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The hypergeometric distribution arises from sampling without replacement. The similar binomial sampling distribution assumes replacement. Hypergeometric distributions are commonly used in quality assurance to determine the acceptable quality limit/level (AQL) given a sampling rate. Given that the inverse CDF function is indispensable in computing confidence intervals and critical values, and given the ubiquity of sampling without replacement in the real world, I was surprised that I could not find a function for the inverse cumulative density function of the hypergeometric distribution. I suppose it is because the exact solution for the normal CDF is very complex, relying on the $_3 F _2$ of the generalized hypergeometric function.

While the normal distribution and binomial distributions approximate confidence intervals and critical values of the hypergeometric distribution, they may result in misleading thresholds, especially for small sample sizes and p values.

What is the ”good” analytical approach (no tables) to finding critical values and/or confidence interval for hypergeometrically distributed variables?

I define “good” here as a either a discrete or continuous method that converges to the true solution.

Also, there’s extra credit if you can improve the root finding algorithm given in the following rough VBA function:

Function inverseHyperGeom_exact(ByVal probability, ByVal Number_sample, ByVal Population_s, _
    ByVal Number_pop, Optional ByVal fromLeft As Boolean = True, Optional ByVal maxIter As Integer = 500)

    Dim trial_prob As Double

    Dim i As Integer

    ' HYPGEOM.DIST(sample_s,number_sample,population_s,number_pop,cumulative)
'    Sample_s     Required. The number of successes in the sample.
'    Number_sample     Required. The size of the sample.
'    Population_s     Required. The number of successes in the population.
'    Number_pop     Required. The population size.
'    Cumulative     Required...

    i = 0

    Do While trial_prob < probability and i <= maxIter

        trial_prob = WorksheetFunction.HypGeom_Dist(i, Number_sample, Population_s, Number_pop, True)

        i = i + 1

    Loop

    If fromLeft = True Then
        inverseHyperGeom_exact = i - 1
    Else
        inverseHyperGeom_exact = i
    End If

End Function
David Addison
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    The R function qhyper is the inverse of the CDF phyper of a hypergeometric distribution. If a lot has 90 good items and 10 defective ones the probability that a sample of size 3 will have $\le 1$ good item is phyper(1, 90,10, 3), which returns 0.0257885. So qhyper(.02, 90,10, 3) returns 1, while qhyper(.03, 90,10, 3) returns 2. // Because intermediate computations can overflow, the programming of such functions has to be done carefully. There are 'log' options for extreme cases, see, for details. – BruceET Jun 10 '19 at 02:16
  • What is your question? – whuber Jun 10 '19 at 14:46
  • See question as amended. – David Addison Jun 10 '19 at 15:44
  • Thank you. The question depends on (a) what you mean by "correct"--would it include any considerations of practicability, for instance?--and (b) the ranges of values for which you need the computation. It's evident, for example, that your VBA code will be useless for any input that requires a huge number of increments of i. (Is there a reason you're not using a better root finder than this sequential search?) Would good approximations qualify as "correct" or not? – whuber Jun 10 '19 at 16:44
  • My rough code is a proof of concept that there may be no integer values that exactly satisfy the critical p value. I would accept a continuous analog distribution of the hypergeometric distribution as a “good approximation”. For example, the critical values for normal distributions are usually used to determine confidence intervals for binomial distributions. This is because the normal distribution is the infinite step limit of the binomial distribution. – David Addison Jun 10 '19 at 19:50

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