REEPRINTED FROhil '~';ATiS]",CAL SCIEWE

JOLUNICE 3.

l 551

GUESSING MODELS it is intuitive that the second guesser should

always guess just a bit higher or a bit lower

Guessing models are statistical structures than the first guesser. This can be proved to

which aim to provide insight into the norma be true under very general circumstances.

tive and optimal behaviors of people who Specifically, if 0, X, and Y are assumed to

must make choices in the apparent absence have a joint distribution that is continuous

of data. Such models are related to aspects and if vi(X,Y) denotes the conditional me

of Bayesian* statistics and the Delphi dian of Oi given X and Y, a key role is played

method', but they have a flavor and a theory by the strategies

of their own.

This is most easily illustrated by the anec Gi' X' + E if X' < I"(")

dote of two statisticians, Bob and Mike, who Xi E otherwise.

engaged in a contest to guess the weights of

people at a party. Bob agreed to always These guesses G' are called Hotelling strate

guess first, and on person 1 Bob guessed 142 gies and the first result in the theory of

pounds. Mike then guessed 142.1 pounds, guessing models is the following:

and the subject declared Mike the winner.

The contest continued; and, when final tal Theorem 1. The Hotelling strategies are E

lies were made, Bob found he had lost al optimal, i.e.,

most threefourths of the time.

To model this scenario, consider a system liniEV(W,0) = supEV(G, 0).

of fourp vectors E>0 G

Target values: (01 02, OP) = 0 Although this result reassures intuitive

feelings, it just makes the first step in telling

First guess: (Xl,X2.... Xp) X the second guesser how to guess. Consider

Second guesser's hunch: (Y1, Y2_ . Y able ingenuity may be required to ferret out

,,) Y those models in which Pi (X, Y) can be calcu

Second guess: (G, , G2, . Gp) G lated, and much of the theory of guessing

strategies resides in the calculation of suit

The 0, denote the real values to be able approximations.

guessed, so, for example, 02 would denote Consider, for example, the strategies

the weight of the second person considered

by Bob and Mike. The Xi are the guesses X'+E if X, < Yj

made by the first guesser, and the Yi repre Xi E if Xi > Yi.

sent the second guesser's best estimate of 0j.

Finally, the G, are the guesses that are an These are the "hunch" guided strategies and

nounced by the second guesser. The first they model the reasonable actions of Mike

problem in this theory is to determine how G in the anecdote. In some cases the hunch

should be determined by X and Y. guided strategies are, in fact, Hotelling strat

As the anecdote suggests, each player egies, but even when these strategies are not

wishes to come closer to each Oi than his E optimal they have surprising power.

opponent, so we consider

p Theorem 2. If for each 1 < i ~ p, Xi and

V(G, 0) 1 Vj(Q 0) Y, are independent and identically distrib

j1 uted with a distribution that is symmetric

where about 0j, then the hunch guided strategy Ji

has a 3 probability of winning the ith con

1 if 1Gj 0.1 < IX. 0.1 4

Effl,0) test as c > 0.

1

{ 0 otherwise. The fact that one shrinks the first guess Yi

With the objective of fflaximizing V(G, 0)5 toward one's hunch Xi anticipates that the

552 GUMBEL DISTRIBUTION

fact that other types of shrinkers are relevant Fisher and Tippett [111, Gumbel [171, and

in guessing models. In particular, when von Mises [46], culminating in the funda

p > 3 and all the Yi's are available to the mental paper by Gnedenko [14] (see EX

second guesser, a very powerful strategy for TREMEVALUE DISTRIBUTIONS). The initial re

the second guesser can be based on the sults concerning the law of large numbers*

JamesStein estimator*. for extremesnot dealt with herecan be

found in de Finetti [91. The basic bibliogra

phy for statistical problems is still Gumbel

NOTE [19]: many results and examples can be

found in this fundamental reference. A large

1. Editor's note: The Delphi method referred to block of references can be found in Johnson

above is the subject of a book [1], and is defined and Kotz [22] and Harter [201. A modern

there as follows: "Delphi may be characterized as and essential reference for probabilistic re

a method for structuring a group communication sults is Galambos [131. Extensions and appli

process so that the process is effective in allowing cations can be found at the end of the entry.

a group of individuals, as a whole, to deal with a

complex problem." [1, p. 31

(See also PUBLIC ADMINISTRATION, STATISTICS BASIC RESULTS

IN, and SOCIOLOGY, STATISTICS IN.)

Consider a sample of k i.i.d. random vari

Reference ables (Y1, . . . , Yk) with cumulative distribu

tion function (CDF) F(x) Pr[ Y xl.

[11 Linstone, H. A. and Turoff, M., eds. (1975). The Then the CW of max(Y,, . .Yk) is

Delphi Method. AddisonWesley, Reading, Mass. Pr 1 max(Y1, Yk) x]

k

Bibliography = I1Pr[Xi x]

Hwang, J. T. and Zidek, J. V. (198 1). J. Appl. Prob., 19, 1

321331. (Investigates the strategy G, = Xi ± E accord = Fk

ing as Xi < V or not.) (X);

Pittinger, A. 0. (1980). J. Appl. Prob., 17, 11331137. in the same way,

(Studies the problem of how the 14 theorem extends to

ndimensional guessing.) Pr[min(Y1, . . . , Yk) x]

Steele, J. M. and Zidek, J. V. (1980). J. A mer. Stat. Ass.,

75, 596601. (Sets the foundation for the theory of F( X))k.

second guessing, and provides the basis for this exposi

tion. Contains much more information on Stein guided In general, to deal with samples of maxima

guessing as well as results of computer simulations.) (or minima), all obtained under the same

J. MICHAEL STEELE conditions (with the same k and F), it would

be necessary to know the form (and parame

ters) of.F. But if k is large, we can try to use

GUMBEL DISTRIBUTION asymptotic distributions in statistical analy

sis. In fact, it can be shown that, in many

The theory of statistical extremes has a short cases, coefficients ak and Pk(> 0) exist such

effective history. Beginning, essentially, with that Pr[(max(Xl, . . . , Xk) ak)lflk < X]

a paper by Dodd [101 on the distribution of = Fk(ak + Pkx) has a nondegenerate limit

the extremes (maxima and minima) of a CDF; note that (ak, Pk) are not uniquely

univariate independent and identically dis defined. There are only three such limit

tributed (i.i.d.) sample, the basic results re forms (Weibull*, Gumbel, and Fr6chet)

garding special properties and asymptotic which can be integrated in a condensed von

behavior are contained in Fr~chet [121, Mises [46]Jenkinson [211 form.

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