The scale and complexity of data sets have increased drastically in the light of development in modern technology. High-dimensional data poses many challenges for statisticians and is becoming the most important research topic in statistics. Sufficient dimension reduction aims at preserving all the relevant information for either regression or classification and effectively transforms high dimensional problem to lower dimensions. When the partial information, such as the mean regression function or higher conditional moment, is of interest, sufficient dimension reduction is tailored for such special targets. This text provides a general treatment of sufficient dimension reduction for the conditional kth moment. The notion of central kth moment central solution space (CKMSS) is introduced and studied through examples, simulation and empirical studies.

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