Consistency vs. unbiasdness [duplicate]

What is the difference between

$\lim_{n \to \infty} \ \mathrm{E}_{\theta}(T_n(X)) = \theta$

and

$T_n(X) \xrightarrow{p} \theta \$ for $\ n \xrightarrow{} \infty$ ?

(unbiasdness vs. Consisteny)

Ups... there seems to be this question already :-( Similar CrossValidated question

Shall I delete this question?

• There's no problem, Michael: duplicates actually serve a useful purpose in that they sometime express the same question in different ways, increasing the chance of good hits in future searches. – whuber Jan 13 '14 at 15:56
• @whuber - This isn't actually a duplicate, as this question is about asymptotic unbiasedness and consistency, not unbiasedness (a small-sample property) and consistency. – jbowman Jan 13 '14 at 16:04
• Hmmm... this leads me to something else: Do you have a nice example, where you don't have unbiasdness for small samples, but asymptotic unbiasdness? I have problems to see why this can happen... – Michael Jan 13 '14 at 16:10
• @jbowman You are right; thank you. I simply accepted the OP's assessment of the duplicate nature. I believe this question has been discussed elsewhere on CV approximately 18 months ago; it's worth a search. – whuber Jan 13 '14 at 16:18
• Michael, it depends on what you mean by "small samples." If you set a fixed threshold between "small" and not "small," then examples are easy to come by: simply switch from an unbiased estimator when the sample is "small" to an asymptotically biased estimator when the sample is not "small." – whuber Jan 13 '14 at 16:22