# Cramér-Rao / Wolfowitz bound with nuisance parameter

Let $$F$$ be a distribution with two parameters, $$\theta$$ and $$\phi$$, whose values are non-random but unknown. Consider a sampling procedure in which $$N$$ samples $$x_1, \ldots x_N$$ are obtained from i.i.d. realizations of the distribution $$F$$. The sample size $$N$$ can be fixed, or can be randomly given by a sequential sampling procedure, with $$N$$ determined by the sequence of observed outcomes $$x_1, x_2, \ldots$$.

Consider the task of estimating $$\theta$$ from $$x_1, \ldots, x_N$$. The parameter $$\phi$$ is a nuisance parameter, i.e. its value is of no interest. Let $$\hat \theta$$ be an estimation of $$\theta$$ computed from $$x_1, \ldots, x_N$$.

I am trying to find a Cramér-Rao-type bound for the variance of $$\hat \theta$$. If it were not for the nuisance parameter $$\phi$$, the answer would be the Cramer-Rao bound for $$N$$ fixed, or Wolfowitz's generalization for the sequential case, subject to suitable regularity conditions.

Are there equivalent bounds in the presence of the nuisance parameter? The papers by Bhapkar (1994) and Godambe (1984) seem to be related to this, but I am afraid I cannot make much sense of them.

I am mostly interested in the sequential case, but a result for fixed $$N$$ would be useful too. Similarly, I am interested in the general case when $$\hat \theta$$ is not restricted to be unbiased, but a result for the unbiased case would be useful. The parameters $$\theta$$ and $$\phi$$ are both one-dimensional.

I think I found the answer, for the case with $$N$$ fixed, using Patrick Breheny's notes on Likelihood Theory, in particular the lecture on efficiency. Thanks to @user1848065, who pointed me to those notes in a comment to another question.
The key is to use the vector form of the Cramér-Rao bound. Let $$\mathrm{\mathbf I}$$ denote the $$2 \times 2$$ information matrix for the vector of parameters $$\theta$$, $$\phi$$ in that order, and let $$\mathrm{\mathbf J} = \mathrm{\mathbf I}^{-1}$$.
• The Cramér-Rao bound for an unbiased estimator of $$\theta$$ when $$\phi$$ is unknown is $$J_{1,1}$$. This applies regardless of wheter $$\phi$$ is considered as a "nuisance" or a "wanted" parameter. In other words, the fact that we do not want to estimate $$\phi$$ cannot be exploited to improve the estimation of $$\theta$$.
• The Cramér-Rao bound for an unbiased estimator of $$\theta$$ when $$\phi$$ is a known number is $$1/I_{1,1}$$, which is less than or equal to $$J_{1,1}$$. In other words, knowledge of the true value of $$\phi$$ is in general helpful for estimating $$\theta$$.