Regression estimation in center sampling when there are measurement and response errors

Introduction 1. Regression estimation in center sampling

The regression model requires adequate information about the auxiliary variable. In the absence of auxiliary variables' information, random selection of a large sample of size ' n is carried out where information on the auxiliary variables is obtained, followed by selecting a subsample of size n from the large sample, and observing the study variable y. This sampling process is the double sampling or the two-phase selection process. In this process, randomization is carried out twice, during the first phase and the second phase. In this work, regression estimation is applied in Center Sampling and two-phase Center Sampling.

Let lrq u be the response of the th q unit with profile r in center l . For the whole population, the population mean is

 is now a parameter of the finite population. The objective will now be of estimating lr B  and not lr β  .

Where The above can be written as σ is a constant and there is an intercept implying 1 1

σ is proportional to one of the' k 2) can be written as

The sample observed value therefore has weights lrqs g and lrq π

. While lrq π is given before sampling, lrqs g depends on the sample obtained.

) For variance estimator, we consider the Taylor linearization approximation of the estimator given by ( )

This is easily proved to be unbiased estimate of l t µ after taking expectation at both stages. Also

∑ ∑ ∑

By Taylor linearization, the approximate estimator is

∑ ∑ ∑

The approximate variance can now be obtained as

Regression estimation in center sampling in the presence of measurement errors

Regression estimation is considered in the presence of measurement and response errors so that we can make use of the information from the auxiliary variable.

Suppose that rather than observing, lrq µ the observed value is Where lrq e is a measurement error.

Let

The regression estimator can now be written as

Using Taylor linearization technique, we have equation ( 13) as ( )

From equation (3), the sampling variance is ( )

lrqs lrq lrqs lrq lrq s lrq lrq lrq lrqq lrq lrq r q q q r lrq lrq lrq

∑ ∑ ∑∑

Also, consider that from equations ( 4) and ( 12), ( )

lrq lrq lrq lrq lrq lrq p lrq lrq lrqq lrq lrq r q q q r lrq lrq lrq

∑ ∑ ∑∑

This depends on lrqs g which is a sample weight and hence difficult to compute. However, using the Taylor linearization gives the approximated value as 1

lrqs lrq lrq lrq lrq lrq lrq lrqq lrq lrq r q q q r lrq lrq lrq

The response variance is given by ( )

Again, the expectation of equation ( 16) is complex due to regression effect. A good approximation for large samples is

The response variance term is

The variance estimator is now From ( )

b) Now consider the special case when 2 p = , that is assumed super population model is

. lrqs g can be obtained as follows

(

1 lrq lr lrq lr lr lr lr lrq lr lr

Two phase regression estimation in center sampling in the presence of measurement errors

Let lrq y be the observed value be modelled as

Also the estimator can be written as

∑ ∑ ∑

Taking expectation under measurement errors, ( )

1 lrq lrq lrq lrq lrq lrq r q q r lrq lrq lrq

∑ ∑ ∑∑

(21) Using simple random sampling without replacement, ( )

1 ˆlr lr lr lr m lr lrq lrq lrq r s q q r lr lr lr

Results and Discussion

Regression estimation in Centre Sampling.

For empirical illustration, simulation of variable X (could be income) and Y (Expenditure) is carried out the objective being to estimate or predict total expenditure which is assumed to regress on income. A population with three centers whose sizes are 1 (1,0,0), (0,1,0), (0,0,

Regression Estimation in Two Phase Center Sampling

A sub sample of 100 is selected from 300 units of the first phase, where b estimates for larger k.

Conclusion

The presence of measurement, response, observational or other kind of errors in survey data is always a hindrance to appropriate conclusions from the analysis which may lead to wrong decisions in policy making. In this paper, it has shown that the presence of measurement and response errors in center sampling under regression estimation results in overestimating and underestimating the regression parameters. This has occurred both under one phase and two phase sampling process. This therefore calls for identification of error sources and their possible minimization so as to reduce their effects on the data in center sampling. This will improve the efficiency on The planning, forecasting as well policy formulation processes will be greatly improved if measurement and response errors are controlled in the data collection process.

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