  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.
