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I have this density function for which I am not able to find a minimal sufficient statistic, as required. It does not belong to the exponential families distribution as the support depend also on the parameter, and with the Lehmann Scheffé approach I am not able to separate the random variable $X$ from the parameter $\theta$.

$$f(x;\theta)=\frac{\beta^3}{2}e^{-\beta(x-\theta)}(x-\theta)^2 \mathbf1_{x\ge\theta}, \quad \beta\text{ known}$$

Michael Hardy
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Lucas
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  • Please include your work, whatever you have tried and stick to one question per post. – StubbornAtom Aug 16 '18 at 14:47
  • Ok, thanks. I posted here the correct question: https://math.stackexchange.com/questions/2886982/minimal-sufficiency-for-fx-%CE%B8-21%CE%B8-x-i%CE%B8-%E2%89%A4-x-%E2%89%A4-%CE%B81 – Lucas Aug 18 '18 at 18:28
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    I have edited this post to include a single question since you already asked the second question in a separate post. – StubbornAtom Aug 19 '18 at 06:16
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    Your distribution is a shifted gamma with shape parameter $\alpha = 3$, rate parameter $\beta$, and location parameter $\theta$. It is my suspicion that no data reduction is possible; that is to say, the minimal sufficient statistic comprises the sample itself. – heropup Aug 19 '18 at 07:48
  • @MichaelHardy Since $\beta$ is known, it it supposed to be fixed. So the family is only indexed by $\theta$. – StubbornAtom Aug 19 '18 at 19:35
  • I will edit my answer further and then un-delete it. – Michael Hardy Aug 19 '18 at 20:31

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