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.

  • $\begingroup$ 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. $\endgroup$ Oct 8, 2018 at 9:49
  • $\begingroup$ @StubbornAtom In the book, it is asked to describe the curve on which the $\theta$ parameter vector lies. $\endgroup$
    – Ankit Seth
    Oct 8, 2018 at 9:52
  • $\begingroup$ 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. $\endgroup$ Oct 8, 2018 at 9:57


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