16(34)
decide to take part in the competition, based on the various factors
previously discussed. In any sample of size
n
of a stochastic variable X,
there is a smallest value X
min
(n). If the distribution function of X is F(.)
and the corresponding frequency function is f(.), the frequency function
g(.) pertaining to X
min
will take the form
g
n
(x) = n [1 – F(x)]
n-1
f(x),
and the expected value of X
min
will be
E{X
min
(n)} = n ∫ x [1 – F(x)]
n-1
f(x) dx.
In simple situations, the frequency function of the extreme value can be
expressed by using elementary functions, but in most cases numerical
approximations are necessary. For example, the minimum value for a
sample of size 2 from a normal distribution can be computed analytically
as
E{X
min
(2)}= μ - σ /√π = μ - 0,564 σ.
In larger samples, extreme values cannot be computed analytically (Harter
1961). Cramér (1945, section 28.6) deduces an asymptotic formula for
large
n
:
E{X
min
(n)} = μ – σ [log(n)
½
- (log(log(n)) + log(4π) –
2C))/2 log(n)
½
+ O(1/log(n)],
where
C
is Euler’s constant (0.5772...). The approximation is also
reasonably good for small to moderate values of
n
, but convergence is
slow (as 1/log(n), when
n
tends to infinity). By using the transformation
from lognormal to normal, Bury (1975) deduces a corresponding formula
for lognormal distributions, which is used in the present analysis.
