Elliptical basis function (EBF) networks are introduced as a new nonparametric method of estimating probability density functions for process data. Unlike Parzen window density estimators that use identical hyperspherical basis functions, the EBF method uses elliptical basis functions adapted to the local character of the data. This technique overcomes the spikiness problem associated with Parzen windows, where in high dimension, they can fail to produce smooth probability density estimates. The EBF estimator produces valid density functions that converage to the underlying distribution of the data in the limit of an infinite number of training examples. A technique based on statistical cross validation is introduced for evaluating different density estimators. The criterion is a measure of how well the density estimator estimates the density of data not used in the training. The EBF density estimation method and the evaluation technique are demonstrated using several examples of fault diagnosis.