OJSOpen Journal of Statistics2161-718XScientific Research Publishing10.4236/ojs.2016.66088OJS-72619ArticlesPhysics&Mathematics Local Polynomial Regression Estimator of the Finite Population Total under Stratified Random Sampling: A Model-Based Approach CharlesK. Syengo1SarahPyeye1GeorgeO. Orwa2RomanusO. Odhiambo2Department of Statistics and Actuarial Science, Jomo Kenyatta University of Agriculture and Technology, Nairobi, KenyaPan African University Institute for Basic Sciences, Technology and Innovation, Nairobi, Kenya14112016060610851097September 5, 2016Accepted: December 3, December 8, 2016© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

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. The performance of the proposed estimator is investigated against some design-based and model-based regression estimators. The simulation experiments show that the resulting estimator exhibits good properties. 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.

Sample Surveys Stratified Random Sampling Auxiliary Information Local Polynomial Regression Model-Based Approach Nonparametric Regression
1. Introduction

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  and  including linear regression estimators and ratio estimators by  , and best linear unbiased estimators by  and  and post-stratification estimators by  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,  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  and  . Nonparametric-based estimation is often more robust and flexible than inference based on parametric regression models or design probabilities (as in designed-based inference)  . 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    .

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  .  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.  improved on  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  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   . 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   and  ).

2. Proposed Estimator

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   and  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.

3. Properties of Proposed Estimator

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  work and are purposely for the convenience of technical arguments and therefore can be relaxed.

3.1. <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/10-1240768x71.png" xlink:type="simple"/></inline-formula>Is Asymptotically Model-Unbiased

Now consider the difference:

and taking expectation yields

since

i.e.

which is the bias associated with.

Approximating by Taylor series expansion about a point and assuming further that and, then observe that

Letting, then

and applying expectations then

Theorem 3 of  allows that under conditions (1)-(4) if and,

So that

It implies that provided that and, and thus is asymptotically model-unbiased.

3.2. Mean Square Error (MSE) of <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/10-1240768x98.png" xlink:type="simple"/></inline-formula>

The estimator (8) has the MSE

which can be decomposed as

Theorem 1 of  allows that under Condition (1), if then

Observe that Equation (24) tends to zero if and and thus

.

This shows that is statistically consistent and thus useful.

4. Simulation Study

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.

4.1. Description of the Population

The working model is taken to be,. In this study, four populations are considered, which are generated from the regression model given by

with the following mean functions

with. They represent a class of correct and incorrect model specifications for the estimators being considered. For, is expected to be the best estimator, since the model assumed is correctly specified. The rest of the mean functions:, and represent various deviations from the linear model,. These populations are plotted in Figure 1. For more on these populations, see  and  .

The errors are assumed to be independent and identically distributed (i.i.d) normal random variables having mean 0 and standard deviation. They contain 2000 units and the population is simulated as i.i.d uniform random variables. The

Estimators being compared in the Simulation study
Horvitz-Thompson
Linear regression , p. 200
Mixed Ratio
Local polynomial with Equation (8)
Plot of linear, sine, bump and jump populations.

population values are generated from the mean functions by adding the errors in each of the cases. Each of the populations is divided into 10 equal, disjoint and mutually exclusive strata which are made as homogeneous as possible to ensure that units in each stratum vary little from each other. A sample of size, is then taken with each stratum contributing a sample size of,. 1000 samples are simulated using simple random sampling without replacement for each case.

Epanechnikov kernel,

is used for kernel smoothing on each of the populations. In each case, bandwidth values (see  ) (with), , and (see  ) are con- sidered.

Data simulations, the estimators and computations were obtained using R Software on a desktop.

To analyze the performance of the proposed estimator against some specified estimators, relative absolute bias (RAB) is computed as

and the relative efficiency (RE) with respect to the Horvitz-Thompson (HT) estimator is computed as

is the estimator of the finite population total being considered; Y is the true population total and R is the number of replications.

The relative efficiency (RE) is meant to examine the robustness of the various estimators against the proposed estimator.

