Neyman-Scott incidental parameter problem

A growing number of nuisance parameters can prevent a maximum-likelihood estimator of a common parameter from being consistent. With two independent normal observations of mean ${\mu }_i$ per group, the profiled variance estimate ${\sum }_i(X_{1i}-X_{2i}{)}^2/(4n)$ tends to half the true variance, because each unknown mean is estimated using only two observations.