Suppose we have a sample of 100 observations, each observation detailing the weekly wage of an individual in a country.
Would the sample mean(estimator) be biased? I know this can be deduced by determining whether or not the observations are iid. Would they satisfy this criteria?
My thoughts are that different wages have varying probabilities (It is extremely unlikely an individual earns a very, very high wage) compared to that of a typical one. This would mean that the observations are not identically distributed and thus not iid, and so this allows for the possibility of bias in the sample mean.
Would this be correct? I am very unsure about this and feel that I may have just described uniform distributions rather than identical ones.
Thanks.
The sample mean is an unbiased estimator of the population mean as long as the expected values of the random variables are the same. We do not need independence nor do we need identical distributions as long as the expected values are the same. It could be that the variances, for example, are not equal, but this is not a problem.
We have that $$ \operatorname E\bar X =\operatorname E\Bigl[\frac1n\sum_{k=1}^nX_k\Bigr] =\frac1n\sum_{k=1}^n\operatorname EX_k =\frac1n\sum_{k=1}^n\operatorname EX_1 =\frac1nn\operatorname EX_1 =\operatorname EX_1. $$ We only used the linearity of the expected value and the assumption that the expected values are the same to show that the estimator is unbiased. Observe that in this case there is a single parameter that we want to estimate. We want to estimate the expected value of these random variables (population mean).
If the expected values are not the same, then it is not clear what we estimate if we use the sample mean. So we have to be careful when we speak about the unbiasedness of the sample mean of the random variables with different expectations because it is not clear what the parameter that we want to estimate is.
I hope this is helpful.