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#1




SOA #250
(Please forgive my makeshift symbols, I'm posting this from my phone and it is cumbersome to use Latex; ¤ is theta). In the answer, we're "shown" that ¤(186^2)exp(¤/186)¤(91^2)exp(¤/91)¤(66^2)exp(¤/66)(exp(¤/60))^7 is proportional to ¤^3(exp(0.148184¤). How? I have no idea what they're doing here (and this isnt the only mle solution where they're doing this).
It's like the coefficients just go away and I'm also not sure what they're doing in the exponents. 
#2




I haven't worker it out, but I assume that is the result if you add the exponents, combine the theta terms, and leave out the constant.

#3




Okay, yes. That's what they did. It's quite obvious in retrospect. I guess now my question is why are we allowed to drop the coefficients? Is it because they go away anyway once we take the derivative of the loglikelihood?

#4




Yes, logarithms turn multiplication into addition. Derivatives make additive constants zero.
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#5




It's also because maximizing 1000 (say) times f occurs at the same point as maximizing f.
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#6




Got it, it all makes sense now. I don't know what my problem was regarding the exponents, I think I was just adding them wrong. Sometimes when I get hyperfocused on the exam material itself, I start having issues with stuff like this that should be obvious.

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