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I am currently trying to learn about the exponential family of distributions. I am trying to understand this question and this answer from Xi'an. I have the same function:

$$f(x; \sigma, \tau)= \begin{cases} \dfrac{1}{\sigma} e^{-(x - \tau)}/\sigma &\text{if}\, x\geq \tau\\ 0 &\text{otherwise} \end{cases}$$

However, it still isn't clear to me why this is not a member of the exponential family. On the other hand, it is said that the function

$$f_{\varphi}(x)= \begin{cases} \varphi x^{\varphi - 1} &\text{if}\, 0 < x < 1, \varphi > 0\\ 0 &\text{otherwise} \end{cases}$$

actually is a member of the (one-parameter) exponential family.

Why is the first function not a member of the exponential family? Furthermore, what is the reasoning behind why one is a member of the family and yet the other is not?

The 'duplicate' does not provide the clarification of the reasoning here Is the negative exponential distribution a member of the exponential family? and here Is the negative exponential distribution a member of the exponential family?. I was really looking for a 'simpler' explanation of things so that I could understand.

The Pointer
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  • Is $\tau$ fixed? – Arya McCarthy Apr 16 '21 at 15:47
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    In the duplicate I provide a general procedure to test whether a family is Exponential or not and I explain the reasoning behind it. It is easy to apply in your case. – whuber Apr 16 '21 at 15:48
  • @whuber But this does not provide the clarification of the reasoning here https://stats.stackexchange.com/questions/355302/is-the-negative-exponential-distribution-a-member-of-the-exponential-family and here https://stats.stackexchange.com/questions/355302/is-the-negative-exponential-distribution-a-member-of-the-exponential-family/355323#355323 . I was really looking for a 'simpler' explanation of things so that I could understand. – The Pointer Apr 16 '21 at 15:54
  • @AryaMcCarthy That's a good question. The example doesn't mention anything about being "fixed", but one would presume that, since the $x$ in $f(x; \sigma, \tau)$ are variable, then the $\sigma$ and $\tau$ must be fixed, no? – The Pointer Apr 16 '21 at 16:04

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