Efficient class of estimators for the estimation of the mean of a sensitive variable using multi-auxiliary variables

The paper proposed an additive optional randomized response technique model that improves upon Gjestvang and Singh (2009) model by effectively balancing respondents privacy protection and statistical estimation efficiency. The proposed model establishes an unbiased estimator of the population mean under both simple random sampling and probability proportional to size sampling schemes. The proposed model effectively balances the privacy protection with statistical efficiency – a key trade-off in survey design involving sensitive variable. For all values of scrambling parameters and sensitivity level, the proposed model recorded high gain in efficiency and the relative efficiency of the proposed model under both sampling scheme is greater than one. As sensitivity level increases, the relative gain in efficiency decreases which is in agreement with theoretical expectations. Nevertheless, even at high sensitivity level W = 0.9, the proposed model maintained acceptable efficiency and unbiasedness. The weighted privacy-efficiency measure established that proposed model out-performed Gjestvang and Singh (2009) model.

With the intention to control a true swapping between the efficiency and the privacy protection this paper introduces a scrambled randomized response (SRR) model to be alternative of Saha’s scrambling mechanism. The basic initiative is to provide an assortment of the additive, the subtractive and the multiplicative models. The simulation and the empirical studies are provided for various sample sizes to compare the efficiency of the proposed model. The results obtained from simulation showed that the proposed model performs better than Pollock and Bek’s additive model. Also, the proposed generalized estimator of mean has been studied using a new SRR model presented in this article and shown that the proposed estimator and its class of estimators are more efficient than existing estimators. It is also shown that gain in efficiency is more when the proposed SRR model is used. The efficiency of the proposed class of estimators over existing estimators using both models is also provided using real data and with a simulation study.

Extreme Ranked Set Sampling (ERSS) is a survey technique which seeks to improve the likelihood that collected sample data provides a good representation of the population and minimizes costs associated with obtaining them. The main goal of a statistical survey is to reduce sampling errors either by devising suitable sampling scheme or by formulating efficient estimator of the population parameters. In an attempt to address the problem of weak or loss of efficiency usually suffered in estimation of population mean under Simple Random Sampling (SRS), a class of ratio-cum-product estimators for population mean of the study variable Y is proposed based on ERSS using information on a single accompanying variable. Members of the proposed class of estimators were obtained by assigning various values to the scalars that helps in designing the estimators. These members were then transformed to a form that can be easily expanded using Taylor’s series approximation, from where various properties such as biases, relative biases, Mean Square Errors (MSEs), and Optimal Mean Square Errors (OMSEs) were derived under large sample approximation. Empirical study was conducted using three natural population data sets in order to investigate the performances and efficiency of the proposed classes of estimators under ERSS over its corresponding counterpart’s estimator based on SRS and some existing ratio and product estimators. This empirical study was followed up with a computer simulation study using R-software. The results revealed that the advocated class of estimators in ERSS produced smaller biases and MSEs which is an indicator of appreciable gain in efficiency and superiority over its corresponding counterpart estimator and some existing ratio type estimators in sample survey for all cases considered in this work and are therefore adjudged to provide a better alternative whenever efficiency is required.

This paper proposes new ratio and product type estimators for estimating the mean of finite population using information on multi-auxiliary variables. The Bias and Mean square error (M. S. E.) of these estimators are calculated. These estimators are compared for their precision with usual mean per unit, ratio and product estimators and are found to be more efficient in many practical situations.

In survey sampling, it is still a great challenge to estimate the proportion of a sensitive characteristic in a population with the assurance of protecting respondent anonymity. Efficiency and privacy protection are key challenges in classical randomized response methods, particularly when applied to sensitive traits. In a bid to overcome this challenge, we propose a more enhanced randomized response model that enhances respondent privacy protection as well as the accuracy of the estimates. Utilizing stratified random sample methods, our model generalizes the use of the pioneering work. Apart from providing unbiased estimators for the sensitive proportion, the proposed model introduces a better sensitivity measure. We demonstrate via theoretical proofs and numerical simulations that our scheme is more efficient and secure compared to existing methods. Our proposed estimator, in specific, has a notably higher percentage relative efficiency (PRE) compared to existing models for both simple and stratified random sampling for certain parameter values, with significant respondent privacy protection and estimating accuracy gains. It is found that, compared to existing methods, the proposed models offer respondents greater efficiency and better privacy protection. Due to these advancements, our approach is particularly beneficial for social and behavioral research where privacy protection is important.

