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I'm trying to find two statistics T1, T2 such that (T1, T2) is jointly sufficient for (λ, θ) for a random sample $X_1,\ldots,X_n$ from a two parameter exponential distribution.

$f(x) = \begin{cases} \lambda e^{-\lambda (x-\theta)}, & \theta < x < \infty, \\ 0, & elsewhere. \end{cases}$

Thanks.

2 Answers 2

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First note that $$ f(x_1 ,x_2 , \ldots ,x_n |\lambda ,\theta ) = \lambda ^n e^{\lambda \theta n} \exp \bigg( - \lambda \sum\limits_{i = 1 }^n {X_i } \bigg) \prod\limits_{i = 1}^n {{\mathbf 1}(x_i > \theta )}, $$ where $\mathbf 1$ denotes the indicator function. Then note that $$ \prod\limits_{i = 1}^n {{\mathbf 1}(x_i > \theta )} = {\mathbf 1}(\min \lbrace x_1 , \ldots ,x_n \rbrace > \theta ), $$ and apply Theorem 2 given here.

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    Here we need to factorize $f(x_1 ,x_2 , \ldots ,x_n |\lambda ,\theta ) = u(x_1,x_2,\ldots,x_n) v(r_{1,\ldots,k}(x_1,x_2,\ldots,x_n | \lambda,\theta)$ right ? but I don't see a function $u(x_1,x_2,\ldots,x_n)$. Isn't $\prod\limits_{i = 1}^n {{\mathbf 1}(x_i > \theta )} = {\mathbf 1}(\min \lbrace x_1 , \ldots ,x_n \rbrace > \theta )$ dependent on $\theta$ ?2011-02-28
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    You can take $u(x_1,\ldots,x_n)=1$ (it is nonnegative and does not depend on the parameters). As for the second question, consider the example given on p. 3 in that link (Uniform population).2011-02-28
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    Also, note the last sentence in Theorem 2.2011-02-28
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    Further, see the examples in http://en.wikipedia.org/wiki/Sufficient_statistic, in particular the example "Uniform distribution (with two parameters)".2011-02-28
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    I kind of see that $T_1 = \sum\limits_{i = 1 }^n {X_i}$ and $T_2 = min\{x_1,\ldots,x_n\}$ are jointly sufficient (please tell me if I'm wrong) but I'm trying to reason it why. I feel like I follow the theorem and exmaples and arrive at the answer. Like always I take the joint pdf and try to separate the function but I don't understand the intuition behind it.2011-02-28
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    BTW your answer and comments helped me a lot to think about the problem rather than giving the answers straight. Thanks.2011-02-28
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    For the intuition behind this example, see, again, http://math.arizona.edu/~tgk/466/sufficient.pdf (for example).2011-02-28
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by neyman factorization theorem X_bar and X(1) are jointly sufficient stats

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    You should include the picture in your post or, better yet, write what it says using MathJax. And please include some words of explanation as well.2016-03-06