Wikipedia:Reference desk/Archives/Mathematics/2023 July 16

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July 16

Name for not-quite Horner's method

The series for exp(x) is 1+x+x²/2!+x³/3!+x⁴/4!+x⁵/5!+... . But as a practical matter if you want to compute a partial sum you wouldn't evaluate it as written. You could use Horner's method, aka synthetic division, but as stated in our article it would rewrite the first 6 terms as 1+x(1+x(1/2!+x(1/3!+x(1/4!+x/5!)))). This reduces the number of multiplications but you'd still need to compute the factorials. It seems easier to factor out those multiplications as well, giving 1+x(1+(x/2)(1+(x/3)(1+(x/4)(1+x/5)))). Horner's method doesn't quite seem to fit, so is there a different name for this type of expansion? They can be used for other partials sums as well. For example (1-4x)^-1/2≈1+2x(1+(6x/2)(1+(10x/3)(1+(14x/4)(1+18x/5)))), π/2≈1+(1/3)(1+(2/5)(1+(3/7)(1+(4/9)(1+5/11)))). --RDBury (talk) 09:03, 16 July 2023 (UTC)

I think this would still be called Horner's method. In practice you would pre-compute the coefficients as floating point numbers and this wouldn't save you any work. –jacobolus (t) 09:09, 16 July 2023 (UTC)

"Nominal confidence level"

Do you agree? (more unambiguously) Hildeoc (talk) 20:00, 16 July 2023 (UTC)

Regardless of whether I or any other editor agrees, a statement like that needs a citation. The fact is that the terms "nominal confidence level" and "nominal coverage probability" are used interchangeably in the literature,^[1] and that some scholars do not consider "nominal confidence level" to mean just the same as "confidence level". For example, we can find statements such as, "the actual CONFIDENCE LEVEL is greater than the nominal confidence level".^[2]  --Lambiam 06:29, 17 July 2023 (UTC)

@Lambiam: I looked into exactly this source among others, and to my mind, this is an example for the mentioned malapropism: In the section above the quotation invoked by you ("the actual confidence level is greater than the nominal confidence level, $(1-1/k^{2})$"), it reads that "The actual coverage probability of the interval [...] is greater than $(1-1/k^{2})$", and later we find: "[...] the probability that the intervals they produce cover the true population mean is greater than the probability they claim, $(1-1/k^{2})$ (the nominal coverage probability)." As "confidence level" as such already refers to the ideal, pre-determined value chosen by the researcher or analyst before conducting a statistical analysis (= nominal coverage probability), what, in terms of semantic logic, would "nominal confidence level" exactly denote then – "nominal nominal coverage probability"? Hildeoc (talk) 11:14, 17 July 2023 (UTC)

The adjective nominal means "in name only", implying that a term thus qualified does, in a given context, not actually have the qualities or properties usually implied by that term. For example, we can read that from 1608 to 1611 John Albert II, Duke of Mecklenburg "was the nominal ruler of Mecklenburg-Schwerin; the actual ruler being the regent, his great-uncle Charles I". John Albert II was ruler in name only; unlike the term would normally imply, he did not actually rule the Duchy. The author of the cited source warns his students that when using Chebyshev's inequality, as advocated in some textbooks,^[3] the claimed confidence level ($1-1/k^{2}$) may not be the true resulting confidence level.  --Lambiam 21:22, 17 July 2023 (UTC)

But still, this doesn't seem to be what most pertinent texts mean when using this spurious synonymy, does it? Hildeoc (talk) 22:40, 17 July 2023 (UTC)

@Lambiam: What about this version? In case of interest, also @Michael Hardy. Hildeoc (talk) 22:11, 19 July 2023 (UTC)