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Cepstral normalization has widely been used as a powerful approach to produce robust features for speech recognition. Good examples of this approach include cepstral mean subtraction, and cepstral mean and variance normalization, in which either the first or both the first and the second moments of the Mel-frequency cepstral coefficients (MFCCs) are normalized. In this paper, we propose the family of higher order cepstral moment normalization, in which the MFCC parameters are normalized with respect to a few moments of orders higher than 1 or 2. The basic idea is that the higher order moments are more dominated by samples with larger values, which are very likely the primary sources of the asymmetry and abnormal flatness or tail size of the parameter distributions. Normalization with respect to these moments therefore puts more emphasis on these signal components and constrains the distributions to be more symmetric with more reasonable flatness and tail size. The fundamental principles behind this approach are also analyzed and discussed based on the statistical properties of the distributions of the MFCC parameters. Experimental results based on the AURORA 2, AURORA 3, AURORA 4, and Resource Management (RM) testing environments show that with the proposed approach, recognition accuracy can be significantly and consistently improved for all types of noise and all SNR conditions.