On the power of private likelihood-ratio tests for goodness-of-fit in frequency tables

Privacy-protecting data analysis is a rising challenge in modern statistics, as the achievement of data confidentiality guarantees, which typically occur through a perturbation of the data, may determine a loss in the “statistical” utility of the data. In this paper, we consider the likelihood-ratio (LR) test for private goodness-of-fit in frequency tables, and study its large sample properties. Under the global (ε, δ)-differential privacy (DP) for frequency tables, with ε,δ ≥ 0 being parameters controlling the level of privacy against intruders, our main result is sharp large deviation principle (LDP) for the power of the LR test, providing a private version of the classical Bahadur–Rao LDP for the Multinomial LR test in the absence of perturbation. This is achieved through a novel sharp LDP for the sum of i.i.d. random vectors, which is of independent interest. The private Bahadur–Rao LDP brings out a critical quantity as a function of the sample size n, the dimension k of the table, and the DP parameters (ε, δ), which determines the loss of power in LR test due to the perturbation of the data, showing how n and k interact with the level of privacy, through ε and δ. This allows to control the (sample) cost of global (ε, δ)-DP, namely the additional sample size to recover the power of the Multinomial LR test. We validate empirically our results on both synthetic data and real data.