Inference in the indeterminate parameters problem

We face an indeterminate parameters problem when there are two sets of parameters, x and g, say, such that the null hypothesis H0:x=x0 makes the likelihood independent of g. A consequence of indeterminacy is the singularity of the information matrix. For this problem the standard results, such as the asymptotic chi-squared distribution of the Wald test statistic, are generally false. In the paper we propose an estimator of the parameters of interest, x, so that a Wald-type test statistic can be used for testing H0. Such an estimator is obtained through the maximization of a modified (penalized) log-likelihood function. We show that a solution to the (penalized) likelihood equation is consistent and asymptotically normally distributed with variance-covariance matrix approximated by the Moore-Penrose pseudoinverse of the information matrix. These properties allow one to construct a Wald-type test statistic useful for inferential purposes.