Postulate a hypothesis for the distribution family.
Estimate the parameters for the selected distribution on the basis of statistical data.
Perform a statistical test to attempt to reject the hypothesis.
If it is not possible to reject the hypothesis the selected distribution function may be
considered to be appropriate for the modelling of the considered random variable. If the
hypothesis is rejected a new hypothesis must be postulated and the process repeated.
This procedure follows closely the classical frequentistic approach to statistical analysis.
However, in many practical engineering applications this procedure has limited value. This
not least due to the fact that the amount of available data most often is too limited to form the
solid basis for a statistical test, but also because the available tests applied in situations with
little frequentistic information may lead to the false conclusions.
In practice it is, however, often the case that physical arguments can be formulated for the
choice of distribution functions and statistical data are therefore merely used for the purpose
of checking whether the anticipated distribution function is plausible.
A practically applicable approach for the selection of the distribution function for the
modelling of a random variable is thus:
first to consider the physical reasons why the quantity at hand may belong to one or the
other distribution family;
thereafter to check whether the statistical evidence is in gross contradiction with the
assumed distribution; by using e.g. probability paper as explained in the subsequent or if
relevant the more formal approaches given in the lecture notes Basic Theory of Probability
and Statistics, (Faber, 2006).
Model Selection by Use of Probability Paper
Having selected a probability distribution family for the probabilistic modelling of a random
variable, probability paper is an extremely useful tool for the purpose of checking the
plausibility of the selected distribution family.
A probability paper for a given distribution family is constructed such that the cumulative
probability density function (or the complement) for that distribution family will have the
shape of a straight line when plotted on the paper. A probability paper is thus constructed by a
non-linear transformation of the y-axis.
For a Normal distributed random variable the cumulative distribution function is given as:
X
X
Xx
Fx,"
$%
!
# (2.85)
where X and X are the mean value and the standard deviation of the Normal distributed
random variable and where is the standard Normal probability distribution function. By
inversion of Equation
,()
(2.85) there is:(^1) (())
xFxXXX
, (2.86)