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We are interested in determining whether the given utterance comes from a member of a given speaker group or an imposter. If it is the former, we are interested in determining the identity of the speaker. The only knowledge available is a set of known utterances from the given group of speakers. The given utterance is manually divided into phonemes without necessarily ascertaining the identity of phonemes. Using statistical decision theory, we will develop various types of tests for speaker verification and identification using only one phoneme segment or the entire utterance. We will consider related problems such as the methods of clustering speakers to aid speaker verification, the optimal choice of phonemes for speaker recognition. Next we consider the role of speaker variability in speech recognition and recognize its complementarity to the problem of optimal choice of phonemes for speaker recognition. We illustrate the efficacy of the various methods developed here by considering the speaker and speech identification problems with three speech data bases.