Workshop leiGH J. HAlliwell
whose Guess is Best?
ACTuARIES AREn’T THE OnLY OnES who forecast. Economists, stock
analysts, meteorologists, odds-makers—we all forecast! Some forecasters
are scored as to their accuracy. But what makes for a good forecast? Or
who in a forecasting contest is the winner? Whose guess is best? The obvious answer is that the best guess is the one that turns out to be closest
with the smallest absolute difference. But be leery of obvious answers.
a Guessing Game
Imagine that the Casualty Actuarial
Society has organized a contest. To the
registrants of an upcoming Loss Reserve
Seminar, it sends a disguised loss triangle
whose ultimate outcome is known to the
contest organizers. Registrants are asked
to provide their guesses of the ultimate
loss, with the winner’s name to be an-
nounced at the seminar. However, to
distinguish the science of loss reserving
from mere guessing, eligible participants
must provide a “two-moment” answer,
i.e., an answer in the form μ ± σ, where μ
and σ are the mean and standard devia-
tion of probability theory. Such a contest
would be in keeping with the recent em-
phasis in actuarial standards on reserve
made the worst guess. However, actu-
aries B and D made “sharper” forecasts,
and they weren’t too far off. Might
their smaller standard deviations offset
their larger differences? And is B better
than D because he (or she) came clos-
er with the same standard deviation?
Whose guess is best?
a likely “normal” solution
At this point, we will propose a solution.
The best guess is that which maximizes
the likelihood of the outcome. This is max-imum-likelihood theory with a twist. To
use an analogy of clothing, rather than
for the parameters to “try on” the sample,
here the sample “tries on” the parameters.
So whose guess is most justified by
the “sample,” or outcome, of $100,000
according to a likelihood based on a nor-
mal distribution of X (For the moment,
indulge me for defaulting to the normal
distribution.) The normal distribution
ion is
fx(x)dx = ———e —
1 2π
1 σ
1
2 x–μ σ
2
.
imizing this function with out-
s equivalent to maximizing its
m: Λ(μ,σ);= lnf (X)
– —ln2π – — ———— –lnσ
1
1
2
2 X – μ σ
2.
the first term of the last line is
for all guesses, we can ignore
seek the guess that maximizes:
– — ———— –lnσ
1
2 X – μ σ
2.
quivalently, we seek the guess
imizes:
— ———— +lnσ
1
2 X – μ σ
2.
