Statistical parameter ($\theta \in \Theta$)

A statistical parameter indexes a family of probability distributions in a statistical model. It is fixed in a frequentist sampling model; a prior distribution makes it random for Bayesian statistics. Different statistical parameter values should produce different observable distributions when identifiability is claimed.

A nuisance parameter is a statistical parameter required to specify the distribution of observations but not itself the target of inference. A profile likelihood maximizes over it; Bayesian model evidence integrates over it using a proper prior distribution. These operations differ and need not give the same inference. A baseline hazard in a Cox proportional-hazards model is an infinite-dimensional example.