  In this paper we outline general considerations on parameter identifiability,
and introduce the notion of weak local identifiability and gradient weak local
identifiability. These are based on local properties of the likelihood, in
particular the rank of the Hessian matrix. We relate these to the notions of
parameter identifiability and redundancy previously introduced by Rothenberg
(Econometrica 39 (1971) 577-591) and Catchpole and Morgan (Biometrika 84 (1997)
187-196). Within the exponential family parameter irredundancy, local
identifiability, gradient weak local identifiability and weak local
identifiability are shown to be equivalent. We consider applications to a
recently developed class of cancer models of Little and Wright (Math
Biosciences 183 (2003) 111-134) and Little et al. (J Theoret Biol 254 (2008)
229-238) that generalize a large number of other recently used quasi-biological
cancer models, in particular those of Armitage and Doll (Br J Cancer 8 (1954)
1-12) and the two-mutation model (Moolgavkar and Venzon Math Biosciences 47
(1979) 55-77).
