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I am reading about MLE (Maximum likelihood Estimation). And I had to also read about the Point estimation, for which the following was said:

"In statistics, point estimation involves the use of sample data to calculate a single value (known as a point estimate since it identifies a point in some parameter space) which is to serve as a "best guess" or "best estimate" of an unknown population parameter (for example, the population mean). More formally, it is the application of a point estimator to the data to obtain a point estimate."

I don't understand the part in bold. Can anyone explain this to me?

I know that in MLE, the value of the parameter, which maximizes the probability of observing the sample at hand, is the point estimator. But I still don't understand the bolted part.

Basically what point estimator vs point estimate?

imbAF
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1 Answers1

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A point estimator is a function that takes as input some data and outputs a point estimate. That is, a point estimate is the output of some estimator for some data set.

For example, suppose a data set $y_1, \ldots, y_n$ is generated by the model $y_i \sim \mathcal{N}(\mu, \sigma^2)$. The quantity $\mu$ is unknown and we want to estimate it. Then an estimator of $\mu$ is a function $f: \mathbb{R}^n \mapsto \mathbb{R}$, $(x_1, \ldots, x_n) \mapsto \frac{1}{n} \sum_{i = 1}^n x_i$, whereas the estimate is the function evaluated at the data set $y_1, \ldots, y_n$, or $f((y_1, \ldots, y_n)) = \frac{1}{n} \sum_{i = 1}^n y_i$. The former is an abstract object, whereas the latter is a real number.

All of this is a little bit simplified, but I hope this gives you the right intuition.

Wittgenstein's Poker
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  • One question, what is the difference between the set of x's and the y's ? – imbAF Nov 24 '22 at 11:20
  • @imbAF The $x$'s are just a stand-in for any input, the $y$'s are an actual data set. As an analogy, take $f(x) = x^2$. There $x$ is a stand in. If we input $2$ into $f(x)$, we get $f(2) = 2^2 = 4$. – Wittgenstein's Poker Nov 24 '22 at 14:47
  • Oh, right, sorry but statistics confuses me so much that some time I have a brain fart. Basically the x is used in the general expression for what the estimator can be – imbAF Nov 24 '22 at 15:06
  • One more thing, and what about the expectation value of the estimator? If the estimator is the mean of a sample, talking about the expectation value of a random variable, means that we must have a certain amount of values to do so. In this case the expectation value of the estimator, is the expectation value of the mean of the sample, but that is a single value, how does it make sense to talk or consider the expectation value of the estimator ? – imbAF Nov 24 '22 at 15:07
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    @imbAF I answered your question in your other stack post :) – Wittgenstein's Poker Nov 24 '22 at 15:16