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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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    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