In
statistics, the Khmaladze transformation is a mathematical tool used in constructing convenient
goodness of fit
The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measure ...
tests for hypothetical
distribution functions. More precisely, suppose
are
i.i.d.
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is us ...
, possibly multi-dimensional, random observations generated from an unknown
probability distribution. A classical problem in statistics is to decide how well a given hypothetical distribution function
, or a given hypothetical parametric family of distribution functions
, fits the set of observations. The Khmaladze transformation allows us to construct goodness of fit tests with desirable properties. It is named after
Estate V. Khmaladze.
Consider the sequence of
empirical distribution functions
based on a sequence of i.i.d random variables,
, as ''n'' increases. Suppose
is the hypothetical
distribution function of each
. To test whether the choice of
is correct or not, statisticians use the normalized difference,
:
This
, as a random process in
, is called the
empirical process
In probability theory, an empirical process is a stochastic process that describes the proportion of objects in a system in a given state.
For a process in a discrete state space a population continuous time Markov chain or Markov population model ...
. Various
functionals of
are used as test statistics. The change of the variable
,
transforms to the so-called uniform empirical process
. The latter is an empirical processes based on independent random variables
, which are
uniformly distributed on