# determine the minimal exponential family form

Suppose we have a random variable $$X$$ having p.d.f of the form $$f(x|\theta)=\exp\{c(\theta)'T(x)-B(\theta)\}h(x),$$ then we say $$X$$ is from exponential family. Further, we say that an exponential family is minimal if the functions $$c(\theta)$$ and the statistics $$T(X)$$, are linearly independent respectively.

I am trying to determine the minimal exponential family form of the following case.

Here, $$X$$ has p.d.f $$f(x|\alpha,\beta)=\exp\left\{\sum_{i=1}^nx_{i}(\alpha+\beta z_i)-\sum_{i=1}^n\log(1+\exp(\alpha+\beta z_i))\right\},$$

which is equal to

$$f(x|\alpha,\beta)=\exp\left\{\sum_{i=1}^nx_{i}\alpha + \sum_{i=1}^n\beta z_ix_i-\sum_{i=1}^n\log(1+\exp(\alpha+\beta z_i))\right\}$$

For the first equation the natural sufficient statistic $$T$$ is , $$T=(X_1, \dots,X_n)$$, while the second is $$T=(\sum_{i=1}^nX_i,\sum_{i=1}^nX_iz_i)$$.

I wonder which one is the minimal exponential family form?

I think if $$n \le 1$$, it's the first form (and the parameter is just $$\alpha + z_1 \beta$$), otherwise it's the second form.