Package 'epsiwal'

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
)

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 $\Sigma \eta$.

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 $p$.

Value

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

Note

An error will be thrown if we do not observe $A y \le b$.

Author(s)

Steven E. Pav [email protected]

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.

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)

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Exact Post Selection Inference with Applications to the Lasso.

Description

Exact Post Selection Inference with Applications to the Lasso.

Details

This simple package supports the simple procedure outlined in Lee et al. where one observes a normal random variable, then performs inference conditional on some linear inequalities.

Suppose $y$ is multivariate normal with mean $\mu$ and covariance $\Sigma$. Conditional on $Ay \le b$, one can perform inference on $\eta^{\top}\mu$ by transforming $y$ to a truncated normal. Similarly one can invert this procedure and find confidence intervals on $\eta^{\top}\mu$.

Legal Mumbo Jumbo

epsiwal is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details.

Note

This package is maintained as a hobby.

Author(s)

Steven E. Pav [email protected]

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

Pav, S. E. "Conditional inference on the asset with maximum Sharpe ratio." Arxiv e-print (2019). http://arxiv.org/abs/1906.00573

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News for package 'epsiwal':

Description

News for package ‘epsiwal’

epsiwal Initial Version 0.1.0 (2019-06-28)

• first CRAN release.

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pconnorm .

Description

CDF of the conditional normal variate.

Usage

pconnorm(
  y,
  A,
  b,
  eta,
  mu = NULL,
  Sigma = NULL,
  Sigma_eta = Sigma %*% eta,
  eta_mu = as.numeric(t(eta) %*% mu),
  lower.tail = TRUE,
  log.p = FALSE
)

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, $\eta$.
mu	an $n$ vector of the population mean, $\mu$. Not needed if eta_mu is given.
Sigma	an $n \times n$ matrix of the population covariance, $\Sigma$. Not needed if Sigma_eta is given.
Sigma_eta	an $n$ vector of $\Sigma \eta$.
eta_mu	the scalar $\eta^{\top}\mu$.
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x]$ otherwise, $P[X > x]$.
log.p	logical; if TRUE, probabilities p are given as log(p).

Details

Computes the CDF of the truncated normal conditional on linear constraints, as described in section 5 of Lee et al.

Let $y$ be multivariate normal with mean $\mu$ and covariance $\Sigma$. Conditional on $Ay \le b$ for conformable matrix $A$ and vector $b$ we compute the CDF of a truncated normal maximally aligned with $\eta$. Inference depends on the population parameters only via $\eta^{\top}\mu$ and $\Sigma \eta$, and only these need to be given.

The test statistic is aligned with $y$, meaning that an output p-value near one casts doubt on the null hypothesis that $\eta^{\top}\mu$ is less than the posited value.

Value

The CDF.

Note

An error will be thrown if we do not observe $A y \le b$.

Author(s)

Steven E. Pav [email protected]

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 confidence interval function, ci_connorm.

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ptruncnorm .

Description

Cumulative distribution of the truncated normal function.

Usage

ptruncnorm(
  q,
  mean = 0,
  sd = 1,
  a = -Inf,
  b = Inf,
  lower.tail = TRUE,
  log.p = FALSE
)

Arguments

q	vector of quantiles.
mean	vector of means.
sd	vector of standard deviations.
a	vector of the left truncation value(s).
b	vector of the right truncation value(s).
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x]$ otherwise, $P[X > x]$.
log.p	logical; if TRUE, probabilities p are given as log(p).

Value

The distribution function of the truncated normal.

Invalid arguments will result in return value NaN with a warning.

Note

Input are recycled as possible.

Author(s)

Steven E. Pav [email protected]

References

Hattaway, James T. "Parameter estimation and hypothesis testing for the truncated normal distribution with applications to introductory statistics grades." BYU Masters Thesis (2010). https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?referer=&httpsredir=1&article=3052&context=etd

Examples

y <- ptruncnorm(seq(-5,5,length.out=101), a=-1, b=2)

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