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$P(X=x|N,D,n)=\frac{^DC_x \times ^{(N-D)}C_{(n-x)}}{^NC_n}$

Now, I was trying to test for $H_0:D\le D_0$ vs $H_1:D>D_0$ using likelihood ratio test.

But to find the maximum likelihood estimate of $D,$ the terms are in $^DC_x$ form, so I could not proceed further.

Can anyone provide me how to find the estimate of $D$ or provide some other test for this problem?

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    $\begingroup$ You can look at $\dfrac{P(X=x\mid N,D+1,n)}{P(X=x\mid N,D,n)}$ with lots of cancellations $\endgroup$
    – Henry
    Commented Nov 26, 2022 at 7:55
  • $\begingroup$ @Henry why should we look at that term ? Can you explain ? $\endgroup$
    – Andrew741
    Commented Nov 26, 2022 at 8:48
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    $\begingroup$ You are looking at how the likelihood changes as $D$ changes, but cannot take the derivative as $D$ can only be an integer. So you might instead take the ratio of successive terms to give you an idea of its behaviour $\endgroup$
    – Henry
    Commented Nov 26, 2022 at 14:41
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    $\begingroup$ @Henry okay . i get it now $\endgroup$
    – Andrew741
    Commented Nov 26, 2022 at 14:59

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Here $X\sim \operatorname{HyperGeo}(N,D,n) . $

We need to find (if it exists) the UMP test of $\mathcal H_0: D\leq D_0$ vs $\mathcal H_1: D> D_0.$

What should be the approach to tackle such problem, in general?

Possible line of approach:

$\bullet$ Check whether the family of distributions possesses the property of Monotone Likelihood Ratio (MLR) through any statistic $T(\mathbf x). $

$\bullet$ If so, hypotheses like above can be assessed via theorem of Karlin-Rubin which does show the existence of a UMP level $\alpha$ test.

Edit: How to calculate the MLE of $D: ~\hat D?$ One needs to solve $ \mathcal L( D|\mathbf x) \geq \mathcal L( D-1|\mathbf x) .$ From this, one would get a bound on $D.$ Can this be MLE of $D?$

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  • $\begingroup$ I was trying to find maximum likelihood ratio but for that I need a mle of the likelihood function . Can you tell how to find the that estimate for hypergeometric distribution ? $\endgroup$
    – Andrew741
    Commented Nov 26, 2022 at 13:08
  • $\begingroup$ @Andrew741 I have edited. Hope you can make insight of the hint. $\endgroup$
    – User1865345
    Commented Nov 26, 2022 at 15:26
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    $\begingroup$ +1 For providing an outline and leaving the formal work for OP. $\endgroup$
    – utobi
    Commented Nov 26, 2022 at 19:59

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