# Why do we need an estimator to be consistent?

I think, I have already understood the mathematical definition of a consistent estimator. Correct me if I’m wrong:

$$W_nW_n$$ is an consistent estimator for $$\theta\theta$$ if $$\forall \epsilon>0\forall \epsilon>0$$

$$\lim_{n\to\infty} P(|W_n – \theta|> \epsilon) = 0, \quad \forall\theta \in \Theta\lim_{n\to\infty} P(|W_n - \theta|> \epsilon) = 0, \quad \forall\theta \in \Theta$$

Where, $$\Theta\Theta$$ is the Parametric Space.  But I want to understand the need for an estimator to be consistent. Why an estimator that is not consistent is bad? Could you give me some examples?

I accept simulations in R or python.

If the estimator is not consistent, it won’t converge to the true value in probability. In other words, there is always a probability that your estimator and true value will have a difference, no matter how many data points you have. This is actually bad, because even if you collect immense amount of data, your estimate will always have a positive probability of being some $$\epsilon>0\epsilon>0$$ different from the true value. Practically, you can consider this situation as if you’re using an estimator of a quantity such that even surveying all the population, instead of a small sample of it, won’t help you.