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Assume a random sample X1, ..., Xn with a normal distribution with mean μ and variance σ2. How do we know the following estimator is unbiased, but inconsistent?

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closed as off-topic by Giskard, Kenny LJ, Adam Bailey, Maarten Punt, Dan Feb 1 at 23:26

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not meet the standards for homework questions as spelled out in the relevant meta posts. For more information, see our policy on homework question and the general FAQ." – Giskard, Kenny LJ, Adam Bailey, Maarten Punt, Dan

  • $\begingroup$ What is the definition of a consistent estimator? $\endgroup$ – Bertrand Jan 24 at 9:21
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When an estimator is consistent, the sampling distribution of the estimator converges to the true parameter value being estimated as the sample size increases.

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Picking some samples from the distribution and calculate the average always gives you unbiased estimate. Even if you only pick the first (i.i.d) observation from the data, it is an unbiased estimator.

To ensure that it is consistent, you need to have $\forall \epsilon$, $P(|\hat{\mu}-\mu|>\epsilon) = 0$ as your sample size goes to infinity. Here, notice that since you are using the first 30 observations from the data, your $\hat{\mu} \sim N(\mu,\frac{\sigma^2}{30})$ even the sample size goes to infinity. Given that normal distribution has a full support over the real line, for any (nonnegative) $\epsilon$ you choose, your probability does not converge to 0. Hence it is not consistent.

If an estimator is unbiased and its variance converges to 0, then your estimator is also consistent but on the converse, we can find funny counterexample that a consistent estimator has positive variance. So we need to think about this question from the definition of consistency and converge in probability.

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