What does it mean when you add a subscript to an estimator?

Question

What does it mean when you add a subscript to an estimator $\Theta$?

Example:

1. $\Theta_n$

2. $\lim \limits_{n \to \infty} E[\Theta_n] = \theta$

3. $\lim \limits_{n \to \infty} \Pr(|\Theta_n - \theta|<\varepsilon)=1$

Here' what I understand so far in my own words. (correct me if i'm wrong about anything.)

• Estimator: An estimator $\Theta = s(X_1, \cdots, X_n)$ is a statistical (ie., a function of random data) that is used to infer the value of an unknown parameter $\theta$ in a statistical model.

  For example, an estimator could be used to predict the value of parameter $\theta$ in the probability distribution:

  $$X \sim f_X(x;~ \theta)$$

  by using the random data of X: $x_1, x_2, \cdots, x_n$.

• Estimate: Being a function of the random data $X$, the estimator $\Theta$ is itself a random variable, and a particular realization of this random variable is called the estimate $\bar{\theta}$.

  $$\bar{\theta} = s(x_1,~ \cdots,~ x_n)$$

• Expected Estimate: This is found by taking the expection of the estimator:

  $$\hat{\theta} = E[\Theta] = E[s(X_1, \cdots, X_n)]$$

Answer 1

Its common to add the subscript 'n' to the estimator $\Theta$ to indicted that it was formed by realizing 'n' random variables of distribution X that are IID. This is useful if we want to take the limit as $n \to \infty$. that is:

$\Theta_n = s(X_1, \cdots, X_n)$ where $X_i \sim f_X(x;~ \theta)$

However, in general the 'n' subscript is not a strictly requirement, and we can use any meaningful subscript on the estimator.