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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:

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

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    $\begingroup$ 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 $\Theta_n = (X_1 + \cdots + X_n)/n$. $\endgroup$
    – littleO
    Aug 19, 2020 at 12:48
  • $\begingroup$ @littleO so basically, in my notes above, the subscript n is implicitly in the equation: $$\Theta_n = s(X_1, \cdots, X_n)$$, and $$\hat{\theta} = E[\Theta_n] = E[s(X_1, \cdots, X_n)]$$ ? $\endgroup$
    – pico
    Aug 19, 2020 at 12:56

1 Answer 1

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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.

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