The Wikipedia entry on this topic is, to me, very confusing. It states that:

If X1, ...., Xn are independent and have a Poisson distribution with parameter λ, then the sum T(X) = X1 + ... + Xn is a sufficient statistic for λ.

I know that $T(X)$ is the "sufficient statistic" function, but I am lost as to what "sum T(X)" means in this context. The sum of my iid observations? What does that even mean if they are N-dimensional?

The demonstration below this line is also confusing (to me) and even mentions a function "h(x)" which is never explained as far as I can see.

Could you explain the concept of "sufficient statistic" for a Poisson distributed sample better?

  • $\begingroup$ Could you clarify for us what an "N-dimensional" Poisson distribution might be? Closely related questions include stats.stackexchange.com/questions/81993 $\endgroup$
    – whuber
    Commented Oct 2, 2018 at 15:27
  • $\begingroup$ It means that T(X) is a sum of your samples that you took. So for example, if you took samples over 5 different time increments, you would have 5 samples and $T(X) = X_1 + X_2 + X_3 + X_4 +X_5$ $\endgroup$
    – Bucephalus
    Commented Oct 2, 2018 at 15:29
  • $\begingroup$ @whuber in astronomy we use what is called a CMD diagram which contains positions of observed stars in a magnitude vs color plot. This diagram can be N-dimensional (adding more magnitudes or colors) and the data in it (observed stars) are thought to follow a Poisson distribution (as their observation depends on a photon-counting process) Does this make sense? $\endgroup$
    – Gabriel
    Commented Oct 2, 2018 at 15:35
  • 1
    $\begingroup$ This is a large collection of univariate Poisson distributions. If the stars are assumed to be selected as if at random, then the values are independent, obviating any apparent need to treat this as a large-dimensional problem. In fact, you can view each pixel's value in the diagram as representing a single ($n=1$) observation from a Poisson distribution. It's unclear whether (for understanding and analyzing this diagram) you need to sum any Poisson variables at all. $\endgroup$
    – whuber
    Commented Oct 2, 2018 at 15:53
  • $\begingroup$ Thanks whuber. I'll read the links and think a bit more about this. $\endgroup$
    – Gabriel
    Commented Oct 2, 2018 at 16:02