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Suppose $T_n$ is a sequence of estimators of $\theta$ and suppose $ET_n\nrightarrow\theta$. Is is true that $T_n$ is NOT consistent for $\theta$?

I don't think unbiasedness is a necessary condition for consistency or, convergence in probability. But, I can't figure a counterexample to support my claim. Any help?

reyna
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    Regarding the title, you probably mean asymptotic unbiasedness. For a counterexample, take any consistent estimator $\hat\theta_n$ of $\theta$ from a parametric model and define $T_n$ such that $T_n=\hat\theta_n$ with high probability (i.e. probability tending to $1$) and $T_n=c_n$ otherwise, where $c_n$ is chosen in a way that ensures $E(T_n)$ does not converge to $\theta$. – StubbornAtom Mar 09 '23 at 19:20

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