Chapter 8 - IIT Kanpurhome.iitk.ac.in/~shalab/sampling/WordFiles-Sampling/... · Web viewDouble Sampling (Two Phase Sampling) The ratio and regression methods of estimation require

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Chapter 8

Double Sampling (Two Phase Sampling)

The ratio and regression methods of estimation require the knowledge of population mean of an

auxiliary variable to estimate the population mean of study variable If the information on the

auxiliary variable is not available, then there are two options – one option is to collect a sample only on

study variable, and use sample mean as an estimator of the population mean.

An alternative solution is to use a part of the budget for collecting information on an auxiliary variable

to collect a large preliminary sample in which alone is measured. The purpose of this sampling is to

furnish a good estimate of . This method is appropriate when the information about is on file

cards that have not been tabulated. After collecting a large preliminary sample of size units from the

population, select a smaller sample of size from it and collect the information on . These two

estimates are then used to obtain an estimator of the population mean This procedure of selecting a

large sample for collecting information on auxiliary variable and then selecting a sub-sample from it

for collecting the information on the study variable y is called double sampling or two-phase sampling.

It is useful when it is considerably cheaper and quicker to collect data on than and there is a high

correlation between and

In this sampling, the randomization is done twice. First, a random sample of size is drawn from a

population of size and then again a random sample of size is drawn from the first sample of size

.

So the sample mean in this sampling is a function of the two phases of sampling. If SRSWOR is

utilized to draw the samples at both the phases, then the

- number of possible samples at the first phase when a sample of size is drawn from a

population of size is , say.

- number of possible samples at the second phase where a sample of size is drawn from the

first phase sample of size is , say.Sampling Theory| Chapter 8 | Double Sampling (Two Phase Sampling) | Shalabh, IIT Kanpur

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Population of X (N units)

Sample (Large) units

Subsample(small)n units

0 samples M

1 samples M

Then the sample mean is a function of two variables. If is the statistic calculated at the second phase

such that being the probability that ith sample is chosen at the

first phase and jth sample is chosen at the second phase, then

where denotes the expectation over the second phase and denotes the expectation over the

first phase. Thus

Sampling Theory| Chapter 8 | Double Sampling (Two Phase Sampling) | Shalabh, IIT Kanpur

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Variance of

Note: The two-phase sampling can be extended to more than two phases depending upon the need and

objective of the experiment. Various expectations can also be extended on similar lines.

Double sampling in ratio method of estimation

If the population mean is not known, then the double sampling technique is applied. Take a large

initial sample of size by SRSWOR to estimate the population mean as

.

Then a second sample is a subsample of size selected from the initial sample by SRSWOR. Let

be the means of and based on the subsample. Then

The ratio estimator under double sampling now becomes

.

The exact expressions for the bias and mean squared error of are difficult to derive. So we find

their approximate expressions using the same approach mentioned while describing the ratio method of

estimation.


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Let

where is the sample mean of based on the sample size


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where is the mean sum of squares of based on an initial sample of size

where is the variance of mean of based on an initial sample of size .

Estimation error of

Write

up to the terms of order two. Other terms of degree higher than two are assumed to be negligible.


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Bias of

The bias is negligible if is large and relative bias vanishes if i.e., the regression line passes through the origin.

MSE of

The second term is the contribution of the second phase of sampling. This method is preferred over the

ratio method if


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Choice of and Write

where contain all the terms containing and respectively.

The cost function is where and are the costs per unit for selecting the samples

and respectively.

Now we find the optimum sample sizes and for fixed cost The Lagrangian function is

Thus

and so


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Comparison with SRS

If is ignored and all resources are used to estimate , then required sample size =

Double sampling in regression method of estimation

When the population mean of the auxiliary variable is not known, then double sampling is used as

follows:

- A large sample of size is taken from of the population by SRSWOR from which the

population mean is estimated as , i.e.

- Then a subsample of size is chosen from the larger sample and both the variables and

are measured from it by taking in place of and treat it as if it is known.

Then . The regression estimate of in this case is given by

where based on the sample of size .

It is difficult to find the exact properties like bias and mean squared error of , so we derive the

approximate expressions.


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Let

Define

Estimation error:Then

Retaining the powers of up to order two assuming (using the same concept as detailed in

the case of ratio method of estimation)


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Bias:

Mean squared error:

Retaining the powers of up to order two, the mean squared error up to the second order of

approximation is


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Clearly, is more efficient than the sample mean SRS, i.e. when no auxiliary variable is used.

Now we address the issue of whether the reduction in variability is worth the extra expenditure

required to observe the auxiliary variable.

Let the total cost of the survey is

where are the costs per unit observing the study variable and auxiliary variable ,

respectively.


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Now minimize the for fixed cost using the Lagrangian function with Lagrangian

multiplier as

Thus

and .

Substituting these values in the cost function, we have

Thus the optimum values of are

The optimum mean squared error of is obtained by substituting as


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The optimum variance of under SRS for SRS where no auxiliary information is used is

which is obtained by substituting in . The relative efficiency is

Thus the double sampling in regression estimator will lead to gain in precision if

.

Double sampling for probability proportional to size estimation:

Suppose it is desired to select the sample with probability proportional to an auxiliary variable but

information on is not available. Then, in this situation, the double sampling can be used. An initial

sample of size is selected with SRSWOR from a population of size and information on is

collected for this sample. Then a second sample of size is selected with replacement and with

probability proportional to from the initial sample of size . Let denote the mean of for the

initial sample of size , Let and denote means respectively of and for the second sample

of size . Then we have the following theorem.


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Theorem:

(1) An unbiased estimator of the population mean is given as

where denotes the total for in the first sample.

(2) and denote the totals of

and respectively in the population.

(3) An unbiased estimator of the variance of is given by

Proof. Before deriving the results, we first mention the following result proved in varying probability

scheme sampling.

Result: In sampling with varying probability scheme for drawing a sample of size from a population

of size and with replacement.

(i) is an unbiased estimator of the population mean being the

probability of selection of ith unit. Note that and can take any one of the N values

with initial probabilities respectively.

(ii) .

(iii) An unbiased estimator of the variance of is


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.

Let denote the expectation of , when the first sample is fixed. The second is selected with

probability proportional to , hence using the result (i) with we find that

where is the mean of for the first sample. Hence

which proves the part (1) of the theorem. Further,

Now, using the result (ii), we get

and hence


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using the probability of a specified pair of units being selected in the sample is So we can

express

Substituting this in , we get

This proves the second part (2) of the theorem.

We now consider the estimation of Given the first sample, we obtain

where Also, given the first sample,

Hence

Substituting

Using


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we get

which further simplifies to

where is the mean sum of squares of for the first sample. Thus, we obtain

which gives an unbiased estimator of Next, since we have

and from this result we obtain

Thus

when gives an unbiased estimator of


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Using (1) and (2) an unbiased estimator of the variance of is obtained as

Thus, the theorem is proved.


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Chapter 8 - IIT Kanpurhome.iitk.ac.in/~shalab/sampling/WordFiles-Sampling/... · Web viewDouble Sampling (Two Phase Sampling) The ratio and regression methods of estimation require