In practice, a hybrid exponentially weighted moving average (HEWMA) control chart for monitoring the process mean based on the normality assumption. The performance of control chart is seriously affected if quality characteristics depart from normality or have the presence of outliers. For such situations, a new robust-HEWMA control chart is proposed for the scenario in which quality characteristics being monitored follow symmetric not necessarily normal as well as skewed phenomena. The generalized least-squares (GLS) algorithm based on order statistics is integrated to determine the control limits for non-normal process. Consequently, the process mean is unbiased and has the minimum variance in spite of non-normality. Moreover, for the long-tailed symmetric and skewed processes, the GLS estimators through their systematic coefficients structures provide robust estimates. To utilize this advantage, we have investigated the impact of data contamination and the estimators used in Phase I on the performance of proposed robust HEWMA control chart in the Phase II. Finally, we provide two real-life examples along with the simulation studies to show the implantation of our proposed chart.

This study introduces a pioneering scrambling response model tailored for handling sensitive variables. Subsequently, a generalized estimator for variance estimation, relying on two auxiliary information sources, is developed following this novel model. Analytical expressions for bias, mean square error, and minimum mean square error are meticulously derived up to the first order of approximation, shedding light on the estimator’s statistical performance. Comprehensive simulation experiments and empirical analysis unveil compelling results. The proposed generalized estimator, operating under both scrambling response models, consistently exhibits minimal mean square error, surpassing existing estimation techniques. Furthermore, this study evaluates the level of privacy protection afforded to respondents using this model, employing a robust framework of simulations and empirical studies.

A flexible scrambled response model using a randomization device for quantitative sensitive data is used to evaluate the protection of respondents’ privacy. A double-sampling regression-cum-exponential estimator is used to estimate the mean of a sensitive variable using the mean of a nonsensitive auxiliary variable under scrambled response. The expected bias, the expected mean square error, and the minimum mean square error of this exponential-type estimator are expressed. Simulations and empirical results show that the proposed estimator under scrambled response model has a lower mean square error and a lower bias than the ratio and the exponential estimators.

Sousa et al. and Gupta et al. presented ratio and regression estimators of population mean of a sensitive variable using auxiliary information in simple random sampling without replacement. In this article, we propose a generalized randomized response technique (RRT) model and use it to develop some exponential estimators in two-phase sampling. We also discuss the privacy protection level of the proposed RRT model. Theoretical and empirical results are presented to examine the performance of the proposed mean estimators in relation to other mean estimators.

Sousa et al. [Improved mean estimation of a sensitive variable using auxiliary information in stratified sampling. J Stat Manage Syst. 2014;17(5–6):503–518] presented a ratio estimator for mean estimation in stratified sampling using an additive scrambled response model for a sensitive variable. In order to improve the estimation of mean, this study is motivated to introduce two difference-cum-exponential ratio estimators for mean estimation of a sensitive variable in stratified sampling. The theoretical discussion is presented to show that the two proposed estimators are more efficient than the available estimators including Sousa et al. [Improved mean estimation of a sensitive variable using auxiliary information in stratified sampling. J Stat Manage Syst. 2014;17(5–6):503–518] ratio and regression estimator in certain situations. A real-life application and simulation studies are presented to express the performance of the proposed estimator and existing estimators. The two studies provide evidence that the two proposed estimators perform more efficiently for mean estimation than the other available estimators.

In this article, we propose new efficient and more generalized difference-cum-exponential type estimator and generalized-difference-cum-generalized exponential type estimators for estimating the mean of sensitivity variable using the auxiliary information. We also discuss theoretically that proposed generalized estimators are more efficient than Sousa et al. ( 2010 Sousa, R. , J. Shabbir, P. C. Real, and S. Gupta. 2010. Ratio estimation of the mean of a sensitive variable in the presence of auxiliary information. Journal of Statistical Theory and Practice 4 (3):495–507. [Taylor & Francis Online] , [Google Scholar] ), Gupta et al. ( 2012 Gupta, S. , J. Shabbir, R. Sousa, and P. C. Real. 2012. Estimation of the mean of a sensitive variable in the presence of auxiliary information. Communications in Statistics-Theory and Methods 41:1–12. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar] ) and Koyuncu, Gupta, and Sousa ( 2014 Koyuncu, N. , S. Gupta, and R. Sousa. 2014. Exponential-type estimators of the mean of a sensitive variable in the presence of non sensitive auxiliary information. Communications in Statistics-Simulation and Computation 43 (7):1583–94. doi: 10.1080/03610918.2012.737492. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar] ) estimators. Results from a real life application and simulation study are presented to demonstrate the performance of the proposed mean estimators in relation to some of the existing mean estimators.