# Rao-Blackwellisation using non-sufficient statistics

The following is given as a remark in chapter 7 of Introduction to mathematical statistics Hogg and Craig, 8th edition. (It is mentioned as "Remark 7.3.1")

Now, I do understand that the distribution of $$Y_1|Y_3$$ would contain $$\theta$$ compulsorily (because $$Y_3$$ is not a sufficient statistic). But I do not understand why $$\theta$$ has to be present compulsorily in the conditional mean $$X=\mathbb E[\varphi(Y_1)|Y_3]$$.

I can agree to the statement "$$\theta$$ can be present in $$X$$". But I do not understand why $$\theta$$ has to be present in $$X$$.
Because the distribution of $$Y_1|Y_3$$ could contain parameters other than $$\theta$$ also. So, it is possible for $$\mathbb E[\varphi(Y_1)|Y_3]=X$$ to contain those other parameters(and not contain $$\theta$$).

So why is it always true that $$\theta$$ will be present in $$X$$?

EDIT:-

I'm not sure, but I think that (going by the definition of sufficiency given in Hogg and Craig) the concept of sufficiency is defined only when the pdf of the random sample ( i.e. $$f_X(x;\theta)$$ ) has exactly one unknown parameter (namely $$\theta$$).
If that's the case, then the only unknown parameter that the distribution of $$Y_1|Y_3$$ contains is $$\theta$$.
Also note that $$X=\mathbb E[\varphi(Y_1)|Y_3]$$, cannot be a constant that is not equal to $$\theta$$( because then $$\mathbb{E}[X]\neq \theta$$ )
So, If $$X$$ is a constant then $$X=\theta$$. But what if $$X$$ is not a constant?
So, in the case when $$X$$ is not a constant, why should $$\theta$$ be present in $$X$$, compulsorily?

Counter-example: If one considers the case of a sample $$(X_1,\ldots,X_n)$$ from a $$N(\theta,\theta^2)$$ distribution, a minimal sufficient statistic is $$T=\left(\sum_i X_i,\sum_i(X_i-\bar X_n)^2\right)$$ Now $$\varphi(T)=\bar X_n$$ is an unbiased estimator of $$\theta$$ that is not sufficient. But the conditional expectation $$\mathbb E_\theta[\varphi(T)|\varphi(T)]=\varphi(T)$$ does not involve $$\theta$$.