Reference Work Entry

International Encyclopedia of Statistical Science

pp 715-718

Date:

Khmaladze Transformation

  • Hira L. KoulAffiliated withPresident of the Indian Statistical Association, Michigan State University
  • , Eustace SwordsonAffiliated withPresident of the Indian Statistical Association, Michigan State University

Background

Consider the problem of testing the null hypothesis that a set of random variables Xi, i = 1, , n, is a random sample from a specified continuous distribution function (d.f.) F. Under the null hypothesis, the empirical d.f.
$${F}_{n}(x) = \frac{1} {n}\sum \limits _{i=1}^{n}\mathbb{I}\{{X}_{ i} \leq x\}$$
must “agree” with F. One way to measure this agreement is to use omnibus test statistics from the empirical process (see Empirical Processes)
$${v}_{n}(x) = \sqrt{n}({F}_{n}(x) - F(x)).$$
The time transformed uniform empirical process
$$\begin{array}{l@{\,}l} {u}_{n}(t) = {v}_{n}(x),\quad t = F(x)\,\end{array}$$
is an empirical process based on random variables Ui = F(Xi), i = 1, , n, that are uniformly distributed on [0, 1] under the null hypothesis. Hence, although the construction of un depends on F, the null distribution of this process does not depend on F any more (Kolmogorov (1933), Doob (1949)). From this sprang a principle, universally accepted in goodnes ...
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