# UMP test for hypergeometric distribution

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

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?

• You can look at $\dfrac{P(X=x\mid N,D+1,n)}{P(X=x\mid N,D,n)}$ with lots of cancellations Nov 26, 2022 at 7:55
• @Henry why should we look at that term ? Can you explain ? Nov 26, 2022 at 8:48
• 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 Nov 26, 2022 at 14:41
• @Henry okay . i get it now Nov 26, 2022 at 14:59

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 $$D?$$

• 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 ? Nov 26, 2022 at 13:08
• @Andrew741 I have edited. Hope you can make insight of the hint. Nov 26, 2022 at 15:26
• +1 For providing an outline and leaving the formal work for OP. Nov 26, 2022 at 19:59