# How to find the curve of parameter of exponential family?

I am having trouble understanding about the parameter curve. I tried looking for an answer, I could not find my problem statement. In an exercise, it is asked to find the curve on which $$\theta$$ parameter vector lies for given exponential families. The exponential families are given as follows-

(i) n($$\theta, a\theta^2$$)
(ii) $$f(x|\theta) = C*exp(-(x-\theta)^4)$$

After converting (i) to exponential form, I get $$(1/\sqrt(2\pi a\theta^2)) *exp(−1/2a )*exp(-x^2/2a\theta^2 + x/a\theta)$$,

so $$w_1(\theta) = 1/2a\theta^2 , w_2(\theta) = 1/a\theta$$ correspondingly with $$t_1(x) = -x^2 , t_2(x) = x$$.

For this one, as far as I understood, $$\theta$$ is mean so $$-\infty < \theta < \infty$$ and $$a\theta^2$$ is variance so $$-\infty < \theta < \infty$$. So combining these two, I get $$-\infty < \theta < \infty$$. But the answer is given as "parabola". I don't understand how the curve of $$\theta$$ is a parabola.

Similarly for (ii), I get exponential form as $$C*exp(x^4)*exp(\theta^4)*exp(-4x^3\theta + 6x^2\theta^2 - 4x\theta^3)$$

The answer for this is given as "spiral in 3D space", while again I get $$0< \theta < \infty$$.

Please tell me the way with which I can determine the curves of parameter these families.

• What is "curve of parameter"? What you are probably referring to is that $N(\theta,a\theta^2)$ is a member of a curved exponential family. You cannot express the pdf in terms of a regular one-parameter exponential family. – StubbornAtom Oct 8 '18 at 9:49
• @StubbornAtom In the book, it is asked to describe the curve on which the $\theta$ parameter vector lies. – Ankit Seth Oct 8 '18 at 9:52
• I would ask you to search using the keyword 'curved exponential family'. See math.stackexchange.com/q/383960/321264, stats.stackexchange.com/q/155628/119261 for example. – StubbornAtom Oct 8 '18 at 9:57