# Asymptotic normal distribution via the central limit theorem

I have a sample $n = 100$ with two "successes" (Two kids having a disease among 100). So we obviously have a binomial distribution.

First I had to compute the maximum likelihood (ML) estimator $\hat{p}$. I got $\hat{p} = \frac{k}{n}$.

Now, I have to derive asymptotic normal distribution for $\hat{p}$ via the central limit theorem (CLT).

I know that the expected value of $\hat{p}$ is not infinite and also variance is not infinite, so I know it will be normally distributed.

I have to know expected value and variance of $\hat{p}$ to get the asymptotic normal distribution, right?

I know that expected value is $\frac{k}{n}$. But what is variance?

• Why is $\hat{p}$ a random variable at all? Also, under almost any model that you might dream up, the CLT will not give you the asymptotic distribution of $\hat{p}$ directly; you have to infer from what the CLT tells you that the asymptotic distribution is that of a degenerate random variable that non-statisticians call a constant. See this answer to a related question. Oct 26, 2013 at 13:37
• Apparently "$k$" is a random variable referring to the number of successes and you are modeling it with a Binomial$(100, \hat{p})$ distribution. What is the mean of that distribution? What is its variance? How are these related to $k/n$?
– whuber
Oct 26, 2013 at 15:14
• "I have a sample n=100 with two "successes" (Two kids having a disease among 100). So we obviously have a binomial distribution" -- this conclusion isn't obvious to me, since we haven't established homogeneity of probability, or independence (kids from one school, or one neighborhood, with a highly contagious disease, for example, wouldn't be expected have independent disease status). How did you arrive at this being obvious without making explicit assumptions to that effect? Oct 28, 2013 at 0:47

Each child can be modeled as a Bernoulli r.v. $X_i$ with probability of having the disease equal to $p_i$, $X_i \sim B(p_i)$, $i=1,\dots ,n$. If you assume that a) $p_1 =p_2=\dots=p_n=p$ and b) that these are independent rv's then their joint density is

$$f(X_1,\dots,X_n) = \prod_{i=1}^{n}p^{x_i}(1-p)^{1-x_i}$$ and their log-likelihood function, viewed as a function of $p$ is

$$\ln L =\sum_{i=1}^{n}\left\{x_i\ln p+(1-x_i)\ln (1-p)\right\}$$

which leads to the MLE for $p$ $$\hat p =\frac 1n\sum_{i=1}^{n}x_i$$ which is unbiased since $$E\hat p =\frac 1n\sum_{i=1}^{n}Ex_i = \frac 1n np =p$$

Consider now the variable $$U_i = X_i - E(X_i) = X_i -p \Rightarrow X_i = U_i + p$$ We have $$EU_i = 0,\qquad Var(u_i) = Var(X_i) = p(1-p)$$ so it is covariance-stationary.

Subsitute for the $x$'s in the estimator

$$\hat p =\frac 1n\sum_{i=1}^{n}(u_i+p) = \frac 1n\sum_{i=1}^{n}u_i +p$$ and consider the quantity $$\sqrt n (\hat p-p) =\sqrt n\frac 1n\sum_{i=1}^{n}u_i= \frac {1}{\sqrt n}\sum_{i=1}^{n}u_i$$

Since the $U$'s are covariance stationary, (and evidently i..i.d) then the CLT certainly applies and so

$$\sqrt n (\hat p-p) \rightarrow_d N\left (0, p(1-p)\right)$$

For approximate statistical inference, we manipulate this expression through $$\sqrt n (\hat p-p) = Z \Rightarrow \hat p = \frac {1} {\sqrt n}Z +p$$

and write that, for "large samples"

$$\hat p \sim_{approx} N\left (p, \frac {p(1-p)}{n}\right)$$

(but not when $n$ truly goes to infinity, since then $\hat p$ does not have a distribution, but collapses to a constant, the true value $p$ since $\hat p$ is a consistent estimator).

• Thanks Alecos! In the meantime I got this as well and it's nice that you confirm my results! :-) But there is something else I want to know: This is the approximation. What is the exact distribution of the estimated p? Oct 27, 2013 at 16:46
• Michael, that's another question. You have the functional form of $\hat p$. Look up how we derive the distribution of a function of a discrete random variable. And if you cannot solve this, post the question. Oct 27, 2013 at 21:44