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As the title states, is there a continuous analogue of a Hypergeometric distribution?

If $ X \sim H(m,n,N)$ is a common Hypergeometric distribution, where $N$ is the population size, $n$ is the number of draws, and $m$ is the number of success. In my case $N,n,m$ are not integer, therefore I could have a not integer number of success.

Austin Mohr
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Simone
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3 Answers3

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Short version, yes.

Long version, it's complicated. There is an interesting link between probability distributions and orthogonal polynomials. For instance, consider the Hermite polynomials, $H_0, H_1, \ldots$. These are orthogonal with respect to the weight function $e^{-x^2/2}$ on the support $(-\infty,\infty)$. In other words, $\int_{-\infty}^{\infty}H_i(x)H_j(x)e^{-x^2/2}dx = \sqrt{2\pi}i!\delta_{ij}$.

The weighting function and the coefficient having a factor $\sqrt{2\pi}$ should ring a bell; specifically, you find these represented in a standard normal distribution.

Norbert Wiener recognized that the two are intricately linked and developed what is called the Gauss-Hermite Polynomial Chaos. In short, we can write a random variable $k$ of any distribution as an infinite series about a random variable $\zeta$ of a normal (Gaussian) distribution by using the Hermite polynomials as basis functions: $$ k = \sum_{i=0}^{\infty}k_i H_i(\zeta).$$

It turns out, you can generalize this to other distributions, and to discrete distributions.

There is something called the Askey Scheme, which is essentially a family tree relating hypergeometric orthogonal polynomials. Many of the weighting functions for these polynomials are distribution functions for probability distributions, meaning we can perform a Wiener-Askey Polynomial Chaos expansion for any random variable about another random variable of almost any distribution of our choosing. See also: http://www.dam.brown.edu/scicomp/media/report_files/BrownSC-2003-07.pdf

The hypergeometric family linked to the discrete hypergeometric distribution is the Hahn family of polynomials. The continuous analogue is uncreatively called the "Continuous Hahn" family. This leads to the following answer:

The weighting function of the Continuous Hahn polynomials will give you the continuous analogue of the discrete hypergeometric distribution. In fact, it is most likely identical to within a scaling parameter. These functions are quite complicated, so it is not intuitive.

The polynomial chaos distribution is quite interesting and powerful. If you look at the series expansion, it is analogous to a Taylor expansion of a random variable. Importantly, it allows you to write a random variable in a possibly unknown or complicated distribution in terms of any distribution of your choosing; as with any series expansion, the challenge is to then compute the deterministic coefficients. This becomes quite potent when you have, say, a dynamical system parameterized about a random parameter -- instead of using complicated stochastic models, or time-consuming Monte-Carlo analysis, you can frame the problem as one of computing a handful of deterministic coefficients, and generate statistical moments and/or Monte Carlo samples from them easily.

Interestingly, the first coefficient, $k_0$ always represents the distribution mean.

Emily
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  • Ah, you're using the physicist's normalization for Hermite polynomials... :) – J. M. ain't a mathematician Aug 04 '12 at 04:56
  • @J.M. In this example yes -- and only because I couldn't recall the exact structure of the coefficients in the inner product (so I wiki'd it :D). In practice, though, I tend to use neither the probabilist's nor the physicist's normalization for generalized Polynomial Chaos. It is much more convenient to choose the normalization such that $\langle H_i^2(\zeta) \rangle = 1$ for all $i = j$. That normalization is somewhere "in between" the two. – Emily Aug 04 '12 at 05:00
  • Interesting. So, the only practical way to do it would be to have a look to polynomial chaos distribution. Furthermore, the current r.v. $X$, which I was looking for a continuous analogue, is used to compute another r.v.. The transformation is very similar to $Y = H(X)$. Where $H$ is the entropy function. It would be really great if I was able to compute the expected value of $Y$. – Simone Aug 07 '12 at 00:32
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    @Simone Generalized polynomial chaos makes it very easy to do that. You can take an intrusive approach, where you replace the random variable $X$ in your transformation with its polynomial chaos expansion. You compute a second polynomial chaos expansion about the same RV for $Y$:

    $$ Y = \sum_{i=0}^P Y_i \Phi_i(\zeta),$$ $$ X = \sum_{i=0}^P X_i \Phi_i(\zeta),$$ $$ Y = H(X) = H\left(\sum_{i=0} X_i(\zeta)\Phi_i(\zeta)\right).$$

    Since it is "very easy" to compute the deterministic coefficients $X_i$, all you need to do is compute $Y_i$ using the Galerkin method. Also, $E[Y] = Y_0$.

