Sufficient statistic for sample without replacement

by Melina   Last Updated May 15, 2018 20:19 PM

This is an exercise that I faced today:

Let $\mathcal{M}$ be a population of ’individuals’ with identification numbers in $\mathbb{Z}$. We assume that the set of all identification numbers equals $\{a, . . . , b\}$ with unknown integers $a, b$. We only know that $b − a + 1 \geq n ∈ \mathbb{N} \setminus \{1\}$.

Now we draw a random sample of size n without replacement from $\mathcal{M}$ and note the tuple $ω = (ω_1, ω_2, . . . , ω_n)$ of identification numbers. Show that $S(ω) =\left (min(ω), max(ω) \right )$ is a sufficient statistic for this experiment. Describe the conditional distribution of $ω$, given $S(ω) = (s_1, s_2)$.

First, I define:

$$\Omega=\{ \omega \in \mathbb{Z}^{n}: \omega_i \neq \omega_j, \text{whenever}\ i \neq j \}$$

$$P_{a,b}=\text{Unif}(\ \Omega \ \cap \{ a,...,b \}^n), $$

$$\Theta=\{ (a,b) \in \mathbb{Z}^{n} : b-a+1 \geq n \}.$$

Any help would be really appreciated.

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