Is the sample mean and variance always unbiased?

[logarithmic]

I'm wondering if the sample mean $$\sum{x_i}/n$$ and sample variance $$\frac{1}{n-1}\sum{(x_i-\bar{x})^2}$$ is always an unbiased estimate of the true expected value and variance of the random variable X, where x_i are iid samples. Or at least asymptotically unbiased.

I don't think it is, since the sample mean (and variance) is only the MLE of a few distributions, like the normal and poisson. So I see no reason for it to be unbiased for all distributions.

However, I've been running some simulations on R, and I cannot seem to find an example of a distribution where the sample mean isn't unbiased, same for the sample variance.

 

[xiaoB]

Is it Hypothesis Testing ?

 

[logarithmic]

Is it Hypothesis Testing ?

No, not hypothesis testing. I just want to know, given some numbers from any unknown distribution, whether if I use the sample mean and sample variance, I will get an unbiased estimate for the true mean and variance.