The paper proposes a new maintenance model for a repairable system. The system is inspected at deterministic times. If a failure occurs between two successive inspections, a minimal repair is performed. At inspection times, two types of maintenance actions can be performed. The first originality of the paper is to consider that the choice between these two types of maintenance depends on the fact that there has been a failure or not since the previous inspection. If no failure has occurred since the last inspection, the maintenance, simply denoted PM, corresponds to a basic preventive maintenance action, whose effect is planned by the initial maintenance plan. If at least one failure has occurred since the last inspection, the maintenance, denoted PCM, includes both the preventive action planned by the initial maintenance plan, and a specific corrective action which aims to fix the defects that have happened since the last inspection. Both types of maintenances are assumed to be imperfect, of the ARA (Arithmetic Reduction of Age) type, with different parameters. Moreover, the paper derives several statistical inference methods for this model. The usual case of complete data is first considered. The second originality of the paper is to analyse interval-censored data in this imperfect maintenance framework. This situation holds when failure times are not observed, and the only information is the fact that failures have occurred or not between two inspections. The quality of the proposed methods is assessed on simulated data. Finally, a real dataset from a gas transmission system operator is provided and analysed with the proposed approach.