Minimal sufficient statistic for $theta$ where $f(x;theta)=fracbeta^32e^-beta(x-theta)(x-theta)^2mathbf1_xgetheta$

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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 $theta$.



$$f(x;theta)=fracbeta^32e^-beta(x-theta)(x-theta)^2 mathbf1_xgetheta, quad betatext known$$




statistics data-sufficiency




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  • Please include your work, whatever you have tried and stick to one question per post.
    – StubbornAtom
    Aug 16 at 14:47










  • Ok, thanks. I posted here the correct question: math.stackexchange.com/questions/2886982/…
    – Lucas
    Aug 18 at 18:28






  • 1




    I have edited this post to include a single question since you already asked the second question in a separate post.
    – StubbornAtom
    Aug 19 at 6:16






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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 at 7:48











  • @MichaelHardy Since $beta$ is known, it it supposed to be fixed. So the family is only indexed by $theta$.
    – StubbornAtom
    Aug 19 at 19:35














up vote
2
down vote

favorite












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 $theta$.



$$f(x;theta)=fracbeta^32e^-beta(x-theta)(x-theta)^2 mathbf1_xgetheta, quad betatext known$$




statistics data-sufficiency




share|cite|improve this question






















  • Please include your work, whatever you have tried and stick to one question per post.
    – StubbornAtom
    Aug 16 at 14:47










  • Ok, thanks. I posted here the correct question: math.stackexchange.com/questions/2886982/…
    – Lucas
    Aug 18 at 18:28






  • 1




    I have edited this post to include a single question since you already asked the second question in a separate post.
    – StubbornAtom
    Aug 19 at 6:16






  • 1




    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 at 7:48











  • @MichaelHardy Since $beta$ is known, it it supposed to be fixed. So the family is only indexed by $theta$.
    – StubbornAtom
    Aug 19 at 19:35












up vote
2
down vote

favorite









up vote
2
down vote

favorite











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 $theta$.



$$f(x;theta)=fracbeta^32e^-beta(x-theta)(x-theta)^2 mathbf1_xgetheta, quad betatext known$$




statistics data-sufficiency




share|cite|improve this question














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 $theta$.



$$f(x;theta)=fracbeta^32e^-beta(x-theta)(x-theta)^2 mathbf1_xgetheta, quad betatext known$$





statistics data-sufficiency





share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 19 at 18:51









Michael Hardy

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asked Aug 16 at 10:27









Lucas

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  • Please include your work, whatever you have tried and stick to one question per post.
    – StubbornAtom
    Aug 16 at 14:47










  • Ok, thanks. I posted here the correct question: math.stackexchange.com/questions/2886982/…
    – Lucas
    Aug 18 at 18:28






  • 1




    I have edited this post to include a single question since you already asked the second question in a separate post.
    – StubbornAtom
    Aug 19 at 6:16






  • 1




    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 at 7:48











  • @MichaelHardy Since $beta$ is known, it it supposed to be fixed. So the family is only indexed by $theta$.
    – StubbornAtom
    Aug 19 at 19:35
















  • Please include your work, whatever you have tried and stick to one question per post.
    – StubbornAtom
    Aug 16 at 14:47










  • Ok, thanks. I posted here the correct question: math.stackexchange.com/questions/2886982/…
    – Lucas
    Aug 18 at 18:28






  • 1




    I have edited this post to include a single question since you already asked the second question in a separate post.
    – StubbornAtom
    Aug 19 at 6:16






  • 1




    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 at 7:48











  • @MichaelHardy Since $beta$ is known, it it supposed to be fixed. So the family is only indexed by $theta$.
    – StubbornAtom
    Aug 19 at 19:35















Please include your work, whatever you have tried and stick to one question per post.
– StubbornAtom
Aug 16 at 14:47




Please include your work, whatever you have tried and stick to one question per post.
– StubbornAtom
Aug 16 at 14:47












Ok, thanks. I posted here the correct question: math.stackexchange.com/questions/2886982/…
– Lucas
Aug 18 at 18:28




Ok, thanks. I posted here the correct question: math.stackexchange.com/questions/2886982/…
– Lucas
Aug 18 at 18:28




1




1




I have edited this post to include a single question since you already asked the second question in a separate post.
– StubbornAtom
Aug 19 at 6:16




I have edited this post to include a single question since you already asked the second question in a separate post.
– StubbornAtom
Aug 19 at 6:16




1




1




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 at 7:48





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 at 7:48













@MichaelHardy Since $beta$ is known, it it supposed to be fixed. So the family is only indexed by $theta$.
– StubbornAtom
Aug 19 at 19:35




@MichaelHardy Since $beta$ is known, it it supposed to be fixed. So the family is only indexed by $theta$.
– StubbornAtom
Aug 19 at 19:35















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