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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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