The confidence intervals (CI) and the average lengths (AL) of the confidence intervals of various estimators are also computed as follows:

where and are the upper and lower confidence limits respectively; and R are as defined earlier.

4.2. Results

The results of this simulation study are summarized in Table 3 and Table 4. For each populations, (), the performance of each estimator is analyzed using the RAB and RE. The RAB indicates the measure of how close the estimator being considered is from the actual value, while the RE is used to check the robustness of the estimator. For instance, an estimator, , will be said to be “better” or more preferable than another one, , if its RE is comparably smaller. That is, if, where and are estimators, then is said to be “better” than.

Summary of the formulae used in computing the respective population totals of the various estimators
EstimatorFormulae
Horvitz-Thompson,
Linear regression estimator,
Mixed Ratio Estimator, ,
Proposed Model-based Local polynomial with,

The confidence intervals and average length of the intervals are also measured for each case. A smaller length is better because it implies that the true population total is captured within a smaller range and therefore results are more precise.

The estimators and are tested under the same bandwidth choice i.e. (with), , and. Results of this simulation are shown in Table 3 and Table 4 below.

Table 3 shows the RAB’s and RE’s of the various estimators with respect to the Horvitz-Thompson estimator (). Table 4 shows the confidence intervals and their average lengths.

In most scenarios, is better than the parametric estimators, but the parametric estimator, , performs best when the model is correctly specified, as Table 3 shows. This occurs both in the linear and the bump populations, where in the former, a strong linear relationship holds between the variables while in the latter, the function is linear over most of its range despite a “bump” for a small part of the range of.

When the model is completely misspecified as in the Sine and Jump populations, a greater efficiency can be achieved by the nonparametric regression estimators. This can be seen in Table 3 for the Sine and Jump populations: the nonparametric estimators (and) are more efficient than their parametric opponent,.

When the underlying superpopulation model is completely unknown, a reasonable choice for finite population total estimation would be the nonparametric estimators such as and with small bandwidth choices. This can be seen in Table 3 and Table 4.

In this study, is sometimes seen to perform much bettter but not as worse as, and hence the proposed estimator, emerges as the best performing among the nonparametric estimators being considered here (see Table 3). A good overall performance is observed with the proposed estimator, with smaller values of RAB and RE than the model-based competitor for every population and fixed bandwidth under consideration.

Despite being relatively the best estimator, its performance is significantly affected by the bandwidth choices. As the bandwidth size increases, some amount of efficiency is lost (see Table 3).

