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From Held, et al., "Appplied Statistical Inference", ex. 1c).

Let $X_{1:3}$ denote a random sample of size $n=3$ from a Cauchy $C(\theta, 1)$ distribution. $\theta \in \mathcal{R}$ is the location parameter of the Cauchy distribution with density

$$ f(x) = \frac{1}{\pi} \frac{1}{1+ (x-\theta)^2}, $$

derive the likelihood function for $\theta$.

Ok, so I would say

$$ L(\theta; x_{1:n}) = \prod_{i=1}^{n=3} f(x_i; \theta) = \frac{1}{\pi^3}\prod_{i=1}^{n=3} \frac{1}{1+ (x_i-\theta)^2}$$

Is that all?

kjetil b halvorsen
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TMOTTM
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  • $1/\pi$ plays no role when maximizing it, so is not needed. – Tim Oct 23 '16 at 19:27
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    Looks right to me – Jon Oct 23 '16 at 19:34
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    Assume your answer is correct. The question "Is that all" would only seem to take the answer "yes". "Yes" is generally not adequate as an answer. If possible, can you reformulate your question to ask something for which a longer-than-one-word answer makes sense? – Glen_b Oct 24 '16 at 00:51
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    The extended version of the question: what applied stats formalism did i miss including? – TMOTTM Oct 24 '16 at 05:32

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