For instance the exponential form of gamma distribution is $e^{(\alpha -1)\ln{x}-\beta x+ (something \ about \ only \ \alpha \ and \ \beta) }$, the natural parameters given on wikipedia are $(\alpha -1)$ and $-\beta$, but why not $\beta $ for the second one. Do constants&sign not affect sufficient statistics and natural parameters?
2026-03-26 17:36:18.1774546578
Does natural parameter of exponential family need the constant or sign?
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Well if i get ur question right u are wondering if the form of the parameters affect the distribution? The distributions have a specific form and u can't change it. If u changes -b to b then $e^{4lnx−3x+(something about only α and β)}$ would belong to the distribution $Gamma(5,-3)$ but in reality it belongs to $Gamma(5,3)$ so yes it is essential that u keep the forms as they are. Of course there are transformations in every distributions formula so u can do different mathematical approaches but they are used intact.