On a functional equation related to Thurstone models

Abstract If f and g are nonvanishing characteristic functions the functional equation g(s)g(t)g(−s − t) = f(as)f(at)f(−as − at) implies g(s) = eibsf(as), i.e., f and g corresponding to probability distributions of the same type. It is shown here that when f and g are allowed to vanish this equation also has solutions in which f and g correspond to distributions of different types. The practical implication is that there are nonequivalent. Thurstone models which cannot be discriminated by any choice experiment with three objects.

If f and g are nonvanishing characteristic functions the functional equation &)&)g(--s -t) = f(as)f(at)f(--as -at) implies g(s) = e'"'f (as), i.e., f and g corresponding to probability distributions of the same type.It is shown here that when f and g are allowed to vanish thig equation also has solutions in which f and g correspond to distributions of different types.The practical implication is that there are nonequivalent Thurstone models which cannot be discriminated by any choice experiment with three objects.
The functional equation (where f and g are complex functions of a real variable) arises in connection with an identifiability problem in choice behavior described recently in this journal (Yellott, 1977).Briefly, if f and g are the characteristic functions of probability distributions F and G, it can be shown that two "Thurstone models" FF and .%c(i.e., models identical to Thurstone's Case V except that distribution F (or G) is substituted for the normal) are equivalent for all choice experiments with three objects iff (1) is satisfied for some positive constant a and all real s and t.If f or g is assumed to be a nonvanishing characteristic function, the unique solution to (1) is readily shown to be where b is any real constant.This means that F(x) = G(UX + b), and so when f is nonvanishing the predictions of Thurstone model FF can be completely duplicated by another model & iffF and G are distributions of the same type, i.e., both normal, both exponential, or whatever.This turns out to be true in particular for the double exponential distribution function F(x) = e-6-", which yields the Thurstone model corresponding to Lute's Choice Axiom.
However without the assumption that f or g is nonvanishing the argument used in the previous paper to establish the uniqueness of solution (2) does not go through, and consultation with J. Aczel indicated that it was not generally known whether any pair of characteristic functions that satisfy (1) must also satisfy (2).l (Examples of complex functions that satisfy (1) but not (2) are not hard to come by, but the question is whether they can also be characteristic functions.)Consequently in that paper I was forced to leave open the general question of whether two Thurstone models can be equivalent for experiments with three objects when F and G are distributions of different types.
The purpose of this note is to lay this problem to rest by exhibiting a pair of characteristic functions!andg that satisfy (1) but not (2).The probability distributions F and G corresponding to this pair of characteristic functions both satisfy the technical requirement for Thurstone models, i.e., the "difference distributions" &(x) = j-m F(x + A OF and --4) 44 = irn G@ + A WY) --m are continuous and strictly increasing, and so this example shows explicitly that two Thurstone models r', and To can be completely equivalent for choice experiments with three objects even though their discriminal process distributions F and G are not of the sm type, i.  , Feller (1966).Actually in probability theory it is customary to suppress the constant factor -2rr, and so the characteristic functions tabled in probability texts correspond to f(-s/2~r).However a pair of characteristic functions defined according to (3) will satisfy functional equation (1) iff f (-s/2?r) and g(-s/2tr) do also, and for present purposes it is convenient to use definition (3) in order to take advantage of Bracewell's (1965) dictionary of transforms.)Recall that if f is absolutely integrable, the probability density function p, corresponding to F (i.e., pF(x) = F'(x)) is given by the inverse transform Now consider the functions SE(-1, l), s E (4,6), s e (-6, -4), = 0, elsewhere; (4) l Subsequently 2. Moszner (1977) has &en a general characterization of the solutions to Eq. ( 1).
In an adjoining no& in this issue he shows how that translates into a characterization of the corresponding distribution functions.
(2) Both s and t ~1,.Then -s -t is either in R, so that both sides of (1) vanish, or in I,, , in which case f and g agree factor by factor on both sides of (1).
This completes the proof.It is worth noting that an infinite number of suchf, g pairs could be constructed by centering the outside triangles at any &o (w > 5), in which case the corresponding densities p, and p, would be Sinc2 (x)[l + cos 2~~x1 and Sines (x)[l + sin 27~x1, all of which yield Thurstone models.Thus there are an infinite number of pairs of nonequivalent Thurstone models that cannot be distinguished by three object choice experiments.
e., F(N) # G(ax + b).C onsequently it is not possible to strengthen the theorems in Yellott (1977), contrary to my conjecture in that paper.Now for specifics.The characteristic function f of a probability distribution (i.e., cumulative distribution function) F is the Fourier-Stieltjes transform f(s) = 1-1 e--dF(x)(3) (e.g.
) are both probability density functions, since both are nonnegative and integrate to 1. (This last is shown by the fact that f(O) = g(0) = 1.)It is also clear that the difference distributions D, and DG are continuous and strictly increasing, since they correspond to a continuous nonvanishing density.(Note that D, = D, , since If I2 = 1 g I".)