This paper aims to construct a valid and efficient confidence interval for the extrema of parameters under privacy protection. The usual statistical inference on the extrema of parameters often suffers from the selection bias issue, and the problem becomes more acute, as in many application scenarios of extrema parameters, we often need to protect the privacy of the data. In this paper, we focus on the exponential family of distributions and propose a privatized parametric bootstrap method to address selection bias in the extrema of parameters problem under the scheme of differential privacy. While the usual privatized parametric bootstrap does not address selection bias appropriately, we prove that with a privatized bias correction term, the proposed parametric bootstrap method can lead to a valid and efficient confidence interval for the extrema of parameters. We illustrate the proposed method with the Gaussian case and regression case and demonstrate the advantages of the proposed method via numerical experiments.

翻译：本文旨在构建隐私保护下参数极值的有效且高效的置信区间。通常对参数极值的统计推断常受选择性偏差问题困扰，而在参数极值的许多应用场景中，由于常需保护数据隐私，此问题变得更加严峻。本文聚焦于指数族分布，提出一种私有化参数自助法，以解决差分隐私框架下参数极值问题中的选择性偏差。尽管常规的私有化参数自助法无法恰当处理选择性偏差，但我们证明，通过加入私有化偏差修正项，所提出的参数自助法能够为参数极值构建有效且高效的置信区间。我们以高斯情形和回归情形为例说明所提方法，并通过数值实验展示其优势。