Local background covariance matrix corruption due to outliers in the sample data may be one of the major causes that limit detection performance of those algorithms that detect local anomalies in hyperspectral images on the basis of the Mahalanobis distance. In this letter, an original detection scheme is presented that efficiently embeds covariance corruption mitigation. A kurtosis-based binary hypothesis test is first applied to each pixel to quickly determine the presence of outliers in the local neighborhood. Rejection of the null hypothesis triggers application of a robust-to-outlier covariance estimation technique. Results on real data exhibit good detection performance and robustness to outliers. Contrary to previous works, this is achieved without an unnecessary increase of the procedural complexity.