# How to estimate the margins distribution using ecdf?

Suppose we have a random variables x.

x <- rnorm(500, 2,3)

Suppose that we do not know the marginal distribution of (x) (I know it is normal).

Then, I would like to estimate the marginal using empirical cumulative distribution function (ecdf).

Then, I can do that easily in R, i.e.,

xecdf <- ecdf(x)

Then, I can plot it as follows:

plot(xecdf).

Now I plot it and everything is great. So, I have estimated the margins non-parametrically.

My question is, how can I know what is the estimated distribution from the result of xecdf? That is, in our example, the margin is normal. I suppose it is unknown just for my question.

Now, how I can derive from the result of ecdf that the margin is normal? How to prove from the result of the ecdf that the margins is normal.

In other words,

I understand the ecdf. however, how I know from ecdf that my margin is normal. That is, when we estimate the margins, we would like to know that is the type of margins. Are they normal or any other distribution? One way to estimate the margins is to use ecdf. So, how can I know the type of margins form ecdf.

• You can’t “derive” functional form of the underlying distribution from empirical distribution. Empirical distribution approximates it, that’s all. Could you be more precise what do you mean? – Tim Jan 13 '18 at 12:18
• @Tim. Thanks for your comment. I understand the ecdf. however, how I know from ecdf that my margin is normal. That is, when we estimate the margins, we would like to know that is the margins. Are they normal or any other distribution. One way is to use ecdf. So, how can I know the type of margins form ecdf. – Silver_80 Jan 13 '18 at 12:20
• @Tim, Thanks again. So, from your comment, we cannot know the type of the margins distribution if we use the ecdf. So, why we use ecdf to estimate them? That is, it does not help use to understand the type of margins. – Silver_80 Jan 13 '18 at 12:23