Intuitive understanding of the difference between consistent and asymptotically unbiased I am trying to to get an intuitive understanding and feel for the difference and practical difference between the term consistent and asymptotically unbiased.  I know their mathematical/statistical definitions, but I'm looking for something intuitive.  To me, looking at their individual definitions, they almost seem to be the same thing.  I realize the difference must be subtle but I just don't see it. I'm try to visualize the differences, but just can't.  Can anyone help?   
 A: I would like to clarify that consistency in general does not imply asymptotic unbiasedness. Consider an estimator for $0$ taking value $0$ with probability $(n-1)/n$ and value $n$ with probability $1/n$. It is a biased estimator since the expected value is always equal to $1$ and the bias does not disappear even if $n\to\infty$. However, it is a consistent estimator since it converges to $0$ in probability as $n\to\infty$.
Asymptotic unbiasedness does not imply consistency either as it is mentioned in other answers. For example, the periodogram is an asymptotically unbiased estimator of the spectral density, but it is not consistent.
Roughly speaking, consistency means that for large values of $n$ we are going to be close to the true value of the parameter with a high probability, i.e. estimates are going to be close to the true value of the parameter. Asymptotic unbiasedness means that for large values of $n$ on average we are going to be close to the true value of the parameter, i.e. the average of estimates is going to be close to the true value of the parameter, but not necessarily the estimates themselves.
A: They are related ideas, but an asymptotically unbiased estimator doesn't have to be consistent.
For example, imagine an i.i.d. sample of size $n$ ($X_1, X_2, ..., X_n$) from some distribution with mean $\mu$ and variance $\sigma^2$. As an estimator of $\mu$ consider $T = X_1 + 1/n$.
(Edit: Note the $X_1$ there, not $\bar{X}$)
The bias is $1/n$ so $T$ is asymptotically unbiased, but it is not consistent.
A: Asymptotic unbiasedness $\impliedby$ consistency + bounded variance
Consider an estimator $\hat{\theta}_n$ for a parameter $\theta$.  Asymptotic unbiasedness means that the bias of the estimator goes to zero as $n \rightarrow \infty$, which means that the expected value of the estimator converges to the true value of the parameter.  Consistency is a stronger condition than this; it requires the estimator (not just its expected value) to converge to the true value of the parameter (with convergence interpreted in various ways).  Since there is generally some non-zero variance in the estimator, it will not generally be equal to (or converge to) its expected value.  Assuming the variance of the estimator is bounded, consistency ensures asymptotic unbiasedness (proof), but asymptotic unbiasedness is not enough to get consistency.  To put it another way, under some mild conditions, asymptotic unbiasedness is a necessary but not sufficient condition for consistency.
Asymptotic unbiasedness + vanishing variance $\implies$ consistency
If you have an asymptotically unbiased estimator, and its variance converges to zero, this is sufficient to give weak consistency.  (This follows from Markov's inequality, which ensures that convergence in mean-square implies convergence in probability).  Intuitively, this reflects the fact that a vanishing variance means that the sequence of random variables is converging closer and closer to the expected value, and if the expected value converges to the true parameter (as it does under asymptotic unbiasedness) then the random variable is converging to the true parameter.
                    
A: There are "unbiased but not consistent" estimators as well as "biased but consistent" estimators:
https://en.wikipedia.org/wiki/Consistent_estimator#Unbiased_but_not_consistent
So, they are not the same thing.
Also, there is a long discussion about this topic here:
What is the difference between a consistent estimator and an unbiased estimator? 
A: If the estimator is bounded then consistency implies asymptotic unbiasness by the dominated convergence theorem.
A: Asymptotic unbiased: As $n \rightarrow \infty$, bias converges to $0$. 
Consistent: As $n \rightarrow \infty$, variance of the estimator converges to $0$.
