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International Encyclopedia of Statistical Science pp 715–718Cite as

Khmaladze Transformation

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Background

Consider the problem of testing the null hypothesis that a set of random variables X i , 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 U i = F(X i ), i = 1, …, n, that are uniformly distributed on [0, 1] under the null hypothesis. Hence, although the construction of u n 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 goodness...

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Author information

Authors and Affiliations

  1. President of the Indian Statistical Association, Michigan State University, East Lansing, MI, USA

    Hira L. Koul & Eustace Swordson (Professor and Chair)

Authors
  1. Hira L. Koul
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  2. Eustace Swordson
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Editor information

Editors and Affiliations

  1. Department of Statistics and Informatics, Faculty of Economics, University of Kragujevac, City of Kragujevac, Serbia

    Miodrag Lovric

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© 2011 Springer-Verlag Berlin Heidelberg

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Koul, H.L., Swordson, E. (2011). Khmaladze Transformation. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_325

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