The following statistics can be used to test the null hypothesis , where is a matrix of known coefficients. Under mild assumptions, each of the following statistics has an asymptotic chisquare distribution with degrees of freedom, where p is the rank of . Let be the maximum likelihood of under the null hypothesis ; that is,


where is the estimated covariance matrix, which can be the modelbased covariance matrix or the sandwich covariance matrix (see the section Robust Sandwich Variance Estimate for details).