# Multivariate canonical exponential family

Consider the canonical d-dimensional exponential family with densities $$p(x)=exp\left(\langle\theta,T(x)\rangle-A(\theta)\right)h(x),\theta\in\Omega$$

with $$\Omega\subset\Omega_0=\{\theta:A(\theta)<\infty\}$$. $$\Omega_0$$ is the natural parameter space. The family is called full if $$\Omega=\Omega_0$$.

Let $$Z\in\mathbb R^{n\times p}$$ be a fixed deterministic matrix and $$Y\sim N(Z\beta,I_n),\beta\in\mathbb R^p$$.

1. Show that the distribution of Y can be represented as an n-dimensional canonical exponential family and specify A, T, $$\Omega,\Omega_0$$. Is this a full family?

2. Under what condition on Z is the above family full rank?

Current work.

We want to write the density of Y in the canonical form. I have:

$$\begin{split}(2\pi)^{-n/2}|I|e^{-\frac 1 2(Y-Z\beta)^TI(Y-Z\beta)} &\propto e^{-\frac 1 2 (Y^T-\beta^TZ^T)(Y-Z\beta)}\\ &=e^{-\frac 1 2(Y^TY-Y^TZ\beta-\beta^TZ^TY+\beta^TZ^TZ\beta)}\\ &=e^{-\frac 1 2 \|Y\|^2}e^{\langle Y,Z\beta\rangle-\frac 1 2\|Z\beta\|^2}\end{split}$$

thus $$h(y)=(2\pi)^{-n/2}e^{-\frac 1 2\|Y\|^2}$$, $$\langle \beta,T(y)\rangle=\langle Z^TY,\beta\rangle$$, and $$A(\beta)=\frac 1 2 \|Z\beta\|^2$$. However, I no idea what $$\Omega,\Omega_0$$ are for part 1 or how to do part 2. Could someone please explain what the "natural parameter space," "parameter space," and "full rank" are for this problem?

• Avoid cross-posting please. – StubbornAtom Feb 13 at 6:58