Successive sampling from a changing population proportion over two successive occasions

Introduction

When population is subject to change a single survey will only yields information about the properties of that population on a given occasion. If the changes in population values are to examined the survey will have to be repeated on several occasions. Repeated sampling processes already in use are extended to estimate the current of the population proportion in the second occasions Successive sampling is devised to deal with such situation. Survey populations are liable to change over time owing to the entry of new units and /or loss of the previous ones. Suppose that of a population comprising units on the fist and the second occasions: 1. Continuing Members (N 1c ): members of population on the first occasion which continue to exist to second occasion. 2. New Members (N 2b ): new members of population which join the population in the time lapse between two occasions. 3. Dropout Members (N 1d ): members of population on the first occasion that dropout before the second occasion

Sample Design

Let a character under study be denoted by x and y on the first and second occasions respectively.

X hi , ( thi Y ) = values of survey binary variable associated with i th (i=1, 2…, N h ) unit of the h th (h=1c, 1d), (h=2c, 2b) subgroup on the first (second) occasion respectively.

Notation: = N1d = Number of members of the population on the first occasion that drop out before the second. N1c = Number of members of the population on the first occasion which continue to exist to the second. N2b = Number of new units which join the population in the time lapse between the two occasions.

N1 = N1d + N1c = Number of units in the population on the first occasion. N2 = N1c+ N2b = Number of units in the population on the second occasions. The of population on the t occasion.

 

The sub group variance of the sample proportion on the first and second occasion respectively.

The proportion of population on the first and second occasion respectively.

The correlation coefficient that describes the association of c X 1 and

 The association between the responses of the continuing members of the first stratum on the two occasions.

Throughout this article, we make the customary simplifying assumption that the variances of the subgroups of continuing members remain unchanged over time, i.e.

S S 

. Furthermore, we assume that the size of the population on the t th occasion (t=1, 2, 3...) and the size of the component subgroup is known. For simplicity we assume that the sizes are large so that the subsampling fractions are negligibly zero. We assume that stratified random sample of units is drawn without replacement from the N 1 members of the population on the first occasion, and their X values are observed. Let n 1c denote the number of the first sample members which continue to exist to the second occasion, then the selection procedure may be given for drawing a sample of n 1 (n 1 is usually different from n 2 i.e. ( ,...

) units from the N 2 members of the population on the second occasion:

-Select a simple random sample without replacement (SRSWOR) of n 1cm from the n 1c continuing members.

-Select a SRSWOR of n 2cu units from the N 1c -n 1c units which were not included in to the first sample to replace (either completely or partially) the n 1cu units not reselected. -Select a SRSWOR of n 2b from the N 2b news members to supplement. n 2c = n 1cm +n 2cu continuing members in the sample.

Based on the responses of the sample members on both occasions the following sub sample indices education poverty may be formed:  and its variance.

Estimation of Population proportion on the Second Occasions

Definition (1-1)

An estimator  is defined as a minimum variance unbiased estimator (mvue) of the parameter  if and only if:

for all values  , and 2. Among all estimators

Consider a generalized estimator c 2  of the population proportion of the sensitive characteristic on the second occasion or current occasion as

Where a, b, c and d are suitable constants its expressed is:

Then the condition of unbiased ness required that

To obtain values of aand c make ) ( 2c

V  a minimum we differentiate with respect to both a and c and set the partial derivatives equal to zero. The optimum values obtained are:

We derive and then obtain:

The mvue of C 2  and its variance are respectively:

We now proceed to derive the estimator of

Corollary (1-1)

We apply corollary (3-1) to the estimation of  ˆ.

Theorem (1-2)

Under the sample design of selection identifiable a mvue of population proportion on the second occasions is given by:

It's worth remarking here that results pertaining to the situations where the population is fixed (i.e. ), or when it contains no new entrants but some of the previous units case to exist (i.e.

