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When JW Tukey retired he wrote an article called Sunset Salvo in which he, among other things, enumerates some Numerical Antihubrisines. These are different rules of thumb for how many observations you need to for eg. make your confidence intervals so-and-so narrow. I struggle with understanding the following antihubrisine:

Measuring a $\sigma^2$ to one significant figure (taking this as, for instance, being reasonably sure for any $u > 0$, whether $2.5u$ or $3.5u$ is correct) [...] In the optimistic, Gaussian case, [requires] about 300 degrees of freedom. In more realistic cases, up to two or three times as many degrees of freedom.

I have many questions about this, maybe because I don't 100% understand the parenthesis in the quote:

  1. How does he define the number of "successfully estimated" significant figures? Something to do with standard errors?
  2. If we can't measure it to one significant figure in most cases, what can we use $\sigma^2$ for at all? Can we do something useful with, say, half a significant figure? Is that a thing?
  3. What exactly is the calculation in this optimistic Gaussian case to arrive at $\approx 300$?
  4. Are we more confident about $\sigma$ than $\sigma^2$?
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  • $\begingroup$ For normal rvs the sample variance is related to the chi-square in that (n-1)S^2/sigma^2 $\endgroup$ Nov 29, 2016 at 13:01
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    $\begingroup$ My edits for the above comment got timed out. I have trouble finishing in 5 mins. To continue (n-1)S^2/sigma^2 is chi square with k-1 dfs. The variance of chi square depends on fourth moments and can be large especially when compared to the variance of the mean. Tukey is almost always right but his writing is often. Velleman and others had to interpret his EDA book (the orange one) for the masses. $\endgroup$ Nov 29, 2016 at 13:16

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