Auxiliary variables correlated with the study variables have been identified to be useful in improving the efficiency of ratio, product and regression estimators both at planning and estimation stages. The existing regression-based estimators are functions of auxiliary variables which are sensitive to outliers. In this paper, a modified class of estimators is proposed using robust non-conventional measures of dispersion which are robust against outliers or extreme values. The properties (Biases and Mean Squared Errors (MSEs)) of the modified class of estimator were derived up to the first order of approximation using Taylor series approach. The empirical studies were conducted using stimulation to investigate the efficiency of the proposed estimators over the efficiency of the existing estimators. The results revealed that the proposed estimators have minimum MSEs and higher Percentage Relative Efficiencies (PREs) among all the competing estimators. These results implied that the proposed estimators are more efficient and can produce better estimate of the population mean compared to other existing estimators considered in the study. Therefore, it can be concluded that proposed estimators have better predictive power for estimating population mean when the study (interest) variables are characterized with outliers or extreme values.

The present thesis analyzes whether and - if so - under which conditions mergers result in merger-specific efficiency gains. The analysis concentrates on manufacturing firms in Europe that participate in horizontal mergers as either buyer or target in the years 2005 to 2014. The result of the present study is that mergers are idiosyncratic processes. Thus, the possibilities to define general conditions that predict merger-specific efficiency gains are limited. However, the results of the present study indicate that efficiency gains are possible as a direct consequence of a merger. Efficiency changes can be measured by a Total Factor Productivity (TFP) approach. Significant merger-specific efficiency gains are more likely for targets than for buyers. Moreover, mergers of firms that mainly operate in the same segment are likely to generate efficiency losses. Efficiency gains most likely result from reductions in material and labor costs, especially on a short- and mid-term perspective. The analysis of conditions that predict efficiency gains indicates that firm that announce the merger themselves are capable to generate efficiency gains in a short- and mid-term perspective. Furthermore, buyers that are mid-sized firms are more likely to generate efficiency gains than small or large buyers. Results also indicate that capital intense firms are likely to generate efficiency gains after a merger. The present study is structured as follows. Chapter 1 motivates the analysis of merger-specific efficiency gains. The definition of conditions that reasonably likely predict when and to which extent mergers will result in merger-specific efficiency gains, would improve the merger approval or denial process. Chapter 2 gives a literature review of some relevant empirical studies that analyzed merger-specific efficiency gains. None of the empirical studies have analyzed horizontal mergers of European firms in the manufacturing sector in the years 2005 to 2014. Thus, the present study contributes to the existing literature by analyzing efficiency gains from those mergers. Chapter 3 focuses on the identification of mergers. The merger term is defined according to the EC Merger Regulation and the Horizontal Merger Guidelines. The definition and the requirements of mergers according to legislation provides the framework of merger identification. Chapter 4 concentrates on the efficiency measurement methodology. Most empirical studies apply a Total Factor Productivity (TFP) approach to estimate efficiency. The TFP approach uses linear regression in combination with a control function approach. The estimation of coefficients is done by a General Method of Moments approach. The resulting efficiency estimates are used in the analysis of merger-specific efficiency gains in chapter 5. This analysis is done separately for buyers and targets by applying a Difference-In-Difference (DID) approach. Chapter 6 concentrates on an alternative approach to estimate efficiency, that is a Stochastic Frontier Analysis (SFA) approach. Comparable to the TFP approach, the SFA approach is a stochastic efficiency estimation methodology. In contrast to TFP, SFA estimates the production function as a frontier function instead of an average function. The frontier function allows to estimate efficiency in percent. Chapter 7 analyses the impact of different merger- and firm-specific characteristics on efficiency changes of buyers and targets. The analysis is based on a multiple regression, which is applied for short-, mid- and long-term efficiency changes of buyers and targets. Chapter 8 concludes.

In this paper, we have developed estimators of finite population mean using Mixture Regression estimators using multi-auxiliary variables and attributes in two-phase sampling and investigated its finite sample properties in full, partial and no information cases. An empirical study using natural data is given to compare the performance of the proposed estimators with the existing estimators that utilizes either auxiliary variables or attributes or both for finite population mean. The Mixture Regression estimators in full information case using multiple auxiliary variables and attributes are more efficient than mean per unit, Regression estimator using one auxiliary variable or attribute, Regression estimator using multiple auxiliary variable or attributes and Mixture Regression estimators in both partial and no information case in two-phase sampling. A Mixture Regression estimator in partial information case is more efficient than Mixture Regression estimators in no information case.

