To understand the general form of the score statistics, let be the vector of first partial derivatives of the log likelihood with respect to the parameter vector , and let be the matrix of second partial derivatives of the log likelihood with respect to . That is, is the gradient vector, and is the Hessian matrix. Let be either or the expected value of . Consider a null hypothesis . Let be the MLE of under . The chi-square score statistic for testing is defined by
It has an asymptotic distribution with r degrees of freedom under , where r is the number of restrictions imposed on by .