This paper by Murray Rosenblatt discusses a transformation that converts a k-variate absolutely continuous distribution into the uniform distribution on the k-dimensional hypercube. The transformation T maps the random vector X to Z = TX, where each component of Z is the cumulative distribution function (CDF) of the corresponding component of X, conditional on the previous components. This transformation results in Z being uniformly distributed on the hypercube. The paper also discusses the implications of this transformation for statistical tests, such as the Kolmogorov-Smirnov and von Mises statistics, and how they can be used to test whether a sample comes from a uniform distribution on the hypercube. The transformation is also useful in setting up regions of equal probability mass in a chi-square test of goodness of fit. The paper provides explicit formulas for the transformation when the distribution is normal. It also notes that there are k! possible transformations corresponding to the k! ways of ordering the coordinates. The paper concludes that while there are multiple transformations, the choice of transformation can be biased by the experimenter, and thus, the use of multiple tests in the same context can lead to such bias. The paper also discusses the use of the transformation in the study of auditory cortex and communication systems.This paper by Murray Rosenblatt discusses a transformation that converts a k-variate absolutely continuous distribution into the uniform distribution on the k-dimensional hypercube. The transformation T maps the random vector X to Z = TX, where each component of Z is the cumulative distribution function (CDF) of the corresponding component of X, conditional on the previous components. This transformation results in Z being uniformly distributed on the hypercube. The paper also discusses the implications of this transformation for statistical tests, such as the Kolmogorov-Smirnov and von Mises statistics, and how they can be used to test whether a sample comes from a uniform distribution on the hypercube. The transformation is also useful in setting up regions of equal probability mass in a chi-square test of goodness of fit. The paper provides explicit formulas for the transformation when the distribution is normal. It also notes that there are k! possible transformations corresponding to the k! ways of ordering the coordinates. The paper concludes that while there are multiple transformations, the choice of transformation can be biased by the experimenter, and thus, the use of multiple tests in the same context can lead to such bias. The paper also discusses the use of the transformation in the study of auditory cortex and communication systems.