Update: Parameters as Population Quantities

May 16, 2012

(This article was first published on BioStatMatt » R, and kindly contributed to R-bloggers)

Some time ago, I had an ineloquent and less-than-cordial online discussion with a commenter on this site, partially about how statisticians define the term “parameter”. This post is just to quote a relevant passage from “Bootstrap Methods and Their Application”, by Davison and Hinkley (1997), that better articulates a point I had made earlier.

2.1.1 Statistical Functions
Many simple statistics can be thought of in terms of properties of the EDF [empirical distribution function]. For example the sample average $\bar{y} = n^{-1} \sigma y_j$ is the mean of the EDF. More generally, the statistic of interest $t$ will be a symmetric function of $y_1,\ldots,y_n$, meaning that $t$ is unaffected by reordering the data. This implies that $t$ depends on the ordered values $y_{(1)} \leq \cdots \leq y_{(n)}$, or equivalently on the EDF $\hat{F}$. Often this can be expressed simply as $t = t(\hat{F})$, where $t(\cdot)$ is a statistical function – essentially just a mathematical expression of the algorithm for computing $t$ from $\hat{F}$. Such a statistical function is of central importance in the nonparametric case because it also defines the parameter of interest $\theta$ through the “algorithm” $\theta = t(F)$. This corresponds to the qualitative idea that $\theta$ is a characteristic of the population described by $F$…

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