An analytical/semianalytical analysis of
the characteristics of the Expected Worst Loss in T realisations for Normal
random variables
[this page  pdf  back links]
Suppose we have which are
independent identically distributed Normal random variables, i.e. . The worst loss in realisations
is .
What, in such circumstances is the expected worst loss (in realisations)?
Without much loss of generality, we can focus on , i.e. unit Normal
random variables, as .
If then all of the must also satisfy . As the are independent and
identically distribution, this means that:
where is the cumulative
distribution function of the unit Normal distribution.
So the probability density function of is where:
The expected value of , i.e. the expected
worst loss in realisations for a
unit Normal distribution, is then , i.e.:
Using a symbolic algebra engine we find that there are
analytical formulae for =
1, 2 or 3 but not thereafter:
As increases becomes more
negative, but only relatively slowly:



1

0


2

0.5641895835

0.5641895835

4

1.029375373

0.4651857895

8

1.423600306

0.394224933

16

1.765991393

0.342391087

32

2.069668828

0.303677435

64

2.343733465

0.274064637

128

2.594597369

0.250863904

256

2.826863279

0.232265910

We may check that the values shown above are reasonable
using a simulation approach, e.g. using VBA code in Microsoft Excel as per VBA code
that can be used to check this analysis.
The worst loss can also be thought of as a specific quantile
of a sample of Normal random variables. It might be viewed as the ’th quantile (i.e.
half way between 0 and , given that the
sample has observations) . We
might therefore expect its expected value to be similar to the corresponding
quantile point of the Normal distribution. However, the Normal probability
density function is not normally approximately flat at the relevant quantile
point (and is instead upward sloping), so the Expected Worst Loss in T
realisations is normally somewhat above the ’th Normal quantile
point:


c.f.

1

0

0

2

0.5641895835

0.67449

4

1.029375373

1.15035

8

1.423600306

1.53412

16

1.765991393

1.86273

32

2.069668828

2.15387

64

2.343733465

2.41756

128

2.594597369

2.66007

256

2.826863279

2.88563
