Khmaladze transformation

Consider empirical distribution function $F\_n$ based on asequence of i.i.d random variables, $X\_1,ldots,\; X\_n$.Suppose $F$ is a hypothetical distribution function ofeach $X\_i$. To test whether the choice of $F$is correct or not, statisticians use the normalized difference,

: $v\_n(x)=sqrt\{n\}\; [F\_n(x)-F(x)]\; .$

This $v\_n$, as a random process in $x$, is called empirical process. Various functional from $v\_n$ are used as test statistics. The change of the time $v\_n(x)=u\_n(t)$, , $t=F(x)$ transforms to the so-called uniform empirical process $u\_n$. The latter is an empirical processes based on independent random variables $U\_i=F(X\_i)$, which are uniformly distributed on $[0,1]$ if $X\_i$s indeed have distribution function $F$.

This fact, discovered and first utilized by Kolmogorov(1933), Wald and Wolfowitz(1936), Smirnov(1937), especially after Doob(1949) and Anderson and Darling(1952) led to the standard rule to choose test statistics from $v\_n$: these are statistics $psi(v\_n,F)$ which possibly depends $F$, in such a way that there exists another statistic $varphi(u\_n)$ from the uniform empirical process, such that $psi(v\_n,F)=varphi(u\_n)$. Examples are

: $sup\_x|v\_n(x)|=sup\_t|u\_n(t)|,quad\; sup\_xfrac\{a(F(x))\}=sup\_tfrac\; \{a(t)\}$

and

: $int\_\{-infty\}^\{infty\}\; v\_n^2(x)d\; F(x)=int\_\{0\}^\{1\}\; u\_n^2(t),dt.$

For all such functionals, their "null" distribution (under the hypothetical $F$) does not depend on $F$, and can be calculated once and then used to test any $F$.

However, it is only rarely that one needs to test simple hypothesis, when a fixed $F$ as a hypothesis is given. Much more often, one needs to verify parametric hypothesis when hypothetical $F=F\_\{\; heta\_n\}$, depends on some parameter $heta\_n$, which the hypothesis does not specify and which has to be estimated from the sample $X\_1,ldots,X\_n$ itself.

Although the estimators $hat\; heta\_n$, most commonly converge to true value of $heta$, it was discovered, Kac, Kiefer and Wolfowitz(1955) and Gikhman(1954), that the parametric, or estimated, empirical process

: $hat\; v\_n(x)=sqrt\{n\}\; [F\_n(x)-F\_\{hat\; heta\_n\}(x)]$

differs significantly from $v\_n$ and the time transformed process $hat\; u\_n(t)=hat\; v\_n(x)$, $t=F\_\{hat\; heta\_n\}(x)$ has the distribution, and the limit distribution as $n\; oinfty$, dependent on parametric form of $F\_\{\; heta\}$ on $hat\; heta\_n$ and, in general, within one parametric family, on the value of $heta$.

From mid-50's to the late-80's, much work was done to clarify the situation and understand the nature of the process $hat\; v\_n$.

In 1981, and then 1987 and 1993, E. V. Khmaladze suggested to replace the parametric empirical process $hat\; v\_n$ by its martingale part $w\_n$ only.

: $hat\; v\_n(x)-K\_n(x)=w\_n(x)$

where $K\_n(x)$ is the compensator of $hat\; v\_n(x)$. Then the following properties of $w\_n$ were established:

* Although the form of $K\_n$, and therefore, of $w\_n$, depends on $F\_\{hat\; heta\_n\}(x)$, as a function of both $x$ and $heta\_n$, the limit distribution of the time transformed process

: $omega\_n(t)=w\_n(x),\; t=F\_\{hat\; heta\_n\}(x)$

is that of standard Brownian motion on $[0,1]$, i.e., isagain standard and independent of the choice of$F\_\{hat\; heta\_n\}$.

* The relationship between $hat\; v\_n$ and $w\_n$ and between their limits, is one to one, so that the statistical inference based on $hat\; v\_n$ or on $w\_n$ are equivalent, and in $w\_n$, nothing is lost compared to $hat\; v\_n$.

* The construction of innovation martingale $w\_n$ could be carried over to the case of vector-valued $X\_1,ldots,X\_n$, giving rise to the definition of the so-called scanning martingales in $mathbb\; R^d$.

For a long time the transformation was, although known, still not used. Later, the work of researchers like R. Koenker, W. Stute, J. Bai, H. Koul, A. Koening, ... and others made it popular in econometrics and other fields of statistics.