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What is "logistic" about the logistic distribution, in a common sense way? What is the etymology of and the lexical rationale for the name, not just pure math definition?

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At en.wikipedia.org/wiki/Logistic_function it is mention that "The function was named in 1844–1845 by Pierre François Verhulst, who studied it in relation to population growth". The lexical rational is still unclear, at least for me – Dan Levin Jan 28 at 19:02
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For me, the association is with the log-odds or logit function $\log\left(\dfrac{p}{1-p}\right)$ which has the inverse $\dfrac{\exp(x)}{1+\exp(x)} = \dfrac{1}{\exp(-x)+1}$ which is the standard logistic function. So is it log- as in "logarithm" and -istic (or -istique in French) as in "related to". – Henry Jan 28 at 19:26
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See Why logistic (sigmoid) ogive and not autocatalytic curve? "Though he [Verhulst] does not explain this choice, there is a connection with the logarithmic basis of the function. Logarithm was coined by John Napier (1550-1617) from Greek logos (ratio, proportion, reckoning) and arithmos (number). Logistic comes from the Greek logistikos (computational). In the 1700's, logarithmic and logistic were synonymous. Since computation is needed to predict the supplies an army requires, logistics has come to be also used for the movement and supply of troops". – amoeba Jan 29 at 12:57

The source document for the name "logistic" seems to be this 1844 presentation by P.-F. Verhulst, "Recherches mathématiques sur la loi d'accroissement de la population," in NOUVEAUX MÉMOIRES DE L'ACADÉMIE ROYALE DES SCIENCES ET BELLES-LETTRES DE BRUXELLES, vol. 18, p 1.

He differentiated what we would now call exponential growth of population when resources are essentially unlimited (as seen for example in the growth of the US population in the late 18th and early 19th centuries) from the slower growth when resource limits begin to be reached.

What we call exponential growth, however, he called a "logarithmique" curve (page 6).

He then developed a formula for population growth in the presence of resource limits, and said of the resulting curve:

"Nous donnerons le nom de logistique à la courbe..." which I translate as "We call the curve logistic..." (original emphasis).

That would seem to be intended to distinguish this growth pattern from the "logarithmique" growth in the absence of resource limits, as the figure at the end of the paper illustrates.

enter image description here

The specific form of the equation presented by Verhulst allows for an arbitrary upper asymptote (eq. 5, page 9), while the form we know and love in statistics is the specific case with an asymptote of 1.

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OED also attributes this sense of "logistic" to Pierre François Verhulst, although the word was used as a synonym of "logarithmic" long before that. – Francis Jan 28 at 22:38
    
But why did Verhulst chose the name "logistique"? Did this word mean something in French that was somehow related to the shape of the curve, or population growth, or limited resources? – amoeba Jan 29 at 0:47
    
@amoeba - Looking purely at etymology (Henry's comment above seems more concrete to me!): within the French language, 'logistique' comes from 'loger', which means to live (to lodge). It however came to French originally through the Greek 'λογιστικός', meaning practical or rational (logical?). So maybe it can be understood that Verhulst saw the version of the model without limits, i.e. unlimited resources, as irrational. Calling the model with limits on resources as the rational model. – Dexter Morgan Jan 29 at 2:34
    
@Dexter: interesting hypothesis! By the way, French Wikipedia says: "Le nom de courbe logistique leur a été donné par Verhulst sans que l'on sache exactement pourquoi. Il écrit en 1845 dans son ouvrage consacré à ce phénomène : « Nous donnerons le terme de logistique à cette courbe ». L'auteur n'explique pas son choix mais « logistique » a même racine que logarithme et logistikos signifie « calcul » en grec." -- so they say that the author did not explain his choice of word and the exact reason remains unknown. – amoeba Jan 29 at 10:55

The logistic distribution is not a common distribution in analysis, but it ties together the notion of a latent underlying continuous variable which is thresholded in binary outcomes. It turns out that thresholding a logistic RV (to 1 if the RV is greater than some unknown value and 0 otherwise) and calculating a maximum likelihood leads to logistic regression. Contrast this approach with thresholding a normally distributed random variable which leads to probit regression. Applying multiple thresholds leads to cumulative link models.

Now, if your question concerned logistic regression, the term was coined by David Cox in 1958 "The regression analysis of binary sequences (with discussion)" in JRRS. He used the term to the logistic, sigmoidal shape of the modeled mean. For describing the process of a curve which models probabilities that accumulate according to a probabilistically sound way, the term "logistic" is an intuitive choice and the nomenclature stuck.

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I don't think I understand your answer at all: why was the term "logistic" an intuitive choice? Because of "the logistic, sigmoidal shape"? But why would the shape be called "logistic" in the first place? – amoeba Jan 29 at 10:44

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