Optimal allocation in estimation of population mean in double sampling for stratification in presence of non-response and measurement error

Introduction 1.1 Nonresponse and Measurement Errors

The presence of nonresponse and measurement errors in a survey contaminates data making analysis to result in under-estimated or over-estimated parameters. This leads to invalid statistical inferences that may have undesirable consequences in policymaking. In a survey, a researcher faces the problem of estimating the population mean and choosing the optimal stratum sample size that minimizes the variance for a specified cost in the presence of errors. In the literature authors who have studied nonresponse under the different sampling schemes include [1][2][3][4][5] and measurement errors include [6][7][8] . The precision of the statistic being estimated in a survey increases with the decrease in the variance. The variance of the statistic depends on the size of the strata which can be determined prior using the allocation method. The optimization technique is used in a survey to obtain a robust estimator of the population parameter for a fixed cost. Double sampling for stratification encounters a major drawback of determining the first phase and the second phase stratum sample sizes that give the desired precision for a specified cost. The problem of optimal allocation in the estimation of the population mean using the auxiliary variable in the presence of errors is not addressed in the literature. The aim of the present study is to use the information on a single auxiliary variable to propose estimators of the population mean in the presence of nonresponse and measurement errors. The expression of biases and mean square errors of the proposed estimators are obtained. The cost of the survey is studied theoretically. The optimum sample sizes and the value of the inverse sampling rate are derived.

Sampling Procedure

In double sampling for stratification, a heterogeneous population of size N is considered. A first phase sample of size 𝑛 ′ is drawn from the population using a simple random sampling without replacement design and the units classified into L homogeneous strata of size 𝑛 ℎ ′ each. The auxiliary variable is studied in the first phase sample. A second phase random sample of size 𝑛 ℎ is drawn from the first phase sample and both the survey variable and the auxiliary variable are studied.

The ℎ 𝑡ℎ stratum first phase sample mean of the auxiliary variable is given as 𝑥

. The second phase sample is divided into responding and nonresponding groups of sizes 𝑛 1ℎ and 𝑛 2ℎ respectively. A random sample of size 𝑟 2ℎ = 𝑛 2ℎ

𝑘 2ℎ

, where 𝑘 2ℎ > 1 is the inverse sampling rate, is drawn from the nonresponding group and used in the estimation of the population mean. Let (𝑦 ℎ𝑖 , 𝑥 ℎ𝑖 ) denote the survey variable and the auxiliary variable respectively. The population mean of the survey variable and the auxiliary variable in the ℎ 𝑡ℎ stratum is given as

respectively. The population mean of the survey variable and the auxiliary variable for the nonresponding units in the ℎ 𝑡ℎ stratum is given as 𝑌 ̅ 2ℎ = 𝑋 ̅ 2ℎ ) 2 . The estimator of the population mean proposed by [9] in the presence of nonresponse is extended to double sampling for stratification and is defined as 𝑦 ̅ ℎ * =

Similarly, the following can be obtained for the auxiliary variable

Since the first phase sample does not suffer from nonresponse and measurement errors,

Square both sides of equations ( 4), ( 5) and (6) , where 𝜃 ℎ ′ = (

) and 𝜃 ℎ * = 𝑊 2ℎ (𝑘 2ℎ -1) 𝑛 ℎ 𝐸(𝜎 𝑌ℎ ) = 𝐸(𝜎 𝑋ℎ ) = 𝐸(𝜎 𝑋1ℎ ) = 0

Bias and Mean Square Error of the Ratio Estimator

To obtain the expression of bias and mean square error of the ratio estimator substitute equations ( 4), ( 5) and ( 6) in 𝑇 𝑅 to obtain

Simplify the right-hand side in terms of power series ignoring terms of order greater than 2 and subtract the population mean to obtain

Take expectations on both sides of equation ( 7) to obtain

The approximation of bias is given as

To obtain the expression of mean square error, square equation ( 7) and ignore terms of order greater than 2

Simplify and take expectations on both sides to obtain the approximation of mean square error as

This can be further simplified into

Bias and Mean Square Error of the Exponential Ratio-ratio Type Estimator

To obtain the expression of bias and mean square error of the exponential ratio-ratio type estimator substitute equation ( 4), ( 5) and (6) in 𝑇 𝐸𝑅𝑅 to obtain

Simplify while ignoring terms of order greater than 2 and subtract the population mean from both sides to obtain

(8) Take expectations on both sides of equation ( 8) to obtain the approximation of bias as

To obtain the expression of mean square error, square both sides of equation ( 8) and ignore terms of order greater than 2

