Confidence interval for the standard deviation on a bimodal distribution

Question

I have a bimodal distribution, and I wish to estimate the mean and standard deviation of the population (well, these could be 2 sub-populations according to the shape).

With the mean I have no problem, the sample is relatively large (n~70), and so the sample mean is approximately normally distributed according to the central limit theorem. I can calculate the corresponding 95% CI.

Now I want to do the same for the standard deviation, but I am not sure if the same theory apply. Can I say that the variance is approximately chi squared distributed when the sample is large ?

Edit:

To test my question, I ran this simulation in R:

mu1 <- 23   
mu2 <- 40
sig1 <- 4.3
sig2 <- 2.8
cpct <- 0.32
m <- 10000
means <-rep(0,m)
sds <-rep(0,m)
vars <-rep(0,m)

bimodalDistFunc <- function (n,cpct, mu1, mu2, sig1, sig2) {
  y0 <- rnorm(n,mean=mu1, sd = sig1)
  y1 <- rnorm(n,mean=mu2, sd = sig2)

  flag <- rbinom(n,size=1,prob=cpct)
  y <- y0*(1 - flag) + y1*flag 
}


for (i in 1:m)
{
    bimodalData <- bimodalDistFunc(n=10000,cpct,mu1,mu2, sig1,sig2)
    means[i] = mean(bimodalData)    
    sds[i] = sqrt(var(bimodalData))
    vars[i] = var(bimodalData)
}

par(mfrow=c(2,2))
hist(bimodalData)
hist(means)
hist(sds)
hist(vars)

And I got the following result:

The distribution of the variances is not chi squared, but why is it normal ? (the bimodal histogram is an example of one of the simulated distributions)

Answer 1

This won't be rigorous, but it should give you a feel for why it might often tend to occur:

1) Imagine you were calculating not the $n-1$ denominator variance, but the $n$-denominator version (this only gives a scaling factor, so it doesn't impact the shape you see... and that scaling factor goes to 1 in the limit)

2) Consider that as sample sizes become large, the distribution of $X_i-\overline{X}$ approaches the distribution of $X_i-\mu$ (e.g. via Slutsky's theorem).

3) Now consider $Y=(X_i-\mu)^2$; by the Central Limit theorem $\sqrt{n}(\overline{Y}-E(Y))$ converges to a normal distribution, as long as the conditions hold (e.g. you need $\text{Var}(Y)$ to exist). Further note that $E(Y)=\sigma^2$.

So - in essence because the sample variance is effectively just a kind of average - you might not be surprised to see sample variance to approach normality (centered at the population variance) as sample sizes become large.

In Asymptotic Statistics, A. W. van der Vaart pursues a somewhat more rigorous argument (see end p26-p27) by writing the $n$-denominator form of variance as a function of $\overline{X}$ and $\overline{X^2}$ and looking at the multivariate CLT applied to the pair.

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However, you have to be careful - in some situations at least - this approximate normality doesn't hold. All the conditions must hold for the argument to go through. (In particular, there's something a little unusual about $(\overline{X}, \overline{X^2})$ for the Bernoulli($\frac{1}{2}$) case.)

Consider the Bernoulli distribution with $p=\frac{_1}{^2}$; for that case, as sample sizes become large, sample variance doesn't approach normality.

In that case, at least, the distribution of the sample variance - even in quite large samples - doesn't become more normal.