Bureau of Social Science Research 1 Tversky (1967) discussed the Bradley-Terry-Luce choice model, which is closely related to the Rasch model, in terms of conjoint measurement. Young (1972) also considered the Bradley-Terry-Luce model in these terms and remarked generally that "the scaling methods in psychometrics conform to the notion of polynomial conjoint measurement."

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It is shown that a version of Mandelbrot's monkey-at-the-typewriter model of Zipf's inverse power law is directly related to two classical areas in probability theory: the central limit theorem and the ''broken stick'' problem, i.e., the random division of the unit interval. The connection to the central limit theorem is proved using a theorem on randomly indexed sums of random variables ͓A. Gut, Stopped Random Walks: Limit Theorems and Applications ͑Springer, New York, 1987͔͒. This reveals an underlying log-normal structure of pseudoword probabilities with an inverse power upper tail that clarifies a point of confusion in Mandelbrot's work. An explicit asymptotic formula for the slope of the log-linear rank-size law in the upper tail of this distribution is also obtained. This formula relates to known asymptotic results concerning the random division of the unit interval that imply a slope value approaching Ϫ1 under quite general conditions. The role of size-biased sampling in obscuring the bottom part of the distribution is explained and connections to related work are noted. ͓S1063-651X͑96͒01007-0͔

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Abstract:The distribution of word probabilities in the monkey model of Zipf's law is associated with two universality properties: (1) the exponent in the approximate power law approaches −1 as the alphabet size increases and the letter probabilities are specified as the spacings from a random division of the unit interval for any distribution with a bounded density function on [0, 1]; and (2), on a logarithmic scale the version of the model with a finite word length cutoff and unequal letter probabilities is approximately normally distributed in the part of the distribution away from the tails. The first property is proved using a remarkably general limit theorem from Shao and Hahn for the logarithm of sample spacings constructed on [0, 1] and the second property follows from Anscombe's central limit theorem for a random number of independent and identically distributed (i.i.d.) random variables. The finite word length model leads to a hybrid Zipf-lognormal mixture distribution closely related to work in other areas.