Question

I know that bias estimator is the difference between the actual and expected value.
It is unbiased when that difference equals to 0, and biased otherwise.

Now my question is, let’s assume I have some set of values (x1,x2,……..,xn) and I want to estimate its mean.
I assign the mean to have the value of the first value (x1).

And the lecturer said it is unbiased…
My question is – why?

The mean of the set of values probably won’t equal to the value of x1, so I would say it should be biased. Why is it UNBIASED?

And if it is Unbiased, (which suppose to be good) why it is bad?

Answer 1

Now my question is, let’s assume I have some set of values
(x1,x2,……..,xn) and I want to estimate its mean. I assign the mean
to have the value of the first value (x1).

The sample mean is the sample mean – you cannot just assign it to be x1

Your sample (x1,…,xn) is a sample of size n from some distribution that is has a probability density function defined by some parameters. Let us just say for simplicity that this is a normal distribution with mean M and variance V. This means that the expected value of EACH x_i is

E(x_i)  = M. for all i since they all are sampled from same population/distribution

The sample mean is Xbar = (x1 + .. + xn) / n

The expectation of this is

 E(Xbar) = (E(x1) + ... + E(xn))/n  = nM/n = M

using simply properties of expectation.
Therefore expected value of your sample mean is M which is same as expected value of population mean. Therefore sample mean is unbiased as the definition of unbiased estimation is that the expectation of the statistic used as estimator is the same as the parameter it is estimating. If you cannot follow the above, I would encourage you to take it up with your lecturer, or post on maths or stats versions of stackexchange.