Relative absolute bias (RAB) and relative efficiency (RE) based on 1000 replications of simple random sampling within strata from four fixed populations of size<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/10-1240768x191.png" xlink:type="simple"/></inline-formula>. Sample size is<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/10-1240768x191.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/10-1240768x192.png" xlink:type="simple"/></inline-formula>
Populationb
RABRERABRERABRERABRE
Linear0.34657240.0321240110.0057789290.031557330.033214961.0678110.032018880.9959899
0.40.0321240110.0057789290.031557330.03353521.0895730.03205330.9965037
10.0321240110.0057789290.031557330.034341221.1449510.032104490.9991698
20.0321240110.0057789290.031557330.032722641.0377530.032120230.9997907
Estimated Totalb = 0.34657241941.4271943.1611939.521941.248
b = 0.41941.4271943.1611938.8071941.167
b = 11941.4271943.1611937.3911941.419
b = 21941.4271943.1611940.3361941.424
Population Total1943.052
Sine0.34657240.0185519310.038364534.2867230.020720861.2435340.016573210.7990398
0.40.0185519310.038364534.2867230.020826491.2559190.016853030.826246
10.0185519310.038364534.2867230.02019471.1838260.018108820.9576443
20.0185519310.038364534.2867230.018953571.0439510.01846070.9908383
Estimated Totalb = 0.34657244071.0664114.0314080.3164056.493
b = 0.44071.0664114.0314081.6854054.513
b = 14071.0664114.0314079.1564066.007
b = 24071.0664114.0314073.044070.166
Population Total4071.383
Bump0.34657240.0310961810.014495690.21309840.032435361.0859120.031009860.9935966
0.40.0310961810.014495690.21309840.032891211.1160630.033193031.123072
10.0310961810.014495690.21309840.033578091.1650750.03213971.061732
20.0310961810.014495690.21309840.031658291.0367390.031063650.9988702
Estimated Totalb = 0.34657242186.492192.7692188.2662172.2
b = 0.42186.492192.7692195.3942151.329
b = 12186.492192.7692200.6892161.91
b = 22186.492192.7692189.3182182.232
Population Total2187.923
Jump0.34657240.00484502210.0248360926.073890.0056168961.3535660.0076769672.274792
0.40.00484502210.0248360926.073890.00562051.350230.0077509742.329744
10.00484502210.0248360926.073890.0051818821.1552660.0055051621.259671
20.00484502210.0248360926.073890.0048525431.0067730.0048727781.006966
Estimated Totalb = 0.34657243299.1853321.6993288.8573322.128
b = 0.43299.1853321.6993288.4153322.202
b = 13299.1853321.6993291.3263309.116
b = 23299.1853321.6993297.4853300.881
Population Total3300.252
Estimated lower and upper confidence limits and corresponding average lengths based on 1000 replications of simple random sampling within strata from four fixed populations of size<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/10-1240768x197.png" xlink:type="simple"/></inline-formula>. Sample size is<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/10-1240768x197.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/10-1240768x198.png" xlink:type="simple"/></inline-formula>. (LCL is the Lower Confidence Limit, UCL is the Upper Confidence Limit and AL is the Average Length)
Populationb
LCLUCLALLCLUCLALLCLUCLALLCLUCLAL
Linear0.34657241905.4311977.42371.9921919.1391967.18348.0441934.861944.189.321936.2491946.2479.998
0.41905.4311977.42371.9921919.1391967.18348.0441934.2501943.3649.1141936.1691946.1659.996
11905.4311977.42371.9921919.1391967.18348.0441933.7111941.0717.3601936.4181946.42010.002
21905.4311977.42371.9921919.1391967.18348.0441936.7331943.9387.2061936.4241946.4249.999
Population Total1943.052
Sine0.34657244026.5804115.55288.9734044.2964183.766139.4704074.6544085.97811.3244050.9374062.04911.113
0.44026.5804115.55288.9734044.2964183.766139.4704076.1564087.21311.0574049.0144060.01210.998
14026.5804115.55288.9734044.2964183.766139.4704074.6504083.6619.0124060.2544071.76011.506
24026.5804115.55288.9734044.2964183.766139.4704068.5894077.4918.9024064.4984075.83411.336
Population Total4071.383
Bump0.34657242146.5452226.43479.8892156.4902229.04872.5582183.2342193.29910.0652166.8392177.56010.721
0.42146.5452226.43479.8892156.4902229.04872.5582190.4732200.3159.8422145.9802156.67810.698
12146.5452226.43479.8892156.4902229.04872.5582196.6212204.7588.1372156.5822167.23810.656
22146.5452226.43479.8892156.4902229.04872.5582185.3202193.3157.9952176.9092187.55410.645
Population Total2187.923
Jump0.34657243290.0273308.34418.3173127.4633515.934388.4713287.9023289.8131.9123321.0783323.1792.101
0.43290.0273308.34418.3173127.4633515.934388.4713287.473289.361.893321.1723323.2322.060
13290.0273308.34418.3173127.4633515.934388.4713290.4093292.2441.8353308.1673310.0651.898
23290.0273308.34418.3173127.4633515.934388.4713296.5693298.4011.8323299.9323301.8291.897
Population Total3300.252

Additionally, a keen look at the estimated totals in Table 3 shows that: as the bandwidth increases, the local linear regression estimator, becomes equivalent to the linear regression estimator,. This shows that the bandwidth has an effect on the mean square error of. Particularly, for whichever bandwidth that is considered in this study, 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 dominates all estimators being considered. The overall performance of is consistently good as long as the bandwidth remains small in this particular study.

5. Conclusion

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.

Acknowledgements

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

Cite this paper

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

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