The user may define any distribution by a DENSITY or a PROBABILITY
specification given in the current edit field. DENSITY is used for
continuous variables and PROBABILITY for discrete ones (with values
0,1,2,...).
A FIT specification may then refer to any such distribution in the same
way as to predefined distributions (explained above).
The DENSITY specification has the form
DENSITY <name of distribution>(<list of parameters>)
Y(X)=<density Y as a function of X and parameters>
The PROBABILITY specification has the form
PROBABILITY <name of distribution>(<list of parameters>)
Y(X)=<probability of value X>
The rules in writing density and probability functions are the same
as those of the VAR operation (see VAR types 2 and 3).
For example, the Beta(a,b) distribution may be defined as follows:
DENSITY Beta(a,b)
Y(X)=if(X<=0)then(0)else(Y2)
Y2=if(X<1)then(X^(a-1)*(1-X)^(b-1))else(0)
Observe that the density function can be given without any normalizing
constant, since the computation procedure of HISTO automatically
rescales it so that the integral of Y is 1 on the range of classi-
fication defined by <variable>=<lower limit>(<step>)<upper limit>.
The trapezoidal rule is used in numeric integration with step length
<step>. Since for nonnormalized densities the integral has to be
recomputed whenever any of the parameter values is changed, giving
a normalized density speeds up the computation.
As an example on discrete distributions the geometric distribution
can be defined as follows:
PROBABILITY Geom(p)
Y(X)=if(X>=0)then(p*(1-p)^X)else(0)
In this case the probabilities are correctly scaled, but in general they
can be rescaled so that their sum is 1 on the range of classification.
The user is responsible for selecting the range wide enough.
When referring to a user-defined distribution in the FIT specification
either form FIT=Beta(3,7) (parameters given)
or FIT=Beta (parameters are to be estimated from the
current grouped data)
can be used. In the latter case the parameter estimates are computed
by ML method using the polytope algorithm (Nelder,Mead 1965) in
maximization of the likelihood function (See Walsh: Methods of
Optimization, p.81-84).
The initial estimates may be given by an INIT specification INIT=2,8
(for example). If INIT is not given, 0's are used as initial values. The
initial step lengths for the optimization algorithm may be given by
STEP=<step1>,<step2>,,, or STEP=<step> where <step> is the common step
length for all the parameters. Default is STEP=0.1 .
H = More information on HISTO
Help System (web edition)