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Let $X1 $, $X2 $, $X3 $...$Xn $ be n observations with distribution function $F $. Let $F^{*} $ be the empirical distribution of the random sample.

$F^{*} = \frac{1}{n} \sum I(X_{i} \le x)$ where I = 1, if the equality holds, 0 otherwise.

i) Find the distribution of $ \sum I(X_{i} \le x)$

ii) Find the limiting distribution of ${\sqrt(n)} [F^{*}(x)-F(x)]$

This is my approach;

Let r observations satisfy the equality and (n-r) observations don't.

$F^{*} = \frac{1}{n} (1+1+..1+0+0+..+0) = \frac{r}{n}$

$ \sum I(X_{i} \le x) = r$

So distribution is the number of successes in the sample.

But I'm not sure if this is the way to approach the problem.

For the 2nd part;

$ Y = {\sqrt(n)} [F^{*}(x)-F(x)] = {\sqrt(n)} [r/n - p] = \frac{1}{\sqrt(n)} [r-np]$

$ n \rightarrow \infty , r\rightarrow np$

therefore,

$ (r-np) \sim N(0, \sigma^{2})$

$ Y = \frac{1}{\sqrt(n)} [r-np] \sim N(0, \frac{\sigma^{2}}{n})$

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    $\begingroup$ Hint. The $X_i$ have to be independent for the problem to have meaning. Assuming that is so, then $\Sigma I$ is binomial distributed (you figure out n and p) and (ii) is easily obtained from the CLT for the binomial. $\endgroup$ – Gordon Smyth Jun 24 at 7:47
  • $\begingroup$ @GordonSmyth Is p = F(x) . But should I relate that to r? $\endgroup$ – Dom Jo Jun 24 at 8:02
  • $\begingroup$ @GordonSmyth I have updated it. Please check $\endgroup$ – Dom Jo Jun 24 at 8:34
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    $\begingroup$ You haven't yet defined what $\sigma^2$ is. $\endgroup$ – Gordon Smyth Jun 24 at 11:38
  • $\begingroup$ @GordonSmyth that would be the variance of binomial RV = npq correct? $\endgroup$ – Dom Jo Jun 24 at 12:33

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