively for the calibration estimator and the GREG-estimator are negligible. But the bias of the GREG-estimator though negligible is the most biased among the estimators considered.

The relative bias for the calibration estimator is relatively smaller than that of the GREG-estimator. The variance for the GREG-estimator is significantly larger than the variance of the calibration estimators, as is indicated by their respective mean square errors in table 4. The average length of the confidence interval for the calibration estimator is significantly smaller than that of the GREG-estimator. The coverage probability of the calibration estimator is also smaller than that of the GREG-estimator. These results showed that there is greater variation in the estimates made by the GREG-estimator than the calibration estimator.

In general, the domain calibration estimator is more efficient than the GREG-estimator and the variance reduction is about 50 percent which is consistent with theory as is reflected by the high population correlation between the study variable and the auxiliary variable .

6. Conclusions

Nonresponse as an aspect in almost every type of sample survey creates problems for estimation which cannot simply be eliminated by increasing sample size. This phenomenon of nonresponse in a sample survey reduces the precision of parameters estimates and increases bias in estimates resulting in larger mean square error, thus ultimately reducing their efficiency.

Sample surveys have long been used as cost-effective means for data collection. Such data is used to provide suitable statistics not only for the population targeted by the survey but also for a variety of subpopulations called domains of study. Sample designs and in particular sample sizes are chosen so as to provide reliable estimates for domains of study.

One of the main objectives of sample survey is the computation of estimates of means and totals for specific domains of interest. The reliability of the associated estimates depends on the variability of the sample size as well as on the study variables, of interest. But budget and other constraints usually prevent the allocation of sufficiently large samples to domains to provide reliable estimates using traditional statistical techniques. This problem of optimal allocation of sample sizes for domain estimation has received less attention than merited in the statistical sample survey theory literature.

This paper addressed these problems by proposing calibration estimators for totals of domains of study and developed an approach for finding the best sample design for the domain calibration estimators subject to a cost constraint in the context of stratified random sampling design (STRS) where domains constitute strata in the sampling design to obtain optimal stratum sample sizes that minimized the variances of the proposed domain calibration estimators and reduced the objective function.

The efficacy of our proposed calibration estimator was tested through a real data analysis. Five performance criteria, namely; bias (B), relative bias (RB), mean square error (MSE), average length of confidence interval (AL) and coverage probability (CP) were used to compare the relative performances of our proposed domain calibration estimator against the traditional GREG-estimator. Results of the analytical study using real data showed that our proposed calibration estimator is substantially superior to the traditional GREG-estimator with relatively small bias, mean square error and average length of confidence interval.