epsiwal (version 0.2.0)

ci_connorm: ci_connorm .

Description

Confidence intervals on normal mean, subject to linear constraints.

Usage

ci_connorm(
  y,
  A,
  b,
  eta,
  Sigma = NULL,
  p = c(level/2, 1 - (level/2)),
  level = 0.05,
  Sigma_eta = Sigma %*% eta
)

Value

The values of \(\eta^{\top}\mu\) which have the corresponding CDF.

Arguments

y

an \(n\) vector, assumed multivariate normal with mean \(\mu\) and covariance \(\Sigma\).

A

an \(k \times n\) matrix of constraints.

b

a \(k\) vector of inequality limits.

eta

an \(n\) vector of the test contrast, \(\eta\).

Sigma

an \(n \times n\) matrix of the population covariance, \(\Sigma\). Not needed if Sigma_eta is given.

p

a vector of probabilities for which we return equivalent \(\eta^{\top}\mu\).

level

if p is not given, we set it by default to c(level/2,1-level/2).

Sigma_eta

an \(n\) vector of \(\Sigma \eta\).

Author

Steven E. Pav

Details

Inverts the constrained normal inference procedure described by Lee et al.

Let \(y\) be multivariate normal with unknown mean \(\mu\) and known covariance \(\Sigma\). Conditional on \(Ay \le b\) for conformable matrix \(A\) and vector \(b\), and given constrast vector \(eta\) and level \(p\), we compute \(\eta^{\top}\mu\) such that the cumulative distribution of \(\eta^{\top}y\) equals \(p\).

References

Lee, J. D., Sun, D. L., Sun, Y. and Taylor, J. E. "Exact post-selection inference, with application to the Lasso." Ann. Statist. 44, no. 3 (2016): 907-927. doi:10.1214/15-AOS1371. https://arxiv.org/abs/1311.6238

See Also

the CDF function, pconnorm, the MLE function, mle_connorm, the special case code for conditioning on the max, ci_connorm_max

Examples

set.seed(1234)
n <- 10
y <- rnorm(n)
A <- matrix(rnorm(n*(n-3)),ncol=n)
b <- A%*%y + runif(nrow(A))
Sigma <- diag(runif(n))
mu <- rnorm(n)
eta <- rnorm(n)

pval <- pconnorm(y=y,A=A,b=b,eta=eta,mu=mu,Sigma=Sigma)
cival <- ci_connorm(y=y,A=A,b=b,eta=eta,Sigma=Sigma,p=pval)
stopifnot(abs(cival - sum(eta*mu)) < 1e-4)