Sample surveys’ main objective is to obtain information about the population, and then use such information to make inference about some population quantities. The information that is mostly sought about the population is usually aggregate values of various population characteristics, total number of units, proportion of units having certain attributes. The information can be collected by either sampling methods or census. One of the approaches to using auxiliary information in construction of estimators is by assuming a working model that describes the relationship between the survey variable and the auxiliary variable. Estimators are then derived based on this model. At this stage, estimators are sought to have good efficiency given that the model is true. In most cases, a linear model is assumed. Generalized regression estimators by [1] and [2] including linear regression estimators and ratio estimators by [3] , and best linear unbiased estimators by [4] and [5] and post-stratification estimators by [6] as well are all derived from the assumption of linear models. Sometimes the linear model fails, and therefore, the resulting estimators do not beat the purely design-based estimators. As a result, [7] proposed a class of estimators in which the working model assumes a nonlinear parametric model. The improvement of the efficiency of such estimators, however, requires prior information about the exact parametric population structure. As a result of these concerns, several researchers have so far considered nonparametric models for. Nonparametric regression may be used in the estimation of unknown finite population quantities such as population totals, means, proportions or averages. The idea of nonparametric regression traces its origin in works by [8] and [9] . Nonparametric-based estimation is often more robust and flexible than inference based on parametric regression models or design probabilities (as in designed-based inference) [10] . In sample surveys, auxiliary information is used at the estimation stage of finite population quantities-population total or mean, say-to increase the precision of estimators of such population quantities [11] [12] [13] .

A variety of approaches exist for construction of more efficient estimators for population total or mean, and they include model-based and design-based methods. Model-based approach in sample surveys is based on superpopulation models, which assumes that the population under study is a realization of a random variable having a superpopulation model. This model is used to predict the nonsampled values of the population, and hence the finite population quantities, total or mean [13] . [14] first considered nonparametric models for within a model-assisted approach and obtained a local polynomial regression estimator as a generalization of the ordinary generalized regression estimator. Their simulation study shows that the proposed estimator performs relatively better than other parametric estimators. [13] improved on [14] estimator and developed a model-based local polynomial regression estimator applicable to direct sampling designs such as simple random sampling and systematic sampling. Their estimator demonstrates better performance than [14] model-assisted estimator. Their estimator also beats other parametric estimators.

In this paper, auxiliary information is used to determine an estimator of finite population total using nonparametric regression under stratified random sampling. To achieve this, a model-based approach is adopted by making use of the local polynomial regression estimation to predict the nonsampled values of the survey variable y. Stratified estimators for finite population total or mean have proved to yield better estimators than those resulting from simple random sampling [15] [16] . Additionally, it has been shown in the literature that local polynomial approximation method has several nice features including satisfactory boundary behaviour, easy interpretability, applicability for a variety of design-circumstances and nice minimax properties (see [17] [18] and [19] ).

Consider a population consisting of N units. Suppose this population is divided into H disjoint strata, each of size.

Let be the survey measurement for the unit in the stra- tum. Further, let be the auxiliary measurement positively correlated with.

From each stratum, a simple random sample of size is selected without replace- ment, where is sufficiently large with respect to and.

Let be the sample in the stratum and be the nonsampled set in the stratum.

The population total is defined as

which can rewritten as

where and.

Once the sample has been observed, the problem of estimating Y becomes the problem of predicting the sum of the nonsampled. Usually, inference is made using the known sample and the model.

The first component in Equation (1) is known while the second requires prediction which is the focus in this paper. In this paper, local polynomial regression method will be used to predict the unknown,.

Suppose the distribution generating is given by the superpopulation model, in which

where are independently distributed random variables with mean 0 and variance.

Then it follows that

where and are assumed to be continuous and twice differentiable fun- ctions of x, and.

In practice, the values of are unknown and so requires prediction. Adopting [13] [14] and [20] ideas, we make use of local polynomial regression of degree p, which is a generalization of the kernel smoothing, to predict the unobserved in Equation (1). Let, where K denotes a continuous kernel function and b is the bandwidth.

Then a model-based local polynomial regression estimator of the nonsampled in the stratum is given by:

where is a column vector of length;;

and. Equation (6)

holds as long as is a nonsingular matrix.

Now denoting the estimator for the finite population total by and the estimator within the stratum by. Therefore, in stratum h, the estimator of the popu- lation total based on local polynomial regression is

and the estimator for the finite population total is

with.

In this section, a study is carried out on various properties of estimator (8), which may be important in practice. In doing so, the following assumptions are made:

1) The regression function has a bounded second derivative.

2) The marginal density, is continuous and.

3) The conditional variance is bounded and continuous.

4) The kernel density function is bounded and continuous satisfying the

following:, , and

for.

These conditions on were imposed and used in [18] work and are purposely for the convenience of technical arguments and therefore can be relaxed.

In this section, a study is carried out on the practical performance of several estimators (see Table 1 and Table 2 for the estimators).

The first estimator is design-based, the second one is parametric and model-based while the last two are nonparametric and model-based.

In this study, performance of the proposed estimator has been investigated against some design-based and model-based regression estimators. The RE values of the proposed estimator are in general close to one. It has been shown that for whichever bandwidth considered, essentially dominates for all the populations except Linear and Bump populations, where is competitive. Further, essentially dominates for all populations except in the Jump population, where it dominates all estimators being considered. Generally, good confidence intervals are seen for the nonparametric regression estimators, and use of the proposed estimator leads to relatively smaller values of RE compared to other estimators. We conclude that non- parametric regression approach under stratified random sampling using the proposed estimator yields good results.

Special thanks to the African Union (AU) for the funding that saw the success of this research.

Syengo, C.K., Pyeye, S., Orwa, G.O. and Odhiambo, R.O. (2016) Local Polynomial Regression Estimator of the Finite Population Total under Stratified Random Sampling: A Model- Based Approach. Open Journal of Statistics, 6, 1085-1097. http://dx.doi.org/10.4236/ojs.2016.66088