checkRd: (-1) eu.Rd:24: Lost braces; missing escapes or markup?
    24 | This function calcualtes the residual $x_{ijk}-bar{x}_{ij.}$ for the kth replication from the ith group at jth time point.
       |                                          ^
checkRd: (-1) eu.Rd:24: Lost braces
    24 | This function calcualtes the residual $x_{ijk}-bar{x}_{ij.}$ for the kth replication from the ith group at jth time point.
       |                                                   ^
checkRd: (-1) eu.Rd:24: Lost braces; missing escapes or markup?
    24 | This function calcualtes the residual $x_{ijk}-bar{x}_{ij.}$ for the kth replication from the ith group at jth time point.
       |                                                       ^
checkRd: (-1) eu.Rd:11: Lost braces; missing escapes or markup?
    11 | and the last column gives the observation $x_{ijk}$.
       |                                              ^
checkRd: (-1) eu.Rd:12: Lost braces; missing escapes or markup?
    12 | For given i and j,  $x_{ijk}, k=1, ..., n_{ij}$ are assumed to be i.i.d. from the same distribution.}
       |                        ^
checkRd: (-1) eu.Rd:12: Lost braces; missing escapes or markup?
    12 | For given i and j,  $x_{ijk}, k=1, ..., n_{ij}$ are assumed to be i.i.d. from the same distribution.}
       |                                           ^
checkRd: (-1) fun.sigijj12.Rd:5: Lost braces; missing escapes or markup?
     5 | \title{Unbiased estimate of  $sigma_{ijj1}^2$}
       |                                     ^
checkRd: (-1) fun.sigijj12.Rd:20: Lost braces; missing escapes or markup?
    20 | This function calculatess an unbiased estimate of  $sigma_{ijj1}^2$ using the u-statisitic of vectors
       |                                                           ^
checkRd: (-1) fun.sigijj12.Rd:21: Lost braces; missing escapes or markup?
    21 | $x=(x_1, x_2, ..., x_{ni})$ and  $y=(y_1, y_2, ..., y_{ni})$,
       |                      ^
checkRd: (-1) fun.sigijj12.Rd:21: Lost braces; missing escapes or markup?
    21 | $x=(x_1, x_2, ..., x_{ni})$ and  $y=(y_1, y_2, ..., y_{ni})$,
       |                                                       ^
checkRd: (-1) fun.sigijj12.Rd:22: Lost braces; missing escapes or markup?
    22 |  where $X_j$ and $Y_j$ are correlated, but $X_j$ and $Y_{j1}$ are independent if $j ne j1$.
       |                                                         ^
checkRd: (-1) fun.sigijj12.Rd:23: Lost braces; missing escapes or markup?
    23 | Note: $sum_{k1 ne k2 ne k3 ne k4} (x_{k1}-x_{k2)})(y_{k1}-y_{k2)}) (x_{k3}-x_{k4)}) (y_{k3}-y_{k4)})$ is an
       |            ^
checkRd: (-1) fun.sigijj12.Rd:23: Lost braces; missing escapes or markup?
    23 | Note: $sum_{k1 ne k2 ne k3 ne k4} (x_{k1}-x_{k2)})(y_{k1}-y_{k2)}) (x_{k3}-x_{k4)}) (y_{k3}-y_{k4)})$ is an
       |                                      ^
checkRd: (-1) fun.sigijj12.Rd:23: Lost braces; missing escapes or markup?
    23 | Note: $sum_{k1 ne k2 ne k3 ne k4} (x_{k1}-x_{k2)})(y_{k1}-y_{k2)}) (x_{k3}-x_{k4)}) (y_{k3}-y_{k4)})$ is an
       |                                             ^
checkRd: (-1) fun.sigijj12.Rd:23: Lost braces; missing escapes or markup?
    23 | Note: $sum_{k1 ne k2 ne k3 ne k4} (x_{k1}-x_{k2)})(y_{k1}-y_{k2)}) (x_{k3}-x_{k4)}) (y_{k3}-y_{k4)})$ is an
       |                                                      ^
checkRd: (-1) fun.sigijj12.Rd:23: Lost braces; missing escapes or markup?
    23 | Note: $sum_{k1 ne k2 ne k3 ne k4} (x_{k1}-x_{k2)})(y_{k1}-y_{k2)}) (x_{k3}-x_{k4)}) (y_{k3}-y_{k4)})$ is an
       |                                                             ^
checkRd: (-1) fun.sigijj12.Rd:23: Lost braces; missing escapes or markup?
