Generalized Extreme-Value Model

McFadden (1978) introduced the generalized extreme-value (GEV) distribu­tion defined by

F(€ue2,. . . ,ej (9.3.67)

= exp {—(/[exp (-eO, exp (-e2), . . . , exp ( Cm)]), where G satisfies the conditions,

(i)

G(ux, u2,. . .

., мт) ё 0,

«і, «2. • • • > мтё0.

(ІІ)

G(olu1,olu2,

. . ., aum)

= otG(ux, u2,. . ., uj.

(iii)

> A

if к is odd

if к is even, k= 1,

duhduh. . .

_ != U

dulk

SO

If Uj — fij + €j and the alternative with the highest utility is chosen as before, (9.3.67) implies the GEV model

„ exp {/ii)G,[ep (fij), exp (fi2. . ., exp (jMm)]

/-УТ / / / | > (У. З.Оо)

3 (/[exp (fj. il exp (fj2),. . ., exp (yUm)]

where Gj is the derivative of G with respect to its yth argument.

Both the nested logit model and the higher-level nested logit model dis­cussed in the preceding sections are special cases of the GEV model. The only known application of the GEV model that is not a nested logit model is in a study by Small (1981).

The multinomial models presented in the subsequent subsections do not belong to the class of GEV models.

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