Rapid advances in speech recognition theory, as well as computing hardware, have led to the development of machines that can take human speech as input, decode the information content of the speech, and respond accordingly. Real-time performance of such systems is often dominated by the evaluation of likelihoods in the statistical modeling component of the system. Statistical models are typically implemented using Gaussian mixture distributions. The primary objective of this thesis was to develop an extension of the Bucket Box Intersection algorithm in which the dimension with the optimal number of splits can be selected when multiple minima are present. The effects of normalization of mixture weights and Gaussian clipping have also been investigated. We show that the Extended BBI algorithm (EBBI) reduces run-time by 21% without introducing any approximation error. EBBI also produced a 12% lower word error rate than Gaussian clipping for the same computational complexity. These approaches were evaluated on a wide variety of tasks including conversational speech.