    23 | Note: $sum_{k1 ne k2 ne k3 ne k4} (x_{k1}-x_{k2)})(y_{k1}-y_{k2)}) (x_{k3}-x_{k4)}) (y_{k3}-y_{k4)})$ is an
       |                                                                       ^
checkRd: (-1) fun.sigijj12.Rd:23: Lost braces; missing escapes or markup?
    23 | Note: $sum_{k1 ne k2 ne k3 ne k4} (x_{k1}-x_{k2)})(y_{k1}-y_{k2)}) (x_{k3}-x_{k4)}) (y_{k3}-y_{k4)})$ is an
       |                                                                              ^
checkRd: (-1) fun.sigijj12.Rd:23: Lost braces; missing escapes or markup?
    23 | Note: $sum_{k1 ne k2 ne k3 ne k4} (x_{k1}-x_{k2)})(y_{k1}-y_{k2)}) (x_{k3}-x_{k4)}) (y_{k3}-y_{k4)})$ is an
       |                                                                                        ^
checkRd: (-1) fun.sigijj12.Rd:23: Lost braces; missing escapes or markup?
    23 | Note: $sum_{k1 ne k2 ne k3 ne k4} (x_{k1}-x_{k2)})(y_{k1}-y_{k2)}) (x_{k3}-x_{k4)}) (y_{k3}-y_{k4)})$ is an
       |                                                                                               ^
checkRd: (-1) fun.sigijj12.Rd:24: Lost braces; missing escapes or markup?
    24 | unbiased est. of  $4*ni*(ni-1)*(ni-2)*(ni-3) [E(X_{ijk}-mu_{ij} u_{ij1k}) ]^2$.
       |                                                   ^
checkRd: (-1) fun.sigijj12.Rd:24: Lost braces; missing escapes or markup?
    24 | unbiased est. of  $4*ni*(ni-1)*(ni-2)*(ni-3) [E(X_{ijk}-mu_{ij} u_{ij1k}) ]^2$.
       |                                                            ^
checkRd: (-1) fun.sigijj12.Rd:24: Lost braces; missing escapes or markup?
    24 | unbiased est. of  $4*ni*(ni-1)*(ni-2)*(ni-3) [E(X_{ijk}-mu_{ij} u_{ij1k}) ]^2$.
       |                                                                   ^
checkRd: (-1) fun.sigijj12.Rd:16: Lost braces; missing escapes or markup?
    16 |       The sigmaijj12 variable gives an unbiased estimate of $sigma_{ijj1}^2$.
       |                                                                    ^
checkRd: (-1) fun.sigijj12.Rd:17: Lost braces; missing escapes or markup?
    17 |       The ssijj1 variable gives an unbiased estimate of $sigma_{ijj} sigma_{ij_1j_1}$.
       |                                                                ^
checkRd: (-1) fun.sigijj12.Rd:17: Lost braces; missing escapes or markup?
    17 |       The ssijj1 variable gives an unbiased estimate of $sigma_{ijj} sigma_{ij_1j_1}$.
       |                                                                            ^
checkRd: (-1) sigma4.Rd:17: Lost braces; missing escapes or markup?
    17 | the U-statistic $sum_{k1 ne k2 ne k3 ne k4} (x_{k1}-x_{k2)})^2 (x_{k3}-x_{k4)})^2$.
       |                      ^
checkRd: (-1) sigma4.Rd:17: Lost braces; missing escapes or markup?
    17 | the U-statistic $sum_{k1 ne k2 ne k3 ne k4} (x_{k1}-x_{k2)})^2 (x_{k3}-x_{k4)})^2$.
       |                                                ^
checkRd: (-1) sigma4.Rd:17: Lost braces; missing escapes or markup?
    17 | the U-statistic $sum_{k1 ne k2 ne k3 ne k4} (x_{k1}-x_{k2)})^2 (x_{k3}-x_{k4)})^2$.
       |                                                       ^
checkRd: (-1) sigma4.Rd:17: Lost braces; missing escapes or markup?
    17 | the U-statistic $sum_{k1 ne k2 ne k3 ne k4} (x_{k1}-x_{k2)})^2 (x_{k3}-x_{k4)})^2$.
       |                                                                   ^
checkRd: (-1) sigma4.Rd:17: Lost braces; missing escapes or markup?
    17 | the U-statistic $sum_{k1 ne k2 ne k3 ne k4} (x_{k1}-x_{k2)})^2 (x_{k3}-x_{k4)})^2$.
       |                                                                          ^