<p>When an estimator is consistent, the sampling distribution of the estimator converges to the true parameter value being estimated as the sample size increases.</p>

CC BY-SA 4.0
2019-01-24T15:15:15.823Z
2019-01-24T15:15:15.823Z
user20883
<p>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. </p>

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

<p>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. </p>

CC BY-SA 4.0
2019-01-26T16:10:34.133Z
2019-01-26T16:10:34.133Z
kokutou_qian
<p>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?</p>

<p><a href="https://i.sstatic.net/ScSm5.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ScSm5.png" alt="enter image description here"></a></p>

CC BY-SA 4.0
2019-01-24T09:21:13.070Z
What is the definition of a consistent estimator?
Bertrand
<econometrics><regression>
Unbiased but inconsistent estimator
Ty_learner
# Unbiased but inconsistent estimator

(Asked by: username_0 on 2019-01-24)

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?

[![enter image description here](https://i.sstatic.net/ScSm5.png)](https://i.sstatic.net/ScSm5.png)

**Tags:** econometrics, regression
> **Comments:**
> **username_1**: What is the definition of a consistent estimator?  

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## 2 answers

### Answer by username_2 on 2019-01-24. Score: 1

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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### Answer by username_3 on 2019-01-26. Score: 0

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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