To estimate the population mean when sampling a heterogeneous population and in the absence of a priori information on auxiliary variables, exponential-ratio multivariate estimators are associated under double stratified sampling with two auxiliary variables. Their biases and mean square errors are expressed and simulated. These mean square errors are smaller (the efficiencies are higher) than those of the sample mean estimator and those of other ratio estimators when the correlation between the study and the auxiliary variables exceeds 0.1 in absolute value. In particular, the proposed estimators are more efficient for low correlations between the study and the auxiliary variables. The gain in efficiency reaches a factor of 230.4% on an empirical dataset where the study variable is weakly correlated with each of the two auxiliary variables, and 182.1% on another empirical dataset where it is strongly correlated.

Due to high convenience and efficiency, electronic auction technology has been developed rapidly and has been applied to many online trading market applications. As more attention has been paid to information security, the privacy issues in the electronic auction have been widely studied. Differential privacy, as a lightweight privacy protection method, is an important direction in privacy preserving auction mechanism designing. However, most of the existing researches on differential privacy-based auction mechanism have not proposed a theoretical privacy inference attack method against the auction market. Therefore, the existence of privacy attacks is questionable, and the necessity and privacy protection performance of the existing differential privacy auction mechanism cannot be verified. To this end, in this paper we addressed the privacy attack issue and privacy protection issue in the auction market simultaneously. First, a Bayesian-based inference attack method against the double auction market was proposed from the perspective of the adversary. Theoretical analysis and evaluation results showed that the proposed inference attack method can effectively infer the bidding information of the target bidders, and attack success rate achieved approximately 95%. Second, an individual differential privacy-based auction mechanism was proposed from the perspective of the auction platform. Since not all the bidders will be attacked, we introduced the concept of individual differential privacy to provide targeted defense for specific bidders. Theoretical analysis demonstrated that the proposed auction mechanism satisfies <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2\varepsilon $ </tex-math></inline-formula> -individual differential privacy. And the extensive evaluation results showed that, compared with the existing differential privacy-based auction mechanism, our proposed mechanism provided the best privacy protection performance, that is, reduced the attack success rate to 20%, and ensured better auction performance, such as social welfare and satisfaction ratio, than the other mechanisms. Note to Practitioners—In this paper, we addressed the non-invasive privacy issues in the widely used electronic auction mechanism. Most of the previous works focused on designing differential privacy based auction mechanism against the non-invasive privacy attack, but neglecting the principle of non-invasive privacy attack methods. This makes it impossible to verify the privacy protection effectiveness of their proposed mechanisms. For this reason, a very large privacy budget may be selected to ensure the efficiency of privacy protection, which will lead to poor auction performance. To this end, we first proposed a Bayesian-based inference attack method against the double auction market. The proposed inference attack method allows the adversary infer the bidders’ private bidding information by the public auction results. Moreover, we then proposed an individual differential privacy auction mechanism, which aimed to achieve effective privacy protection while minimizing the added noise, thereby improving auction performance. The experiments demonstrate that the proposed Bayesian-based inference attack method achieves a good attack successful rate, and the proposed individual differential privacy auction mechanism will achieve the better efficiency of privacy protection as well as auction performance comparing with the exist differential privacy-based auction mechanism. In conclusion, this paper provides a verification method for the future research on the privacy protection of electronic auction mechanism. Meanwhile, this paper proposes an efficient privacy protection auction mechanism, which can be used in various trading scenarios.

A randomized response model for sensitive attribute with privacy measure using Poisson distribution

A Concise Theory of Randomized Response Techniques for Privacy and Confidentiality ProtectionaaThe views expressed in this chapter are those of the authors and not necessarily those of the US Census Bureau. The analysis and conclusions contained in this chapter are those of the authors and do not represent the official position of the US Energy Information Administration (EIA) or the

Moving from the scrambling mechanism recently suggested by Saha [25], three scrambled randomized response (SRR) models are introduced with the intent to realize a right trade-off between efficiency and privacy protection. The models perturb the true response on the sensitive variable by resorting to the multiplicative and additive approaches in different ways. Some analytical and numerical comparisons of efficiency are performed to set up the conditions under which improvements upon Saha's model can be obtained and to quantify the efficiency gain. The use of auxiliary information is also discussed in a class of estimators for the sensitive mean under a generic randomization scheme. The class includes also the three proposed SRR models. Finally, some graphical comparisons are carried out from the double perspective of the accuracy in the estimates and respondents’ privacy protection.

