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According to MathWorld (link): "The sample raw moments are unbiased estimators of the population raw moments".

While in Wikipedia (link) it is said:

...the $k$-th raw moment of a population can be estimated using the $k$-th raw sample moment $$\frac{1}{n}\sum_{i = 1}^{n} X^k_i$$

applied to a sample $X_1\dots X_n$ drawn from the population.

It confuses me because to my understanding estimators are statistics (link) used to estimate some parameter. As such, they are functions of random variables. But the expressions found in the articles seems to be functions of outcomes of random variables, and not of the the variables themselves.

I tend to think that a "sample moment" is 'something' that can be computed directly from a sample (or their corresponding outcomes as I understand from this answer). That means, from specific values. One can define an statistic, and then modify it by replacing the random variables with specific outcomes, and perform a computation. But I feel that this is not the same that the original object.

Can anyone clarify me these subtle differences to understand what is a "sample moment"?

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  • $\begingroup$ You appear to think that $X_i$ is an "outcome of" a random variable, and not a random variable itself. Am I understanding you correctly? $\endgroup$
    – jbowman
    Apr 19 at 1:30
  • $\begingroup$ @jbowman Yes, I think so because the text of "applied to a sample" (however, math is not my field) $\endgroup$ Apr 19 at 1:42
  • $\begingroup$ If a distribution has a moment generating function, then the raw moments uniquely define the distribution, so the raw moments could be seen as parameters. Even with a narrower family of distributions with fewer parameters, each raw moment is typically a function of these parameters. $\endgroup$
    – Henry
    Apr 19 at 12:58

1 Answer 1

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Here we use a distinction between an "estimator" and the corresponding "estimate"

You are correct that estimators are statistics, and a statistic is a function of the observable data values. These values can either be considered prior to observation (when we treat them as being random variables) or after observation (when they are now known constants). Thus, any statistic can be considered in its pre-observational random variable form or as a post-observational constant (see this related answer).

When we are talking about estimating things with an estimator, we actually have a terminological difference for this distinction. An estimator is a function of the observable values considered as random variables (and so is itself random) and an estimate is the corresponding function of the observed values considered as known constants (and so is itself a constant). Thus, if you have observable random variables $X_1,...,X_n$ with corresponding observed values $x_1,...,x_n$ (using the standard notational distinction based on capitalisation) then you might refer to the estimator $T(X_1,...,X_n)$ (which is a random variable) and the corresponding estimate $T(x_1,...,x_n)$ (which is a constant).

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  • $\begingroup$ +1 Thank you. I think I understand what you explained me. Excuse me, but I still have a doubt. The "sample raw moment" is an estimator with which I can compute a unnamed estimate??. Or "the sample raw moment" is the estimate computed with an unnamed estimator??. Or "sample raw moment" refers to both, the estimator and each estimate computed by applying it (using its functional form to compute an estimate from some specific sample)??. Or is not a rigorous term convention and there is some freedom when speaking about it in field of statistics?? $\endgroup$ Apr 19 at 14:48
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    $\begingroup$ We would call them both the "sample raw moment" nd hope the notation or context makes the distinction clear. So $\hat{M}_k \equiv \sum_i X_i^k / n$ is the $k$th sample raw moment (as an estimator) and $\hat{m}_k \equiv \sum_i x_i^k / n$ is the $k$th sample raw moment of the data (as a estimate). Ideally we would us a notation like this that distinguishes the random and constant versions, but sometimes we wouldn't even make that distinction in the notation and would instead rely on textual context. $\endgroup$
    – Ben
    Apr 19 at 20:34

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