tauestimate: Tau estimate for the Principal Fitted Components Model (PFC)
In meszre/tauPFC: Computes Robust Estimators for the PFC Model

Description Usage Arguments Details Value References Examples

Computes the tau estimators for the PFC model

1	tauestimate(X, Fy, d, aux, inic)

`X`	vector of response variables in the inverse model, n x p matrix, each row is a response vector
`Fy`	vector of covariates in the inverse problem, vector containing functions of the response variable in the original multiple regression problem. Is a n x r matrix, each row is the corresponding response vector
`d`	number indicating the reduction subspace dimension
`aux`	list containing the constants for the tau-scale
`inic`	list containing initial estimators for the coefficient matrix and the covariance matrix

We consider the Principal Fitted Components (PFC) model given by X = μ + Γβ f(y) +Δ^1/2ε where the variables are

y is an observed response variable of the original model. f is a known vector valued function, that takes values in R^r
X is the correspondent p x 1 observed covariates vector
ε is unobserved p dimensional vector, Δ^1/2ε is the error vector

and the unknown parameters (to be estimated) are

μ a p x 1 vector of intercepts
Γ is a full-rank p x d matrix whose columns span the dimension reduction subspace
β is a full-rank d x r matrix
cov(ε) = Δ, is a p x p positive definite matrix

Both coefficient matrices Γ and β are not unique, but their product p x r matrix is unique, with rank d ≤ min(p,r). The notation refers to Cook and Forzani (2008).

List with the following components

`mu`	tau-estimator for the term μ in the PFC model
`beta`	tau-estimator for the parameter β in the PFC model
`gamma`	tau-estimator for the parameter Γ in the PFC model
`delta`	tau-estimator for the covariance matrix Δ in the PFC model

Bergesio, A., Szretter Noste, M. E. and Yohai, V. J. (2020). A robust proposal of estimation for the sufficient dimension reduction problem

Cook, R. D. and Forzani, L. (2008). Principal Fitted components for dimension reduction in regression. Statistical Science, 23(4):485-501.

p=10
n=200
mutrue=rep(0,p)
gamatrue=as.matrix(c(1,rep(0,p-1)))
betatrue=t(as.matrix(c(1,1)))

# to generate a dataset of n observations following the
# above PFC model, r=2
data_sim=generate(p,n,mutrue,gamatrue,betatrue,sigmatrue=1)
Fy=data_sim$Fy
X=data_sim$X

aux=kappa_and_c(p,efficiency=.9)
inic=initial(X,Fy,aux,efficiency=0.85)
tauestimate(X,Fy,d=1,aux,inic)