In statistics, the Khmaladze transformation is a mathematical tool used in constructing convenient goodness of fit tests for hypothetical distribution functions.
are i.i.d., 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,
is the hypothetical distribution function of each
is correct or not, statisticians use the normalized difference, This
, is called the empirical process.
transforms to the so-called uniform empirical process
The latter is an empirical processes based on independent random variables
This fact was discovered and first utilized by Kolmogorov (1933), Wald and Wolfowitz (1936) and Smirnov (1937) and, especially after Doob (1949) and Anderson and Darling (1952),[1] it led to the standard rule to choose test statistics based on
are defined (which possibly depend on the
being tested) in such a way that there exists another statistic
derived from the uniform empirical process, such that
Examples are and For all such functionals, their null distribution (under the hypothetical
However, it is only rarely that one needs to test a simple hypothesis, when a fixed
Much more often, one needs to verify parametric hypotheses where the hypothetical
, which the hypothesis does not specify and which have to be estimated from the sample
, most commonly converge to true value of
, it was discovered that the parametric,[2][3] or estimated, empirical process differs significantly from
, is dependent on the parametric form of
and, in general, within one parametric family, on the value of
From mid-1950s to the late-1980s, much work was done to clarify the situation and understand the nature of the process
In 1981,[4] and then 1987 and 1993,[5] Khmaladze suggested to replace the parametric empirical process
were established: For a long time the transformation was, although known, still not used.
Later, the work of researchers like Koenker, Stute, Bai, Koul, Koening, and others made it popular in econometrics and other fields of statistics.