env.tcond: Fit the envelope model with t-distributed errors

In Renvlp: Computing Envelope Estimators

Fit the envelope model with t-distributed errors

Description

For fixed envelope dimension u, fit the envelope model in multivariate linear regression, where the errors follow a multivariate t-distribution.

Usage

env.tcond(X, Y, u, df, asy = TRUE)

Arguments

X	Predictors. An n by p matrix, p is the number of predictors. The predictors can be univariate or multivariate, discrete or continuous.
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 envelope. An integer between 0 and r.
df	Degrees of freedom of the t-distribution. A positive number that is greater than 2.
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.

Details

This function fits the envelope model to the responses and predictors,

Y_{i} = \alpha + \Gamma\eta X_{i}+\varepsilon_{i}, \Sigma=\Gamma\Omega\Gamma'+\Gamma_{0}\Omega_{0}\Gamma'_{0}, i=1, ..., n,

using the maximum likelihood estimation. The errors \varepsilon_{i} follow a multivariate t-distribution with scale matrix \Sigma and degrees of freedom df. When the dimension of the envelope is between 1 and r-1, the alternating algorithm in Forzani and Su (2021) is implemented. When the dimension is r, then the envelope model degenerates to the standard multivariate linear regression. When the dimension is 0, it means that X and Y are uncorrelated, and the fitting is different.

Value

The output is a list that contains the following components:

beta	The envelope estimator of the regression coefficients.
Sigma	The envelope estimator of the error covariance matrix.
Gamma	An orthogonal basis of the envelope subspace.
Gamma0	An orthogonal basis of the complement of the envelope subspace.
eta	The coordinates of beta with respect to Gamma.
Omega	The coordinates of Sigma with respect to Gamma.
Omega0	The coordinates of Sigma with respect to Gamma0.
mu	The estimated intercept.
loglik	The maximized log likelihood function.
covMatrix	The asymptotic covariance of vec(beta). 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 envelope model. 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 estimator (with consideration of nonconstant variance) over the envelope estimator, for each element in beta.
n	The number of observations in the data.

References

Forzani, L. and Su, Z. (2021). Envelopes for elliptical multivariate linear regression. Statist. Sinica 31, 301-332.

Examples

data(concrete)
X <- concrete[1:78, 1:7]  # The first 78 observations are training data
Y <- concrete[1:78, 8:10]
## Not run: u <- u.env.tcond(X, Y, 6)
## Not run: u

m <- env.tcond(X, Y, 2, 6)
m
m$beta

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