SEPaLS {SEPaLS}

Function to estimate SEPaLS estimators

Description

Function to estimate SEPaLS estimators

Usage

SEPaLS(
  X,
  Y,
  yn,
  type = c("vMF", "Laplace"),
  mu0 = NULL,
  kappa0 = NULL,
  lambda = NULL
)

Arguments

X

(n\times p)-dimensional matrix of the covariates.

Y

(n)-dimensional vector of the response.

yn

y_n the quantile correponding to lowest values of Ys to put in th e tail.

type

character, wether vMF for von Mises-Fisher prior or Laplace for Laplace prior. See details.

mu0

\mu_0, unitary (p)-dimensional vector. The direction parameter for the vMF prior.

kappa0

\kappa_0, positive. The concentration parameter for the vMF prior.

lambda

\lambda, positive. The concentration parameter for the Laplace prior.

Details

The SEPaLS estimators are built depending on the value given to type:

• vMF: then the estimator is proportional to

  \hat{\beta}_{ml}(y_n) + \kappa_0\mu_0,

  where \hat{\beta}_{ml}(y_n) is the EPLS estimator, which coincides with the maximum-likelihood estimator of SEPaLS for a threshold y_n.

• Laplace: then the estimator is proportional to

  S_\lambda\left(\hat{\beta}_{ml}(y_n)\right),

  where S_\lambda is the soft-thresholding operator of threshold \lambda.

Value

A SEPaLS estimator

See Also

bootstrap.SEPaLS

Examples

set.seed(1)
n <- 3000
p <- 10
X <- matrix(rnorm(n*p),n,p)
beta <- c(5:1,rep(0,p-5)) ; beta <- beta/sqrt(sum(beta^2))
Y <- (X%*%beta)^3 + rnorm(n,sd=1/3)
mu0 <- rnorm(p) ; mu0 <- mu0/sqrt(sum(mu0^2))
sepals_vMF <- SEPaLS(X,Y,yn=1,type="vMF",mu0=mu0,kappa0=1)
sepals_Laplace <- SEPaLS(X,Y,yn=1,type="Laplace",lambda=0.01)