Optimum Allocation of the second sample

The usual problem of choosing sample for more precise, we consider the allocation of the second sample optimum for measuring population proportion  ˆ, under this case, the sampling units are identifiable prior to the selection of the sample, then, the number of the first sample members which dropout in the time span between the two occasions, n 1d , and consequently n 1c

 

), will be known in advance thus given the variance expression (7) we are in a position to determine the optimum allocation of the second sample by obtaining an explicit mathematical statement of the best values of , to be under taken since the rest of the parameters involved are assumed known.

Theorem (1-3)

If under the sample identifiable the required estimator is w 2  , then, the minimum sampling variance is attained when:



And opt-short for optimum.

The unconstrained problem may now be solved by following the usual minimization procedures. Equating the partial derivatives of ) ( w v  with respect to n 1cm and n 2b to zero and solving gives the optimum solution:

As evident from 3-6, opt cm n .

1

does not depend on the continuing subgroup size or variance except at the boundary of the solution space these parameters (i.e.W 2c and S 1c ) play predominant role in determining the optimum number of units allocated to subgroup, whereas the number of members of the sample on the preceding occasion with continue to exist and the correlation between their responses on the two occasions give the number of matched units in the current sample.

The an optimally allocated cross sectional sample of size n in is drawn from the two subgroups comprising the population on the second occasion then the resulting estimator st 2  (St, stands for stratified) of 2  will have an optimum variance of form:

  

It may therefore be concluded that for obtaining a more precise estimator of the current proportion of a changing population successive sampling with some degree of overlap continues to be favorable approach.

Definition (1-2)

T and 2

T be two estimators of population parameter T based on two different samples of the same size: denote their respective variances by

T is more precise i.e. has smaller variance than 2 T then 1 ) , ( 2

Results and Discussions

Values of

0.00 0.20 0.40 0.60 0.80 0.99 0.00 𝜆 1𝑐𝑚𝑜𝑝𝑡 0.50 0.49 0.48 0.44 0.38 0.12 𝜆 2𝑏𝑜𝑝𝑡 0.00 0.00 0.00 0.00 0.00 0.00 gp 0.00 0.01 0.04 0.11 0.25 0.75 0.40 𝜆 1𝑐𝑚𝑜𝑝𝑡 0.30 0.30 0.29 0.27 0.23 0.07 𝜆 2𝑏𝑜𝑝𝑡 0.00 0.00 0.00 0.00 0.00 0.00 gp 0.00 0.01 0.03 0.07 0.15 0.45 0.80 𝜆 1𝑐𝑚𝑜𝑝𝑡 0.10 0.10 0.10 0.09 0.08 0.02 𝜆 2𝑏𝑜𝑝𝑡 0.00 0.00 0.00 0.00 0.00 0.00 gp 0.00 0.00 0.01 0.02 0.05 0.15 0.95 𝜆 1𝑐𝑚𝑜𝑝𝑡 0.03 0.00 0.00 0.00 0.00 0.00 𝜆 2𝑏𝑜𝑝𝑡 0.00 0.00 0.00 0.00 0.00 0.00 gp 0.00 0.00 0.00 0.01 0.03 0.08 Source: own calculations -When  tends to zero, the gain in precision compared with no overlap. Also become negligible however, large the values of c 1

 may be. Thus impels that under such situations the two estimators

Conclusions

To estimate the current population proportion over population change repeated sampling survey processes already in use are extended to cater for the two dimensional nonstationarity inherent in the population. The numbers of members of the sample on the preceding occasion which continue to exist and the correlation between their responses on the two occasions gives the number of overlap in the current sample; the correlation between values of the two occasions was the most influential on deciding the optimum allocation to be undertaken and achievable level of precision. The sampling scheme developed and the estimators drawn therefrom were tested empirically using different values of the design parameters, namely the correlation and the subgroup weights, this led to the main finding that optimum overlap proportion for estimating current population proportion 0.5.

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