Likelihood-based Inference with Singular Information Matrix

We consider likelihood-based asymptotic inference for a p-dimensional parameter theta of an identifiable parametric model with singular information matrix of rank p - 1 at theta=theta* and likelihood differentiable up to a specific order. We derive the asymptotic distribution of the likelihood ratio test statistics for the simple null hypothesis that theta = theta* and of the maximum likelihood estimator (MLE) of theta when theta = theta*. We show that there exists a reparametrization such that the MLE of the last p - 1 components of theta converges at rate O-p(n(-1/2)). For the first component theta(1) of theta the rate of convergence depends on the order s of the first non-zero partial derivative of the log-likelihood with respect to theta(1) evaluated at theta*, When s is odd the rate of convergence of the MLE of theta(1) is O-p(n(-1/2s)). When s is even, the rate of convergence of the MLE of \theta(1)- theta(1)*\ is O-p(n(-1/2s)) and moreover, the asymptotic distribution of the sign of the MLE of theta(1) - theta(1)* is non-standard. When p = 1 it is determined by the sign of the sum of the residuals from the population least-squares regression of the (s + l)th derivative of the individual contributions to the log-likelihood on their derivatives of order s. For p>1, it is determined by a linear combination of the sum of residuals of a multivariate population least-squares regression involving partial and mixed derivatives of the log-likelihood of a specific order. Thus although the MLE of \theta(1) - theta(1)*\ has a uniform rate of convergence of O-p(n(-1/2s)), the uniform convergence rate for the MLE of theta(1) in suitable shrinking neighbourhoods of theta(1)* is only O-p(n(-1/(2s+2))).