The growing attention paid to industrial condition-based maintenance during the last decade has increased the interest in incipient fault detection and diagnosis of complex systems. For these types of faults leading to slight changes in an early stage, the detection and isolation are subtle and can easily be covered by noise. It is then a challenging problem in fault diagnosis to obtain sufficiently sensitive, accurate, and robust techniques. This paper proposes a sensitive fault diagnosis method referred to as Local Mahalanobis Distance Analysis (LMDA) for incipient fault detection in multivariate nonlinear systems. In this method, we define the local Mahalanobis distance to recognize outliers in an unknown distribution. To speed up its computation, which depends on the sample size, we propose a distance-based down-sampling algorithm that can remove redundancy information from samples and hence reduce sample size. Based on this operation, a local Mahalanobis distance signal can be computed with acceptable time consumption for online monitoring. Considering the difficulty of detecting incipient faults, we finally use the LMD envelope as the discriminate value in the detection procedure to improve the detection sensitivity. A case study using the Continuous-flow Stirred Tank Reactor (CSTR) is proposed to check and validate the proposed methodology's effectiveness. The performances evaluated in terms of detection delay, false alarm rate, missed detection rate, and area under the receiver operating characteristic curve (AUC) show that our proposal outperforms state-of-the-art-based solutions in sensitivity, accuracy, and robustness.