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I am doing exercise 3.18 of "The Bayesian Choice":

Give a sufficient statistic associated with a sample $x_1,...,x_n$ from a truncated normal distribution $$ f (x|\theta) \propto \exp(-(x - \theta)^2)/2) \mathbb I[\theta - c,\theta+c](x) $$ when $c$ is known.

Using the factorization criterion I found that the following statistics are sufficient for $\theta$. $$ T_1 = \sum_{i = 1}^{n} x_i , \quad T_2 = \{x_{(1)} + c , x_{(n)} -c \} $$ I would like to confirm if the statistics are sufficient, especially for $T_2$. Suppose $T^* = x_{(1)} + c $ and $T^{**} = x_{(n)} - c$, would that also be sufficient? greetings.

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    $\begingroup$ Your $T_2$ can be treated as two statistics or as one two-dimensional statistic. Indeed you could have one three-dimensional statistic and (since you know $n$ and $c$) it could be $(\bar x, x_{(1)}, x_{(n)})$ $\endgroup$
    – Henry
    Commented Apr 20 at 17:56
  • $\begingroup$ I confirm @Henry's point that the sufficient statistic is of dimension $3$ when $n\ge 3$. $\endgroup$
    – Xi'an
    Commented Apr 21 at 9:25

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