Confidence Ellipsoids of a Multivariate Normal Mean Vector Based on Noise Perturbed and Synthetic Data with Applications

Abstract

In this paper we address the problem of constructing a confidence ellipsoid of a multivariate normal mean vector based on a random sample from it. The central issue at hand is the sensitivity of the original data and hence the data cannot be directly used/analyzed. We consider a few perturbations of the original data, namely, noise addition and creation of synthetic data based on the plug-in sampling (PIS) method and the posterior predictive sampling (PPS) method. We review some theoretical results under PIS and PPS which are already available based on both frequentist and Bayesian analysis (Klein and Sinha, 2015, 2016; Guin et al., 2023) and derive the necessary results under noise addition. A theoretical comparison of all the methods based on expected volumes of the confidence ellipsoids is provided. A measure of privacy protection (PP) is discussed and its formulas under PIS, PPS and noise addition are derived and the different methods are compared based on PP. Applications include analysis of two multivariate datasets. The first dataset, with p = 2, is obtained from the latest Annual Social and Economic Supplement (ASEC) conducted by the US Census Bureau in 2023. The second dataset, with p = 3, pertains to renal variables obtained from the book by Harris and Boyd (1995). Using a synthetic version of the original data generated through PIS and PPS methods and also the noise added data, we produce and display the confidence ellipsoids for the unknown mean vector under various scenarios. Finally, the privacy protection measure is evaluated for various methods and different features.