Sufficient estimators question

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February 20th 2010, 11:31 AM
jass10816

Sufficient estimators question

A random sample of size $n$ is drawn from the pdf,

$f_Y(y;\theta) = e^{-(y-\theta)}$ , $\theta\le y$

I can show that $Y_{max}$ is not sufficient by an example, but I am having a hard time showing that $Y_{min}$ is sufficient.
February 20th 2010, 04:16 PM
jass10816

So I got that the $f_{y_{min}}(y_{min};\theta) = n[e^{-(y-\theta)}]^n$ , but I can't seem to factor this out of $\prod{e^{-(y-\theta)}}$

Hope this helps clarify my question.
February 20th 2010, 07:05 PM
matheagle

It's sufficient by the Factorization Theorem.

$f_Y(y;\theta) = e^{-(y-\theta)}$ , $\theta\le y$

should be rewritten with indicators.

BUT you need a sample here

$L(y_1,...,y_n) =\prod_{i=1}^n e^{-(y_i-\theta)}I(y_i>\theta)$

$=e^{n\theta}e^{-\sum_{i=1}^ny_i}I(y_{min}>\theta)$
February 21st 2010, 12:12 PM
jass10816

I am not familiar with what you mean with $I$ ? Thanks for your help.
February 21st 2010, 12:24 PM
Moo

Indicator/characteristic function :

$I(y_i>\theta)=1$ if $y_i$ is indeed $>\theta$ . 0 otherwise.

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