Sufficient Statistic Basics

Question

If I know the value of a sufficient statistic, but not the sample that generated it, am I right to suspect that the conditional distribution of any other statistic given the sufficient statistic will not depend on the parameter of interest? Formally speaking:

Let $\theta$ be the parameter of interest. $T(x)$ is the known sufficient statistic. Now, for any other statistic $\tilde{T}(x)$, we (would; conjecturing) have:

$$ f_{\tilde{T}\mid T}(\tilde{t}\mathbb\mid\theta,t)=f_{\tilde{T}\mid T}(\tilde{t}\mid t) $$

Thanks in advance.

EDIT: just to add to my line of thought. I am thinking of the new statistic as equivalent to the sample points, since they differ just by a function. So if the if I have a sufficient statistic for the distribution, it will automatically be sufficient to any other statistic.

Answer 1

That is correct, PROVIDED that the statistical model is right. But the sufficient statistic is not where you will find evidence that the model doesn't fit. For example, in estimating the mean and variance of a normally distributed population, the sufficient statistic is the pair whose components are the sum of the observations and the sum of their squares, whereas evidence of non-normality will be found in the residuals.