Khmaladze transformation

In statistics, the Khmaladze transformation is a mathematical tool used in constructing convenient goodness of fit tests for hypothetical distribution functions. More precisely, suppose formula_1 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 formula_2, or a given hypothetical parametric family of distribution functions formula_3, 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 formula_4 based on a sequence of i.i.d random variables, formula_1, as "n" increases. Suppose formula_2 is the hypothetical distribution function of each formula_7. To test whether the choice of formula_2 is correct or not, statisticians use the normalized difference,
This formula_10, as a random process in formula_11, is called the empirical process. Various functionals of formula_10 are used as test statistics. The change of the variable formula_13, formula_14 transforms to the so-called uniform empirical process formula_15. The latter is an empirical processes based on independent random variables formula_16, which are uniformly distributed on formula_17 if the formula_7s do indeed have distribution function formula_2.
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), it led to the standard rule to choose test statistics based on formula_10. That is, test statistics formula_21 are defined (which possibly depend on the formula_2 being tested) in such a way that there exists another statistic formula_23 derived from the uniform empirical process, such that formula_24. Examples are
and
For all such functionals, their null distribution (under the hypothetical formula_2) does not depend on formula_2, and can be calculated once and then used to test any formula_2.
However, it is only rarely that one needs to test a simple hypothesis, when a fixed formula_2 as a hypothesis is given. Much more often, one needs to verify parametric hypotheses where the hypothetical formula_31, depends on some parameters formula_32, which the hypothesis does not specify and which have to be estimated from the sample formula_33 itself.
Although the estimators formula_34, most commonly converge to true value of formula_35, it was discovered that the parametric, or estimated, empirical process
differs significantly from formula_10 and that the transformed process formula_38, formula_39 has a distribution for which the limit distribution, as formula_40, is dependent on the parametric form of formula_41 and on the particular estimator formula_42 and, in general, within one parametric family, on the value of formula_35.
From mid-1950s to the late-1980s, much work was done to clarify the situation and understand the nature of the process formula_44.
In 1981, and then 1987 and 1993, Khmaladze suggested to replace the parametric empirical process formula_44 by its martingale part formula_46 only.
where formula_48 is the compensator of formula_49. Then the following properties of formula_46 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.