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Let X be a random variable with the following PMF

$ f(x;\theta)=\frac{1}{4}, x=1,2 $

$ f(x;\theta)=\frac{1+\theta}{4}, x=3 $

$ f(x;\theta)=\frac{1-\theta}{4}, x=4 $

and $0\leq\theta\leq1$

Find a sufficient statistic for $\theta$

ATTEMPTED SOLUTION

I am familiar with finding the sufficient statistic when given a PDF with X already present. However, X is not here. So, I began by finding the likelihood function with the following method

$L(\theta)=\frac{1}{4}*\frac{1}{4}*\frac{1+\theta}{4}*\frac{1-\theta}{4}$\

$L(\theta)=\frac{1}{256}*(1-\theta^{2})$

From here I am not sure what to do. There is no X present and I and unsure how to introduce one. Any help would be appreciated.

Statsdude
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    This is the same question as https://stats.stackexchange.com/questions/496468/sufficient-statistics-and-discrete-distributions, with only minor variations. Maybe a concrete example will help: you take a sample of three values and they turn out to equal $1,$ $3,$ and $3$ (in that order). What is the likelihood of this sample? – whuber Nov 17 '20 at 21:45
  • @whuber Thanks for the link but that question wasn't answered either so I don't know if what was attempted is the correct way of thinking – Statsdude Nov 17 '20 at 22:29
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    Although the duplicate is not answered, there are very useful hints in the question and its comments. – whuber Nov 18 '20 at 13:44

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