Simplify and take expectations on both sides to obtain 𝐸(𝑇 𝐸𝑅𝑅 -𝑌 ̅ ) 2

Bias and Mean Square Error of the Exponential Product-ratio Type Estimator

To obtain the expression of bias and mean square error of the exponential product-ratio type estimator substitute equation ( 4), ( 5) and (6) in 𝑇 𝐸𝑃𝑅 to obtain

Simplify the right-hand side in terms of power series ignoring terms of order greater than two and subtract the population mean from both sides to obtain

Take expectations on both sides of equation ( 9) to obtain the approximation of bias as

To obtain the expression of mean square error, square equation ( 9) and ignore terms of order greater than 2

Simplify and take expectations on both sides to obtain the approximation of mean square error as 𝐸(𝑇 𝐸𝑃𝑅 -𝑌 ̅ ) 2

Optimal Allocation in the Presence of Nonresponse and Measurement Error

Define the total cost of the survey in the ℎ 𝑡ℎ stratum as . Therefore the expected cost function is defined as

The expected total cost function is given as

To obtain the optimal allocation for the ratio estimator of the population mean define the Lagrangian function for optimization as

, where 𝜆 is a Lagrange's multiplier. The optimum values of 𝑛 ℎ ′ , 𝑛 ℎ and 𝑘 2ℎ are obtained by differentiating the Lagrangian function partially with respect to 𝑛 ℎ ′ , 𝑛 ℎ and 𝑘 2ℎ and equating the derivatives to zero as follows (11)

The optimum value of 𝑛 ℎ is obtained by substituting equations ( 10) and ( 11 The Lagrange's multiplier is obtained by substituting the values of 𝑛 ℎ ′ , 𝑛 ℎ and 𝑘 2ℎ * in the expected cost function and is defined as

The minimum mean square error of the ratio estimator is given as

To obtain the optimal allocation for the exponential ratio-ratio type estimator define the Lagrangian function for optimization as

, where 𝜆 is a Lagrange's multiplier. To obtain the optimum values of 𝑛 ℎ ′ , 𝑛 ℎ and 𝑘 2ℎ differentiate the Lagrangian function partially with respect to 𝑛 ℎ ′ , 𝑛 ℎ and 𝑘 2ℎ then equate the derivatives to zero To obtain the optimal allocation for the exponential product-ratio type estimator define the Lagrangian function for optimization as

, where 𝜆 is a Lagrange's multiplier.

Conclusion

The present study has proposed a ratio, exponential ratio-ratio type and exponential product-ratio type estimators of the population mean in the presence of nonresponse and measurement errors under double sampling for stratification. The expression of mean square errors and biases of the estimators have been derived. The cost of the survey has been studied theoretically. The optimum values of the sample sizes and the inverse sampling rate have been derived. It is noted that the optimum value of the sample size depends on the value of the Lagrange's multiplier. The mean square error of the estimators decreases with the increase in the cost of the survey. A high cost of the survey results in the selection of a large sample size.

References

1. Javid S, Sat G, Shakeel A. A generalized class of estimators under two-phase stratified sampling for non-response. Communications in Statistics-Theory and Methods, 2018. DOI:10.1080/03610926.2018.1481969.
2. Searls DT. The utilization of known coefficient of variation in the estimation procedure. Journal of American Statistical Association. 1964; 308:1225-1226.
3. Zahid E, Shabbir J. Estimation of population mean in the presence of measurement error and non-response under stratified random sampling. PLoS ONE. 2018; 2:e0191572. https://doi.org/10.1371/journal.pone.0191572.
4. Onyango OR, Ouma OC. Application of auxiliary variables in two-step semi-parametric multiple imputation procedure in estimation of population mean. International Journal of Science and Research (IJSR). 2017; 6: 77-83. DOI: 10.21275/ART20176978.
5. Cochran W. Sampling technique, Third Edition, John Wiley and Sons, New York, 1977.
6. Shalabh. Ratio method of estimation in the presence of measurement errors. Journal of Indian Society of Agricultural Statistics. 1997; 50:150-155.
7. Shukla D, Pathak S, Thakur N. An estimator for mean estimation in presence of measurement error. Research and Reviews: A Journal of Statistics. 2012; 1:1-8.54.
8. Singh H, Karpe N. Estimation of population variance using auxiliary information in the presence of measurement error. Statistics in Transition-New Series. 2008; 9:443-470.
9. Hansen M, Hurwitz W. The problem of non-response in sample surveys. Journal of American Statistical Association. 1946; 41:517-529.