Nonexistence of Maximum Likelihood Estimation of Variance Components in Some Stochastic Models

Although maximum likelihood has been widely used to estimate unknown parameters in stochastic models of random errors, we show that the method cannot be used to estimate variance components for some stochastic models of routine measurement systems under some conditions, because the likelihood function is unbounded for such stochastic models. No optimal solution of variance components exists for these likelihood functions from the point of view of global optimization, implying that variance components for such stochastic models of practical importance cannot be estimated by maximizing the likelihood function.