#### Score Statistics and Tests

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 .