Calibration approach for finite population parameter estimation

1. SOME IMPROVED METHODS OF ESTIMATION FOR FINITE
POPULATION PARAMETERS
Presentation
(For JRF to SRF Upgradation)
Under the Supervision of:- Submitted by:-
Dr. M. K. Chaudhary Basant Kumar Ray
(Associate Professor) (Research Scholar)
Department of Statistics
Institute of Science
Banaras Hindu University
Varanasi – 221005

2. Introduction
 The auxiliary information can be used in many ways to obtain an
improved estimator of an unknown finite population parameter in
survey sampling. Some well-known techniques that use auxiliary
information are the ratio method of estimation, product method of
estimation, regression method of estimation, combined ratio-type
of estimation, unbiased ratio-type of estimation, etc. These
techniques incorporate auxiliary information to improve the
precision of the estimation procedure. In our study, we have used
calibration approach to incorporate the auxiliary information and
find the improved estimators for the finite population parameters
(such as mean, total, variance, etc).

3.  What is Calibration Approach:-The calibration estimation approach
is a reweighting technique that incorporate the auxiliary information
in order to find the improved estimators for the finite population
parameters.
 The first question is "how to incorporate the auxiliary variable into
the estimation process." The answer to this question is "by using
some constraints based on the auxiliary variable."
 The calibration approach produces a set of optimum weights, and
these calibrated weights are supposed as close as possible to the
design weights. The first requirement for the calibration approach is
"the calibrated weights must be as close as possible to the design
weights." The first requirement is satisfied by taking a distance
function (such as chi-square type distance, modified chi-square type
distance function, Hellinger type distance function, etc.).”

4. This distance function gives the distance between calibrated weights and
design weights. In our study, we have considered only the chi-square type
distance function to find the expression of calibrated weights in explicit
form and reduce the complexity of the computation. For our purpose, we
minimize the appropriate distance function mathematically with respect to
calibrated weights.
The calibrated weights are such that it gives the exactly known
population characteristics (such as mean, total, variance, etc.) when applied
to the sample values of auxiliary variables.
The other interesting question is, "Why are we using the calibration
approach only and not the other methods." The answer is "because of its
benefits." Some of its merits are listed below:-

5. Some Merits of the Calibration Approach:
 The calibrated weights are such that these give the exactly known
population characteristics (such as mean, total, variance etc) when
applied to the sample auxiliary variable. Since calibrated weights are
giving an exact estimate of the population characteristics for auxiliary
variables and auxiliary variables are strongly correlated with the
study variable, it should also work well for the study variable.
 Traditional methods use response models in the non-response
estimation process. There is no need for a response model in the
calibration approach.
 Traditional methods are sometimes complex, time-consuming, and
inconvenient. The calibration approach method is a simple, effective,
and convenient method for the estimation of the finite population
parameters.

6.  The calibration approach involves easy computation for calibrated
weights.
 Statistical organizations of some countries have developed software to
compute the calibrated weights like GES (Statistics Canada’s general
software, CLAN (Statistics Sweden).
Some Demerits of the Calibration Approach:
 Sometimes we get negative calibrated weights for the chi-square type
distance function. There is no sense of these negative calibrated weights
in some situations.
 Sometimes there do not always exist solutions for some distance
functions. There always exist solutions for chi-square distance function
and modified minimum entropy distance function.
 For some distance functions, we cannot obtain a linear form or closed-
form expression for the calibrated weights. For these types of expression,
it is not easy to further treatment.

7. Literature Review
The first attempt for the calibration estimation approach was made by
Deville and Särndal (1992) in the survey sampling. Singh et al. (1998)
proposed a calibration estimator for population mean in stratified
random sampling by using one constraint based on a single auxiliary
variable. Tracy et al. (2003) proposed a calibration estimator for
population mean in stratified random sampling using two constraints
based on a single auxiliary variable. Rao et al. (2012) used the
calibration approach to find the calibration estimator of the population
mean in stratified random sampling using two constraints based on two
auxiliary variables.

8. Nidhi et al. (2017) proposed calibration estimators of population mean in
stratified random sampling and stratified double sampling. Some other
authors who worked on the calibration approach are Estevao and Säarndal
(2006), Kim (2007), Särndal, C. E. (2007), Kim and Park (2010), Koyuncu
and Kadilar (2014), Clement and Enang (2015), Koyuncu and Kadilar
(2016) etc.
Several attempts were made by researchers to deal with the problem
of non-response. The first effort was made by Hansen and Hurwitz (1946).
Qasim (2014) used the calibration approach to estimate the population total
in presence non-response in simple random sampling and pareto sampling.
Some other names of authors working on non-response problem are Khare
and Srivastava (1993), Lundström and Särndal (1999), Chang and Kott
(2007), Chaudhary et al. (2014), Andersson and Särndal (2016), Chaudhary
et al. (2020) etc.

