Maximum likelihood + Kernel Density Estimation Here my problem. I have a limited data sets of random variables $x: x_1, x_2, ..., x_N$. I can estimate the probability density function of $x$ by mean of Kernel Density Estimation method. (It works quite well in comparing with the complete data sets).
But how can I determine the "true" mean value of $x$? Which hypotheses must be added?
Thanks
 A: You can't determine the population mean from a sample.
However, you can estimate it.
Unless you know (or can assume) a distribution, why would you not simply estimate the population mean by the sample mean?
A: For a reasonably large set of observations, the Weak Law of Large Numbers ensures that the sample mean is a consistent estimator of the population mean. The requirements are quite mild, we only have to assume that the observations are iid from a population with finite mean but not finite variance.
Alternatively, again for reasonably large samples, a Confidence Interval can be formed by exploiting the Central Limit Theorem, namely that the random variable:
$$\frac{\bar{X}-\mu}{S/\sqrt{n}}\to^D N\left(0,1 \right)$$
Then you can "pivot" it and obtain the desired CI for a specified value of $\alpha$:
$$1-\alpha=P\left(\bar{X}-z_{\alpha/2}\frac{S}{\sqrt{n}}<\mu <\bar{X}+z_{\alpha/2} \frac{S}{\sqrt{n}}\right) $$
where $z_{\alpha/2}$ is the upper $\alpha/2$ critical point of a standard normal distribution, i.e. $P \left(Z>z_{\alpha/2}\right)=\alpha/2$.
Hope this helps.