In this paper, we propose a calibration estimator of population mean in stratified sampling using the known mean and variance information from multi-auxiliary variables. The problem of determining the optimum calibrated weights is formulated as a Nonlinear Programming Problem (NLPP) that is solved using the Lagrange multiplier technique. Numerical example with real data is presented to illustrate the computational details of the proposed estimator. A comparison study is also carried out using real and simulated data to evaluate the performance and the usefulness of the proposed estimator. The study reveals that the proposed estimator with multi-auxiliary information is more efficient estimator of the population mean as it provides least estimated variance and highest gain in relative efficiency (RE). References Jean Claude Deville and Carl Erik Sarndal. Calibration estimators in survey sampling. Journal of the American statistical Association, 87(418):376–382, 1992. doi:10.1080/01621459.1992.10475217 . Victor M Estevao and Carl Erik Sarndal. Survey estimates by calibration on complex auxiliary information. International Statistical Review, 74(2):127–147, 2006. doi:110.1111/j.1751-5823.2006.tb00165.x Patrick J Farrell and Sarjinder Singh. Model-assisted higher-order calibration of estimators of variance. Australian and New Zealand Journal of Statistics, 47(3):375–383, 2005. doi:10.1111/j.1467-842X.2005.00402.x Wolfram Research, Inc. Mathematica, Version 11.3. Champaign, IL, 2018. Jong Min Kim, Engin A Sungur, and Tae Young Heo. Calibration approach estimators in stratified sampling. Statistics and probability letters, 77(1):99–103, 2007. doi:10.1016/j.spl.2006.05.015 Phillip S Kott. Using calibration weighting to adjust for nonresponse and coverage errors. Survey Methodology, 32(2):133, 2006. Dinesh K Rao. Mathematical programing in stratified random sampling. PhD thesis, School of Computing, Information and Mathematical Sciences, The University of the South Pacific, Fiji, February 2017. Dinesh K. Rao, Tokaua. Tekabu, and Mohammad G M Khan. New calibration estimators in stratified sampling. In Proceedings of Asia-Pacific World Congress on Computer Science and Engineering, pages 66–70. IEEE, 2016. Gurmindar K Singh, Dinesh K Rao, and Mohammed GM Khan. Calibration estimator of population mean in stratified random sampling. In Proceedings of Asia-Pacific World Congress on Computer Science and Engineering (APWC on CSE), pages 1–5. IEEE, 2014. Sarjindar Singh, Stephen Horn, and Frank Yu. Estimation of variance of general regression estimator: Higher level calibration approach. Survey Methodology, 24:41–50, 1998. Sarjinder Singh. Advanced Sampling Theory With Applications: How Michael Selected Amy, volume I and II. Kluwer Academic Publishers, Netherlands, 2003. Sarjinder Singh. On the calibration of design weights using a displacement function. Metrika, 75(1):85–107, 2012. doi:10.1007/s00184-010-0316-6 Sarjinder Singh, Stephen Horn, Sadeq Chowdhury, and Frank Yu. Theory and methods: Calibration of the estimators of variance. Australian and New Zealand Journal of Statistics, 41(2):199–212, 1999. doi:10.1111/1467-842X.00074 D S Tracy, S Singh, and R Arnab. Note on calibration in stratified and double sampling. Survey Methodology, 29(1):99–104, 2003. Changbao Wu and Randy R Sitter. A model-calibration approach to using complete auxiliary information from survey data. Journal of the American Statistical Association, 96(453):185–193, 2001. doi:10.1198/016214501750333054

In this paper, we have considered stratified two-phase sampling design with sub-sampling the non-respondents in the presence of non-response for estimating population mean considering the information of two auxiliary variables. The proposed estimators are the exponential function of two auxiliary variables when means of the two auxiliary variables are not known in prior. Further the proposed estimators are provided with their generalized form. The bias and mean square error expressions of the proposed estimators have been derived in two different cases of non-response. The conditions for which proposed estimators are more efficient as compared to some other estimators, have also been discussed in each case of non-response. It is shown that the proposed estimators are more efficient as compared to Hansen and Hurwitz (J Am Stat Assoc 41:517–529, 1946) unbiased estimator and Tabasum and Khan (Assam Stat Rev 20(1):73–83, 2006) two-phase ratio and product estimators modified to the stratified sampling. An empirical study has also been carried out to demonstrate the performances of the estimators.