     – Emily Aug 07 '12 at 01:36
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    The Galerkin approach exploits orthogonality of the polynomial basis family $\Phi(\zeta)$ on its support to allow for the generation of $P+1$ equations to compute the $P+1$ $Y_i$'s. Simply project the expansion of $Y$ onto each orthogonal basis polynomial, $\langle Y \Phi_i(\zeta)$, and then (depending on the form of your equation $H(X)$) orthogonality should wipe out all of the "cross" terms, leaving you to compute something along the lines of $$Y_i = \frac{\langle Y \Phi_i(\zeta) \rangle}{\langle \Phi_i^2(\zeta)\rangle}.$$ – Emily Aug 07 '12 at 01:40
  • Is it also possible to use the Gauss-Hermite polynomial chaos for the cumulative hypergeometric distribution? – linello Jul 09 '14 at 14:13
  • can you please point me to some good explanation of the continuous Hahn polynomials? I'm interested in understanding if it is possible to compute the hahn polynomial from the following $3F_2({1,mi-m,mi-pi},{mi+1,mi+p-pi-m+1},1)$ which comes a simplification of the hypergeometric cumulative distribution $\sum{i=mi}^m \binom{pi}{i} \binom{p-pi}{m-i}/\binom{p}{m}$ – linello Jul 10 '14 at 11:18
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    It is possible, though not necessarily practical, to use generalized polynomial chaos to represent any distribution with finite second moments. The best reference is the paper by Xiu and Karniadakis entitled "The Wiener-Askey polynomial chaos for stochastic differential equations. " It is very fundamental, and may even address what you're looking for. – Emily Jul 10 '14 at 18:17
  • Thanks for the good reference. What I'm still missing is how is possible to compute the cumulative hypergeometric distribution given the Wiener-Askey chaos expansion. Why isn't the sum approximable with an integral as follows?

    $\sum_{i=m_i}^m \binom{p_i}{i}\binom{p-p_i}{m-i}/\binom{p}{m} \approx \int_{m_i-1/2}^{m+1/2} \binom{p_i}{x}\binom{p-p_i}{m-x}/\binom{p}{m} dx $ where the binomial coefficients are evaluated with the help of $\Gamma$ function as $\binom{a}{b} = \frac{\Gamma(a+1)}{\Gamma(b+1)\Gamma(a-b+1)}$

     – linello Jul 10 '14 at 18:55
  • I got here since I came to the same question, but I wonder why no one refers here to the analogy of the binomial distribution (BD) and the Gauß function (GF). I wonder if one imposes a completely analogous "continuation" procedure which leads from BD to GF, where would one end for the hypergeometric distribution, as well the continuous Hahn family? – Raphael J.F. Berger Feb 14 '18 at 13:17
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It is also the Pearson distributions. In the original article the change of the probabilities (at $k+1$ wrt $k$) of hypergeometric distribution divided by the current probability is a rational function of the index $k$.

Pearson noted that if the $k+1$ and k go close, then at the limit this is a derivative of the pdf devided by pdf and it is equal to the rational function (1st order devided by a second order polynomial). He solved the system and created a Cartesian product (and a chart) of any parameter entering. I would see that his distributions are simply the scaled variable times a normalized Appell $F_1$ function. I would say that a generalization to that would be a generalized Lauricella $F^{(n)}_D$ function, normalized, times the scaled variable.

I like very much this simple thought of Pearson. The reason he developed this function system is that the real life probabilities have not an infinite parameter range and they are not symmetrical like the Normal. Normal would not fit the biological data of the day and in this world nothing is without limits and asymmetry like the height and the armslength of people and crabs. But the original article (series) is more interesting to read. We owe this generalization to anthropology.

Simone
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A.S.Stefanakis
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Pearson has published many articles. Try the last one: http://rsta.royalsocietypublishing.org/content/216/538-548/429 But there are nice reasons to check the previoys ones.

The Appel and Lauricella generalizations cannot be known to Pearson but they are mentioned!! and in the name of absence of experimental data they are left off (he says they are left in abeyance).

You can work it out when you will notice that for real roots of the binomial in his equation, the integral of the cumulative distribution function can be put in the form of an integral representation of these functions. Put them side to side to tell.

A.S.Stefanakis
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