In linear discriminant analysis, the assumption of equality of the dispersion matrices of different classes leads to a classification rule based on minimum Mahalanobis distance from the class centres. However, without this assumption, the resulting quadratic discriminant classifier involves, in addition to the Mahalanobis distances, the ratio of the determinants of the dispersion matrices as a factor. In fact, it has been observed that, for discriminating between populations with underlying elliptically symmetric distributions, such classifiers also incorporate similar factors, apart from the Mahalanobis distances. In this paper, a nonparametric classification technique which generalizes discriminant analysis has been proposed. The method of cross-validation is used to make the technique adaptive to a given dataset. An extensive simulation study is presented to illustrate the potential of the method. Finally, through implementation on a number of real-life data sets, it has been demonstrated that the proposed generalized quadratic discriminant analysis (GQDA) compares very favourably with other nonparametric methods, and is computationally cost-effective. •A nonparametric classification technique has been proposed.•It generalizes Fisher׳s discriminant analysis.•Cross-validation is used to make the technique adaptive to given data.•Found empirically to compare very favourably with other nonparametric methods.•Computationally cost-effective.