This paper treats an aspect of the learning or estimation phase of statistical pattern recognition (and adaptive statistical decision making in general). Simple mathematical expressions are derived for the improvement in supervised learning provided by additional nonsupervised learning when the number of learning samples is large so that asymptotic approximations are appropriate. The paper consists largely of the examination of a specific example, but, as is briefly discussed, the same procedure can be applied to other parametric problems and generalization to nonparametric problems seems possible. The example treated has the additional interesting aspect that the data does not have structure that would enable the machine to learn in the nonsupervised mode alone; but the additional nonsupervised learning can provide substantial improvement over the results obtainable by supervised learning alone. A second purpose of the paper is to suggest that a new fruitful area of research is the analytical study of the possible benefits of combining supervised and nonsupervised learning.