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

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:

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

• Often you have a sequence $X_1, X_2, \ldots$ of independent random variables that all have the same distribution, and $\Theta_n$ is an estimator that uses only the first $n$ random variables $X_1, \ldots, X_n$ to obtain an estimate of a parameter $\theta$ . For example, if we're estimating the mean of the underlying distribution, then perhaps $\Theta_n = (X_1 + \cdots + X_n)/n$. Aug 19, 2020 at 12:48
• @littleO so basically, in my notes above, the subscript n is implicitly in the equation: $$\Theta_n = s(X_1, \cdots, X_n)$$, and $$\hat{\theta} = E[\Theta_n] = E[s(X_1, \cdots, X_n)]$$ ?
– pico
Aug 19, 2020 at 12:56

## 1 Answer

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.