Mathematics Asked by tc1729 on December 16, 2020

Consider choosing $Mn/6$ balls from a population consisting of $M$ balls of each of $n$ colors (so $Mn$ balls in total). So the density function of the sample is given by a multivariate hypergeometric distribution: $$f(x_1,ldots, x_n) = frac{binom{M}{x_1}cdotsbinom{M}{x_n}}{binom{Mn}{Mn/6}}.$$ Can one say anything about the limiting behavior of the distribution as $Mtoinfty$, where the number of colors $n$ is fixed? Since the sample size grows at the same rate as the population size, this wouldn’t converge to a binomial/multinomial distribution as it would if the sample size were fixed. Any help is appreciated! (The $1/6$ in $Mn/6$ is arbitrary, I’m just curious in general about the case where the sample size is always a fixed fraction of the population size).

I guess it wouldn’t surprise me if nothing really useful can be said, in which case I have a related question. Suppose you consider the same scenario, but instead of starting off with $M$ balls of each color, we only started off with, say, $5M/6$ balls of each color. So the modified density function would be: $$g(x_1,ldots, x_n) = frac{binom{5M/6}{x_1}cdotsbinom{5M/6}{x_n}}{binom{5Mn/6}{Mn/6}}.$$ As $Mtoinfty$, is there any meaningful relationship between $f$ and $g$ that can be made? It vaguely seems to me like as $M$ grows large the two densities should look more and more alike, but it’s possible that that intuition is awry.

For the $m^{th}$ ball of color $n$ let $X_{m}^{n}$ be the indicator random variable for whether it was drawn. Suppose we are drawing fraction $mu in (0,1)$ of the balls in the population (e.g. $mu = 1/6$), then:

$$mathbb{E}[X_{m}^{n}] = mu$$

$$Var(X_{m}^{n}) = mu(1-mu) equiv sigma^{2}$$

For any $(m,n) neq (m',n')$:

$$begin{align} Cov(X^{n}_{m}, X^{n'}_{m'}) &= mathbb{E}[X_{m}^{n}X_{m'}^{n'}]-mu^{2} \ &= -mu (1-mu)/(MN-1) \ &= -sigma^{2}/(MN-1) end{align}$$

Fixing $N$, for any $M$ denote: $$bar{X}^{n}_{M} = frac{1}{M}sum_{m=1}^{M} X_{m}^{n}$$ Which has the following properties: $$mathbb{E}[bar{X}^{n}_{M}] = mu$$

$$begin{align} Var(bar{X}^{n}_{M}) &= frac{1}{M^{2}} left[ M Var(X_{m}^{n}) + M(M-1)Cov(X_{m}^{n}) right] \ &= frac{1}{M} left[ Var(X_{m}^{n}) + (M-1)Cov(X_{m}^{n}) right] \ &= frac{1}{M} left[ sigma^{2} - (M-1)sigma^{2}/(MN-1) right] \ &= frac{sigma^{2}}{M}left( frac{M(N-1)}{MN-1} right) end{align}$$

Define $Y^{n}_{M} = sqrt{M}(bar{X}^{n}_{M} - mu)$, then by the central limit theorem $Y^{n}_{M}$ converges in distribution to $N(0, sigma^{2}(N-1)/N)$. (Note the central limit theorem still applies here though the random variables are slightly dependent. Cite Theorem 1 of "The Central Limit Theorem For Dependent Random Variables" by Wassily Hoeffding and Herbert Robbins.)

The covariance for $n neq n'$ is:

$$Cov(bar{X}^{n}_{M}, bar{X}^{n'}_{M}) = Cov(X^{n}_{m}, X^{n'}_{m'}) = -sigma^{2}/(MN-1)$$

$$Rightarrow Cov(Y^{n}_{M}, Y^{n'}_{M}) = Msigma^{2}/(MN-1) rightarrow -sigma^{2}/(N-1)$$

Thus, $(Y^{1}_{M}, ldots , Y^{N}_{M})$ converges in distribution to a multivariate normal centered around $0$ with a covariance matrix that has $sigma^{2}(N-1)/N$ on the diagonal and $-sigma^{2}/(N-1)$ on the off-diagonal. (Note, this covariance matrix has rank $N-1$.)

(To prove $(Y^{1}_{M}, ldots , Y^{N}_{M})$ does indeed converge to a multivariate normal, we would have to show any linear combination of them converges to a normal, which follows via the same argument used to show $Y^{n}_{M}$ converges to a normal.)

Correct answer by Sherwin Lott on December 16, 2020

I don't think that in the present case a limiting distribution exists in the strict sense as $Mtoinfty$. However, it seems to be the case that the hypergeometric distribution approaches a normal distribution in this limit, with diminishing height, increasing average and deviation. More explicitly, consider the case $n=2$, for which the hypergeometric distribution reads:

$$P(x)=frac{binom{m}{x}binom{M-m}{N-x}}{binom{M}{N}}$$

and to tackle the particular problem at hand set $m=frac{M}{2}~,~N=fM~,~ f< 1/2$. Note that if the sampling fraction exceeds the critical value $1/2$ it becomes more complicated to obtain a simple estimate using the Stirling approximation for the factorial, so I will work with the previously mentioned restricted case. In this case it is clear that $xin [0,fM]$. After plugging in the Stirling approximation $$x!approx x^xe^{-x}sqrt{2pi x}$$

and simplifying we obtain a monstrous expression for $P(x)$ in the limit $Mtoinfty$ which I will omit for now. The limit of this expression as one lets $M$ grow is, strictly speaking, zero. However, it turns out that $ln P(x=fMt)$ is proportional to $M$. This points to the fact that as $Mtoinfty$, since $ln P<0$ only points near the maximum of $P$ will attain non-zero values. We see that the maximum is attained at $t=1/2$. With this, we conclude after simplification that

$$P(x)approxsqrt{frac{2}{pi f(1-f)M}}expleft[-frac{2}{f(1-f)M}(x-fM/2)^2right]$$

This means that the distribution moves further along the x-axis as $Mtoinfty$ but also shortens and broadens to keep the normalization constant. Numerical evidence supports this result as shown in the plot below.

Answered by DinosaurEgg on December 16, 2020

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