# Unbiased but inconsistent estimator [closed]

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?

• What is the definition of a consistent estimator? – Bertrand Jan 24 '19 at 9:21

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.