epsiwal (version 0.2.0)

mle_connorm: mle_connorm .

Description

Maximum likelihood estimate of normal mean, subject to linear constraints.

Usage

mle_connorm(y, A, b, eta, Sigma = NULL, Sigma_eta = Sigma %*% eta, ...)

Value

The maximum likelihood estimate of \(\eta^{\top}\mu\).

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.

Sigma_eta

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

...

dots are passed to uniroot.

Author

Steven E. Pav

Details

Computes the maximum likelihood estimate of unknown mean of a normal vector conditional on linear constraints.

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 \(eta\), we compute the maximum likelihood estimate of \(\eta^{\top}\mu\).

References

Reid, S., Taylor, J. and Tibshirani, R. "Post-selection point and interval estimation of signal sizes in Gaussian samples." Can. J. Statistics. 45, no. 2 (2017): 128-148. doi:10.1002/cjs.11320. https://arxiv.org/abs/1405.3340

See Also

the confidence interval function, ci_connorm, the CDF function, pconnorm, the special case code for conditioning on the max, mle_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)

mval <- mle_connorm(y=y,A=A,b=b,eta=eta,Sigma=Sigma)
# try again, but control tolerance:
mval <- mle_connorm(y=y,A=A,b=b,eta=eta,Sigma=Sigma,tol=1e-8)