diff --git a/CVE/R/directions.R b/CVE/R/directions.R
index 23f1b8b..b74f2bc 100644
--- a/CVE/R/directions.R
+++ b/CVE/R/directions.R
@@ -5,8 +5,8 @@ directions <- function(object, k, ...) {
 
 #' Computes projected training data \code{X} for given dimension `k`.
 #'
-#' Returns \eqn{B'X}. That is the dimensional design matrix \eqn{X} on the
-#' columnspace of the cve-estimate for given dimension \eqn{k}.
+#' Returns \eqn{B'X}. That is, it computes the projection of the \eqn{n x p}
+#' design matrix \eqn{X} on the column space of \eqn{B} of dimension \eqn{k}.
 #'
 #' @param object an object of class \code{"cve"}, usually, a result of a call to
 #'          \code{\link{cve}} or \code{\link{cve.call}}.
diff --git a/CVE/R/estimateBandwidth.R b/CVE/R/estimateBandwidth.R
index d0a3326..4dbd73c 100644
--- a/CVE/R/estimateBandwidth.R
+++ b/CVE/R/estimateBandwidth.R
@@ -37,7 +37,6 @@
 #' # calculate cve with method 'simple' for k = 1
 #' set.seed(21)
 #' cve.obj.simple <- cve(y ~ x, k = k)
-#' print(cve.obj.simple$res$'1'$h)
 #' print(estimate.bandwidth(x, k = k))
 #' @export
 estimate.bandwidth <- function (X, k, nObs, version = 1L) {
diff --git a/CVE/R/plot.R b/CVE/R/plot.R
index 087d4b6..0c48223 100644
--- a/CVE/R/plot.R
+++ b/CVE/R/plot.R
@@ -2,9 +2,9 @@
 #'
 #' Boxplots of the output \code{L} from \code{\link{cve}} over \code{k} from
 #' \code{min.dim} to \code{max.dim}. For given \code{k}, \code{L} corresponds
-#' to \eqn{L_n(V, X_i)} where \eqn{V} is a stiefel manifold element as 
-#' minimizer of
-#' \eqn{L_n(V)}, for further details see Fertl, L. and Bura, E. (2019).
+#' to \eqn{L_n(V, X_i)} where \eqn{V} is the minimizer of \eqn{L_n(V)} where
+#' \eqn{V} is an element of a Stiefel manifold (see
+#' Fertl, L. and Bura, E. (2019)).
 #'
 #' @param x an object of class \code{"cve"}, usually, a result of a call to
 #'          \code{\link{cve}} or \code{\link{cve.call}}.
diff --git a/CVE/R/predict.R b/CVE/R/predict.R
index 10f6049..a25defb 100644
--- a/CVE/R/predict.R
+++ b/CVE/R/predict.R
@@ -1,7 +1,7 @@
 #' Predict method for CVE Fits.
 #'
-#' Predict response using projected data \eqn{B'C} by fitting
-#' \eqn{g(B'C) + \epsilon} using \code{\link{mars}}.
+#' Predict response using projected data. The forward model \eqn{g(B' X)} is
+#' estimated with \code{\link{mars}} in the \code{\pkg{mda}} package.
 #'
 #' @param object an object of class \code{"cve"}, usually, a result of a call to
 #'          \code{\link{cve}} or \code{\link{cve.call}}.
@@ -9,7 +9,7 @@
 #' @param k dimension of SDR space to be used for data projection.
 #' @param ... further arguments passed to \code{\link{mars}}.
 #'
-#' @return prediced response at \code{newdata}.
+#' @return prediced respone(s) for \code{newdata}.
 #'
 #' @examples
 #' # create B for simulation
diff --git a/CVE/R/predict_dim.R b/CVE/R/predict_dim.R
index 3c8e4fa..e5e38f2 100644
--- a/CVE/R/predict_dim.R
+++ b/CVE/R/predict_dim.R
@@ -122,31 +122,32 @@ predict_dim_wilcoxon <- function(object, p.value = 0.05) {
     ))
 }
 
-#' Estimate Dimension of Reduction Space.
+#' Estimate Dimension of the Sufficient Reduction.
