henv: Fit the heteroscedastic envelope model

In Renvlp: Computing Envelope Estimators

Fit the heteroscedastic envelope model

Description

Fit the heteroscedastic envelope model derived to incorporate heteroscedastic error structure in the context of estimating multivariate means for different groups with dimension u.

Usage

henv(X, Y, u, asy = TRUE, fit = TRUE, init = NULL)

Arguments

X	A group indicator vector of length n, where n denotes the number of observations.
Y	Multivariate responses. An n by r matrix, r is the number of responses and n is number of observations. The responses must be continuous variables.
u	Dimension of the heteroscedastic envelope. An integer between 0 and r.
asy	Flag for computing the asymptotic variance of the envelope estimator. The default is TRUE. When p and r are large, computing the asymptotic variance can take much time and memory. If only the envelope estimators are needed, the flag can be set to asy = FALSE.
fit	Flag for computing the fitted response. The default is TRUE.
init	The user-specified value of Gamma for the heteroscedastic envelope subspace. An r by u matrix. The default is the one generated by function henvMU.

Details

This function fits the heteroscedastic envelope model to the responses,

Y_{(i)j} = \mu + \Gamma\eta_{(i)} +\varepsilon_{(i)j}, \Sigma_{(i)}=\Gamma\Omega_{(i)}\Gamma'+\Gamma_{0}\Omega_{0}\Gamma'_{0}

for i = 1, ..., p, using the maximum likelihood estimation. When the dimension of the heteroscedastic envelope is between 1 and r-1, the starting value and blockwise coordinate descent algorithm in Cook et al. (2016) is implemented. When the dimension is r, then the envelope model degenerates to the standard multivariate linear regression for comparing group means. When the dimension is 0, it means there is no any group effect, and the fitting is different.

Value

The output is a list that contains the following components:

beta	The heteroscedastic envelope estimator of the group main effect. An r by p matix, the ith column of the matrix contains the main effect for the ith group.
Sigma	A list of the heteroscedastic envelope estimator of the error covariance matrix. Sigma[[i]] contains the estimated covariance matrix for the ith group.
Gamma	An orthonormal basis of the heteroscedastic envelope subspace.
Gamma0	An orthonormal basis of the complement of the heteroscedastic envelope subspace.
eta	A list of the coordinates of beta with respect to Gamma. eta [[i]] indicates the coordinates of the main effect of the ith group with respect to Gamma.
Omega	A list of the coordinates of Sigma with respect to Gamma. Omega[[i]] indicates the coordinates of the covariance matrix of the ith group with respect to Gamma.
Omega0	The coordinates of Sigma with respect to Gamma0.
mu	The heteroscedastic envelope estimator of the grand mean. A r by 1 matrix.
mug	A list of the heteroscedastic envelope estimator of the group mean. An r by p matix, the ith column of the matrix contains the mean for the ith group.
loglik	The maximized log likelihood function.
covMatrix	The asymptotic covariance of (mu, vec(beta)')'. An r(p + 1) by r(p + 1) matrix. The covariance matrix returned are asymptotic. For the actual standard errors, multiply by 1 / n.
asySE	The asymptotic standard error for elements in beta under the heteroscedastic envelope model. An r by p matrix. The standard errors returned are asymptotic, for actual standard errors, multiply by 1 / sqrt(n).
ratio	The asymptotic standard error ratio of the standard multivariate linear regression for comparing group means over the heteroscedastic envelope estimator, for each element in beta. An r by p matrix.
groupInd	A matrix containing the unique values of group indicators. The matrix has p rows.
n	The number of observations in the data.
ng	The number of observations in each group.
Yfit	Fitted responses.

References

Su, Z. and Cook, R. D. (2013) Estimation of Multivariate Means with Heteroscedastic Error Using Envelope Models. Statistica Sinica, 23, 213-230.

Cook, R. D., Li, B. and Chiaromente, F. (2010). Envelope Models for Parsimonious and Efficient Multivariate Linear Regression (with discussion). Statist. Sinica 20, 927- 1010.

Cook, R. D., Forzani, L. and Su, Z. (2016) A Note on Fast Envelope Estimation. Journal of Multivariate Analysis. 150, 42-54.

Examples

data(waterstrider)
X <- waterstrider[ , 1]
Y <- waterstrider[ , 2:5]

## Not run: u <- u.henv(X, Y)
## Not run: u

## Not run: m <- henv(X, Y, 2)

Renvlp documentation built on Oct. 11, 2023, 1:06 a.m.