The problem is to test independence in a contingency table. Specifically, suppose X1,..., Xn are i.i.d., where each Xi is cross-classified, so that Xi = (r,s) with probability pr,s, r = 1,..., R, s = 1,..., S. Under the full model, the pr,s vary freely, except they are nonnegative and sum to 1. Let pr· =

s pr,s and p·s =

r pr,s. The null hypothesis asserts pr,s = pr· p·s for all r and s. Determine the likelihood ratio test and its limiting null distribution.