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?

  • $\begingroup$ Avoid cross-posting please. $\endgroup$ – StubbornAtom Feb 13 at 6:58

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