# Definition of sufficient statistic when the support of the statistic depends on the unknown parameter?

Suppose we have a random vector $$\mathbf{X} = (X_1,\ldots,X_n)$$ or sample with pdf or pmf $$f(\mathbf{x};\theta)$$, where $$\theta \in \Theta$$ is an unknown parameter (or vector of parameters), whose specification completely determines $$f(\mathbf{x};\theta)$$.

As far as I understand, a statistic is just a a random variable $$T=T(\mathbf{x})$$ that is a function (that does not depend on $$\theta$$) of $$\mathbf{X}$$. That's not to say the distribution of $$T$$ doesn't depend on $$\theta$$ of course, just that the function that relates $$\mathbf{X}$$ and $$T$$ does not.

Now, usually from what I've seen, the statistic $$T$$ is defined to be a sufficient statistic if the distribution of $$\mathbf{X}$$ conditional on $$T=t$$ does not depend on $$\theta$$.

Now, my issue with this definition (which I don't understand how is it not addressed anywhere I've looked) is that if the support of $$T$$ depends on $$\theta$$, then we can't really pick a $$t$$ in the support of $$T$$ and then claim that the conditional distribution does not depend on $$\theta$$, because that same conditional distribution may be ill-defined for some values of $$\theta$$ for which $$t$$ is not in the support of $$T$$, right?

I kinda came up with my own definition which I'm hoping someone could comment on to see if this is what's more formally meant by that usual definition:

A statistic $$T(\mathbf{X})$$ is sufficient for $$\theta$$ if there exists a non-negative function $$C:\mathbb{R}^n\times \mathbb{R} \to \mathbb{R}$$ that does not depend on $$\theta$$ and such that $$p(\mathbf{x},t;\theta) = C(\mathbf{x},t) \cdot q(t;\theta)$$ for all $$\mathbf{x} \in \mathbb{R}^n, t\in \mathbb{R}, \theta \in \Theta$$, where $$p$$ and $$q$$ denote the joint pdf/pmf of $$(\mathbf{X},T(\mathbf{X}))$$ and the pdf/pmf of $$T(\mathbf{X})$$, respectively.

• (a) You're welcome to invent any kind of definition you want, but a definition by itself is without any utility. What matters is what you can use your definition for. Yours does not deserve to be called a "sufficient statistic" until you can show that all--or at least most--of the important properties and theorems enjoyed by sufficient statistics also hold for your definition. (b) I cannot make sense of the paragraph about "my issue." Could you elaborate on what you think the problem is? Perhaps offer a simple example?
– whuber
Oct 2, 2016 at 22:07
• Sure, for instance, consider $X_1,\ldots,X_n$ to be iid variables following a uniform distribution on the interval $[0,\theta]$. Now, if we consider the statistic $T(X_1,\ldots,X_n) = \sum_{i=1}^n X_i$, then it's clear that the support of $T$ is $[0,n\theta]$, which clearly depends on $\theta$. Now, if I pick any $t$, even if $t>0$, then how can I claim the conditional distribution of $\mathbf{X}=(X_1,\ldots,X_n)$ conditional on $T=t$ is a constant function of $\theta$, if that conditional distribution isn't even defined for, say $\theta = \frac{t}{2n}$? Oct 2, 2016 at 22:29
• Maybe you should consult the Wikipedia article.
– whuber
Oct 2, 2016 at 22:44
• If you are interested, I found the answer to my question I suppose. The more rigorous definition (in the measure theory sense) does address the issue I mentioned by basically imposing that it's not the conditional distribution of $\mathbf{X}|T(\mathbf{X})=t$ itself that needs to be independent of $\theta$, but what they call a version of it. Anyways, here's a link on an article that rigorously defines it this way (page 232, definition 5). Oct 2, 2016 at 23:44
• @user45453: You could write those details up and then answer your Q yourself Jan 26, 2019 at 4:27

The most frequently seen example of this is $$X_1,\ldots,X_n\sim\operatorname{iid} \operatorname{Uniform}(0,\theta).$$ It is said that the conditional distribution of $$(X_1,\ldots,X_n)$$ given $$\max\{X_1,\ldots,X_n\}$$ does not depend on $$\theta.$$
But to be precise, it ought to be said that it does not depend on which of the values of $$\theta$$ consistent with the observed value of $$\max\{X_1,\ldots,X_n\}$$ is the true value. Then one would take that to be the definition of sufficiency of the maximum.