Integral Transformation Methods for SDR in Regression
The routine, itdr(), which allows to estimate the sufficient dimension reduction subspaces, i.e., central mean subspace or central subspace in regression, using Fourier transformation proposed by Zhu and Zeng (2006) , convolution transformation proposed by Zeng and Zhu (2010)
and iterative Hessian transformation methods proposed by Cook and Li (2002) . The predictor variables can be consider to have a multivariate normal distribution or an elliptical contoured distribution. If the distribution of the predictor variables is unknown, then the predictors' distribution can be estimated by the kernel density estimation method. Moreover, each of these routines is supported with a bootstrap procedure to estimate their tuning parameters. That is, wx() estimates the tuning parameter for the predictor variables, wy() estimates the tuning parameter for the response variable, and wh() estimates the bandwidth parameter for the kernel density estimation method. The function invFM() estimates the central subspace using Fourier transform approach for inverse dimension reduction method proposed by Weng and Yin (2018) . The function d.test() estimates the dimension of the central mean subspace using hypothesis under invFM(). Moreover, the dsp() function provides the two distance measures between two subspaces spanned by the columns of two matrices; Vector correlation proposed by Hooper (1959) , and Trace correlation proposed by Hotelling (1936) .