\(\epsilon \) -local differential privacy (LDP) is a strict privacy definition used to protect data providers in a distributed statistical survey against an untrusted curator. The trade-off between privacy and utility as it relates to LDP has been well studied; however, the existing studies do not consider the issue of missing data, which is endemic in statistical surveys. One of the main reasons for data missingness is the privacy concern of data providers who believe that non-response is a way to protect their sensitive records. Protecting privacy and addressing non-response are inseparable. In this study, we examine the trade-off between privacy and data utility in locally private estimation problems for data containing non-responses. To evaluate the trade-off, we analyze minimax risk with three non-response models: missing completely at random (MCAR), missing not at random (MNAR), and missing at random (MAR). We derive the lower bounds of minimax risk, which measures the difficulty of an estimation problem, for locally private estimations with the three non-response models. Moreover, we provide order-optimal estimators for the MCAR and MAR models in the mean estimation problem. The results of these analyses imply that the effective sample size is \(\epsilon ^2(1-\gamma )^2n\) , where \(\gamma \) is the non-response rate, rather than n when we need to hide non-responses. If hiding non-responses is unnecessary, the effective sample size is \(\epsilon ^2(1-\gamma )n\) .
