Is $\theta$ a location or a scale parameter in the $\mathcal N(\theta,\theta)$ and $\mathcal N(\theta,\theta^2)$ densities?

Is this a valid question to ask or does the question not arise? I don't know what happens to the parameters in curved exponential families. Moreover, is there a standard parameter space to be taken in these two cases when it is not mentioned explicitly? As $\theta$ can be both the population mean and sd (or variance), should I take $\theta\in\mathbb R$ or $\theta\in\mathbb R^{+}$ or $\theta\in\mathbb R-\{0\}$? Or does this depend on my sample?

I am trying to find the UMVUE for $\theta$ in both these cases, but first I need to get this question out of my head.

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    $\begingroup$ Ask yourself this: when you change $\theta$, does the distribution merely shift its location? If not, it's not a location parameter. When you change $\theta$, is the distribution really the very same as before, but expressed in different units of measurement? If not, then it's not a scale parameter. $\endgroup$
    – whuber
    May 28 '18 at 12:52

Since, when $X\sim{\cal N}(\theta,\theta^2)$,$$Z=\dfrac{X-\theta}{\theta}=\dfrac{X}{\theta}-1\sim{\cal N}(0,1)$$and assuming $\theta\ne 0$, since $\theta=0$ is a special case that results in a Dirac mass at zero, the parameter $\theta$ is a scale parameter as $$X=\theta(Z+1)$$ is the scaled version of $Z+1$ that has a fixed distribution. (Note that applying $\theta=0$ to the above results in the correct Dirac mass at zero.)

When $X\sim{\cal N}(\theta,\theta)$, with $\theta>0$, $$Z=\dfrac{X-\theta}{\theta^{1/2}}=\dfrac{X}{\theta^{1/2}}-\theta^{1/2}\sim{\cal N}(0,1)$$the parameter $\theta$ is neither scale nor location. (Again, applying $\theta=0$ to the above results in the correct Dirac mass at zero.)

  • $\begingroup$ So in your first case, $\theta$ has to be positive and in the second case it just has to be non-zero? $\endgroup$ May 29 '18 at 9:27
  • $\begingroup$ But you perform a division by $\theta$ while writing $Z$ in both cases. So, shouldn't the zero point be excluded? $\endgroup$ May 29 '18 at 9:38

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