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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 $T = X_1 + 1/n$.

The bias is $1/n$ so $T$ is asymptotically unbiased, but it is not consistent.

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.

They are similar; a consistent estimator that's biased must nevertheless be asymptotically unbiased (otherwise it could not be consistent), 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 $T = X_1 + 1/n$.

The bias is $1/n$ so $T$ is asymptotically unbiased, but it is not consistent.

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 $T = X_1 + 1/n$.

The bias is $1/n$ so $T$ is asymptotically unbiased, but it is not consistent.

They are similar; a consistent estimator that's biased must nevertheless be asymptotically unbiased (otherwise it could not be consistent), but an asymptotically unbiased estimator doesn't have to be consistent.

For example, imagine an i.i.d. sample of size $n$ from some distribution with mean $\mu$ and variance $\sigma^2$. As an estimator of $\mu$ consider $T = X_1 + 1/n$.

The bias is $1/n$ so $T$ is asymptotically unbiased, but it is not consistent.

They are similar; a consistent estimator that's biased must nevertheless be asymptotically unbiased (otherwise it could not be consistent), 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 $T = X_1 + 1/n$.

The bias is $1/n$ so $T$ is asymptotically unbiased, but it is not consistent.