What is an estimator?

Question

If $p_y$ is a probability function for a density, which depends on the value of $y$ (for example, $y$ might be the mean in the poisson distribution).

Assuming that $y$ is random -- i.e. unknown -- how would one estimate the probability that $X < x$ for $X$ being distributed with $p_y$ as the probability function?

My approach so far has been to just generate a large set of $y$ and then using those values to generate an equally large sets of $X$ and then counting the number of values less than $x$.

Would/could this be considered a classic estimator?

score 2 · Accepted Answer · answered Apr 19 '16 at 10:24

2

What you are looking at, basically, is the empirical distribution function. This is a very classical thing.

According to the law of large numbers, you will eventually end up with very good estimates for $P(X < x)$. Further, by CLT, your estimator for this probability is asymptotically normal. There is also a result on the uniformity of the convergence and other results related to testing whether the underlying distribution has a specific form.

answered Apr 19 '16 at 10:24

Matias Heikkilä

2,542

Am I correct in assuming that this could be considered "the" classic estimator? – Clearer Apr 19 '16 at 11:35
For the problem of estimating $P(X < x)$, in general, sure. I can't even think of any other way of doing that, unless you have some extra information about $X$. – Matias Heikkilä Apr 19 '16 at 12:02
That's what I thought. Awesome – Clearer Apr 20 '16 at 05:21

What is an estimator?

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