9. Objectives of the Present Study:-
The objectives of the present study are
 To propose efficient calibration estimators for the finite population
mean in stratified sampling and stratified double sampling motivated
by Nidhi et al. (2017). For this purpose, we have used the chi-square
type distance function and two calibration constraints for each
stratum. Empirical study has been also performed to check efficiency
of proposed calibration estimators.
 To propose efficient calibration estimators for the finite population
mean in stratified sampling under the presence of non-response
motivated by Dykes et al. (2015) and Qasim (2014). For this purpose,
we have used the chi-square type distance function and some
calibration constraints. Empirical studies have been also performed to
check efficiency of proposed calibration estimators.

10.  To propose calibration estimators for finite population mean in
stratified sampling by improving the efficiency of the Hansen and
Hurwitz's (1946) estimator when some units don't respond.
Empirical studies have been also performed to check efficiency of
proposed calibration estimators.
 To propose generalized type of calibration estimators for the finite
population mean in stratified sampling and stratified double
sampling. We have used the chi-square type distance function and n
calibration constraints in this study.
 To propose efficient calibration estimators for the finite population
variance in stratified sampling using different calibration constraints
and chi-square type distance function. Empirical studies have been
also performed to check efficiency of proposed calibration
estimators.

11. Research Paper Accepted for Publication:
 Accepted a research paper entitled, “A Calibration Based Approach on
Estimation of Mean of a Stratified Population in the Presence of Non-
Response'' in Journal “Communications in Statistics – Theory and
Methods”.
Published Research Papers:
 Chaudhary, M. K., Prajapati, A., & Ray, B. K. (2021). Some convex-
type classes of estimators of finite population mean under random
non-response. Journal of Information and Optimization
Sciences, 42(8), 1951-1965.
 Chaudhary, M. K., Dutta, T., & Ray, B. K. (2021). A New Calibration
Estimator of Population Mean in Two-Stage Sampling Design using
Population Level Auxiliary Information. International Journal of
Statistics and Reliability Engineering, 8(1), 112-120.

12. Conference Attended:
 Presented a research paper entitled, “Calibration Approach for
Estimating the Mean of a Stratified Population in the Presence of
Non-response '' in conference RASTA-2021 held at Department of
Statistics, Institute of Science, Banaras Hindu University, during
December 15-17, 2021.
 Presented a research paper entitled, “Calibration approach for
estimating the population mean under stratified single/ two phase
sampling scheme'' in conference MMA-2022 held at Department of
Mathematics, Institute of Science, Banaras Hindu University, during
29-30 January 2022.

13. References
[1] Andersson, P. G., & Särndal, C. E. (2016). Calibration for non-response
treatment using auxiliary information at different levels. In The Fifth
International Conference on Establishment Surveys (ICES-V), Geneva,
Switzerland, June 20–23, 2016.
[2] Chang, T., & Kott, P. S. (2007). Using Calibration Weighting to Adjust
for Nonresponse Under a Plausible Model (with full appendices) (No.
1496-2016-130586).
[3] Chaudhary, M. K., Kumar, A., Vishwakarma, G. K., & Kadilar, C. (2020).
Family of combined-type estimators for population mean using stratified
two-phase sampling scheme under non-response. Journal of Statistics and
Management Systems, 1-14.
[4] Chaudhary, M. K., Prajapati, A., & Singh, R. (2014). Two-phase sampling
in estimation of population mean in the presence of non-response. Infinite
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14. [5] Dykes, L., Singh, S., A. Sedory, S., & Louis, V. (2015). Calibrated
estimators of population mean for a mail survey
design. Communications in Statistics-Theory and Methods, 44(16),
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[6] Khare, B. B., & Srivastava, S. (1993). Estimation of population mean
using auxiliary character in presence of non-response. National
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[7] Kim, J. M., Sungur, E. A., & Heo, T. Y. (2007). Calibration approach
estimators in stratified sampling. Statistics & probability letters, 77(1),
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[8] Koyuncu, N., & Kadilar, C. (2014). A new calibration estimator in
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[9] Lundström, S., & Särndal, C. E. (1999). Calibration as a standard
method for treatment of nonresponse. Journal of official
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15. [10] Nidhi, Sisodia, B. V. S., Singh, S., & Singh, S. K. (2017).
Calibration approach estimation of the mean in stratified sampling
and stratified double sampling. Communications in Statistics-Theory
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[11] Qasim, M. (2014). Calibration Estimation under Nonresponse
based on Simple Random Sampling Vs Pareto Sampling.
[12] Reddy, M. K., Rao, K. R., & Boiroju, N. K. (2010). Comparison of
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[14] Särndal, C. E., & Lundström, S. (2005). Estimation in surveys with
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[15] Singh S., Horn S., Yu F. (1998). Estimation variance of general
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[16] Singh, S. (2003). Advanced Sampling Theory With Applications:
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