 #' 
-#' This function estimates the dimension of the mean dimension reduction space,
-#' i.e. number of columns of \eqn{B} matrix. The default method \code{'CV'}
-#' performs l.o.o cross-validation using \code{mars}. Given
-#' \code{k = min.dim, ..., max.dim} a cross-validation via \code{mars} is
+#' This function estimates the dimension, i.e. the rank of \eqn{B}. The default
+#' method \code{'CV'} performs leave-one-out (LOO) cross-validation using
+#' \code{mars} as follows for \code{k = min.dim, ..., max.dim} a
+#' cross-validation via \code{mars} is
 #' performed on the dataset \eqn{(Y_i, B_k' X_i)_{i = 1, ..., n}} where
 #' \eqn{B_k} is the \eqn{p \times k}{p x k} dimensional CVE estimate. The
 #' estimated SDR dimension is the \eqn{k} where the
 #' cross-validation mean squared error is minimal. The method \code{'elbow'}
 #' estimates the dimension via \eqn{k = argmin_k L_n(V_{p - k})} where
-#' \eqn{V_{p - k}} is space that is orthogonal to the columns-space of the CVE estimate of \eqn{B_k}. Method \code{'wilcoxon'} is similar to \code{'elbow'}
-#' but finds the minimum using the wilcoxon-test.
+#' \eqn{V_{p - k}} is the space that is orthogonal to the column space of the 
+#' CVE estimate of \eqn{B_k}. Method \code{'wilcoxon'}  finds the minimum using
+#' the Wilcoxon test.
 #' 
 #' @param object an object of class \code{"cve"}, usually, a result of a call to
 #'          \code{\link{cve}} or \code{\link{cve.call}}.
-#' @param method This parameter specify which method will be used in dimension
-#'  estimation. It provides three methods \code{'CV'} (default), \code{'elbow'},
-#'  and \code{'wilcoxon'} to estimate the dimension of the SDR.
+#' @param method This parameter specifies which method is used in dimension
+#'  estimation. It provides three options: \code{'CV'} (default),
+#' \code{'elbow'} and \code{'wilcoxon'}.
 #' @param ... ignored.
 #'
-#' @return list with
+#' @return A \code{list} with
 #' \describe{
-#'    \item{}{cretirion of method for \code{k = min.dim, ..., max.dim}.}
-#'    \item{k}{estimated dimension as argmin over \eqn{k} of criterion.}
+#'    \item{}{criterion for method and \code{k = min.dim, ..., max.dim}.}
+#'    \item{k}{estimated dimension is the minimizer of the criterion.}
 #' }
 #'
 #' @examples
diff --git a/CVE/R/summary.R b/CVE/R/summary.R
index 84588b0..ca3c4c4 100644
--- a/CVE/R/summary.R
+++ b/CVE/R/summary.R
@@ -1,7 +1,6 @@
-#' Prints a summary of a \code{cve} result.
+#' Prints summary statistics of the \eqn{L} \code{cve} component.
 #'
-#' Prints a summary statistics of output \code{L} from \code{cve} for
-#' \code{k = min.dim, ..., max.dim}.
+#' Prints a summary statistics of the \code{L} component of a \code{cve} object #' for \code{k = min.dim, ..., max.dim}.
 #'
 #' @param object an object of class \code{"cve"}, usually, a result of a call to
 #'          \code{\link{cve}} or \code{\link{cve.call}}.
diff --git a/CVE/R/util.R b/CVE/R/util.R
index c91954e..3627d51 100644
--- a/CVE/R/util.R
+++ b/CVE/R/util.R
@@ -1,11 +1,13 @@
-#' Draws a sample from the invariant measure on the Stiefel manifold
+#' Random sample from Stiefel manifold.
+#'
+#' Draws a random sample from the invariant measure on the Stiefel manifold
 #' \eqn{S(p, q)}.
 #'
 #' @param p row dimension
 #' @param q col dimension
-#' @return \eqn{p \times q}{p x q} semi-orthogonal matrix.
+#' @return A \eqn{p \times q}{p x q} semi-orthogonal matrix.
 #' @examples
-#'  V <- rStiefel(6, 4)
+#' V <- rStiefel(6, 4)
 #' @export
 rStiefel <- function(p, q) {
     return(qr.Q(qr(matrix(rnorm(p * q, 0, 1), p, q))))