This book presents a new research topic in statistics — vertical density representation (VDR). The theory of VDR has been found to be useful for developing new ideas and methodologies in statistics and management science. The first paper related to VDR appeared in 1991. Several World Scientific Publishing Company have since been published and work is continuing on the topic. The purpose of this book is to survey the results presented in those papers and provide some new, unpublished results.VDR may be regarded as a special kind of transformation. By assuming that a variate is uniformly distributed on the contours of a given function in real n-dimensional space, and considering the density of the ordinate of the given function, the density of the original variate can be represented. The book discusses basic results and extensions. In particular, the uniform assumption on contours is relaxed to the general case. Applications are presented in Monte Carlo simulation, chaos-based uniform random number generation, and what may be called behavioral estimation. In addition, the authors include a new result in analyzing correlation into two separate components, which provides flexibility in modeling correlated phenomena, such as when combining expert estimates.

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