I am currently studying the concepts of likelihoods and sufficient statistics in mathematical statistics. The following theorem was presented to me:
Let $Y_1, \dots, Y_n$ be a random sample from a distribution $f_\theta (y)$, $\theta = (\theta_1, \dots, \theta_p) \in \Theta$, belonging to the $k$-parameter exponential family with the natural parameters $(c_1(\theta), \dots, c_k (\theta))$; that is, $f_\theta (y) = \exp \left[ \sum_{j = 1}^k c_j(\theta) T_j(y) + d(\theta) + S(y) \right], y \in \operatorname{supp}(f_\theta)$.
Assume that the set $\{ (c_1(\theta), \dots, c_k(\theta)), \theta \in \Theta \}$ contains an open $k$-dimensional rectangle.
Then the sufficient statistic $T(\mathbf{Y}) = (\sum_{i = 1}^n T_1(Y_i), \dots, \sum_{i = 1}^n T_k(Y_i))$ is complete (and, therefore, minimally sufficient if the latter exists).
I don't understand this concept of a "$k$-dimensional rectangle". I was wondering if people would please take the time to explain this.
Best Answer
A $2$-dimensional rectangle is a rectangle in the plane.
A $3$-dimensional rectangle is a right rectangular prism, i.e., a cuboid or rectangular parallelepiped.
In the context of the discussion, a $k$-dimensional rectangle is a set of the form $$\{(x_1, x_2, \ldots, x_k) \in \mathbb R^k : a_i \le x_i \le b_i \; \forall \; i = 1, 2, \ldots, k\}$$ which can be thought of as the Cartesian product of the intervals $[a_i, b_i]$ for each $i = 1, 2, \ldots, k$.