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I have a question regarding maximum likelihood estimators/functions. I thought I understood the concept. But now i found an example of an Maximum likelihood function that i don’t get.

The question is to find the MLE for a hypergeometric distribution of M

I would have said that the Likelihood function is for $x_{1},.....,x_{k}$

$$L(M)=\prod_{i=1}^{k}\frac{\binom{M}{x_i}\binom{N-M}{n-x_i}}{\binom{N}{n}}$$

But now i found a solution just stating it to be

$$L(M)=\frac{\binom{M}{x}\binom{N-M}{n-x}}{\binom{N}{n}}$$

I don’t get why i can loose the Product sign...

Thanks for your help!

StubbornAtom
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Ang
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1 Answers1

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Your second formula is the probability mass function of hypergeometric distribution. Likelihood function $\mathcal{L}$ is defined as probability mass (or density) function $f$ evaluated on some point $x$ that is maximized in terms of parameter $\theta$

$$ \mathcal{L}(\theta|X) = f(X, \theta) $$

if you have more then one such point, you usually assume that they are independent and identically distributed (all come from the same distribution), what leads to likelihood defined in terms of all your data

$$ \mathcal{L}(\theta|X_1,X_2,\dots,X_n) = \prod_{i=1}^n f(X_i, \theta) $$

This is what the first formula is.

Tim
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  • The problem question, does state neither. that is why i would try to solve for any k. (X1....Xk)... my problem is that i have no idea how to solve it with the product sign.... i know i have to look at L(M)/L(M+1) and i can do that for the second formula without a problem. But i cant seem to get anywhere with the general form. (My math knowledge is not that great though....) – Ang Apr 23 '19 at 08:47
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    @Ang sorry but I don't understand your comment. What exactly is the problem? Your question was asking about meaning of the $\prod$ symbol, so I explained it. – Tim Apr 23 '19 at 08:50
  • many thanks for that. i am sorry if i sounded rude, this was not my intent. how would one solve for the estimator with the product sign? i am getting nowhere when trying to solve.... – Ang Apr 23 '19 at 08:56
  • @Ang this seems to be separate question. Maybe post it as a separate question, with describing in greater detail how did you try solving it and where are you stuck. – Tim Apr 23 '19 at 09:01
  • thanks! i will attempt a few more times and then post :) maybe i get lucky! – Ang Apr 23 '19 at 09:02