This paper is concerned with a design method of a failure diagnosis system for a linear discrete-time system with a parametric failure whose variation and its occurrence time cannot be estimated in advance. The following diagnosis system is proposed: i) A normal mode filter (Kalman filter) feeds the information of its innovation sequence to a detector. ii) The detector which uses the WSSR (Weighted Sum-Squared Residual) test detects anomaly states affected by the parametric failure. iii) An adaptive extended Kalman filter is constructed on the basis of a hypothesis test for parameter estimates, and simultaneously estimates unknown parameters and the states under the failure mode. iv) A discriminator which employs the cost function of approximate local sensitivities discriminates a failure parameter. Numerical results for a simple plant model illustrate the effectiveness of the proposed failure diagnosis system.