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, 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)

---

ci_connorm_max .

Description

Confidence intervals on normal mean, conditioning on the max.

Usage

ci_connorm_max(
  yk,
  yk1,
  sigma = 1,
  rho = 0,
  p = c(level/2, 1 - (level/2)),
  level = 0.05
)

Arguments

yk	the observed maximum value, $y_k$.
yk1	a vector of the other observed values, $y_{k1}$, or just the scalar second largest value.
sigma	the common standard deviation.
rho	the common correlation.
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).

Details

Computes the confidence interval of unknown mean of a normal vector conditional on the one element being the maximum.

Let $y$ be multivariate normal with unknown mean $\mu$ and known covariance $\Sigma$. We assume that $\Sigma$ is compound symmetric with common variance $\sigma^2$ and common correlation $\rho$.

Conditional on $y_k \ge y_i$ for all $i$, we compute the confidence interval of $\mu_k$.

Value

The values of $\mu_k$ which have the corresponding CDF.

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, the MLE function, mle_connorm_max, the more general version, ci_connorm.

---

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]

Maintainer: Steven E. Pav [email protected] (ORCID)

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). https://arxiv.org/abs/1906.00573

Pav, S. E. "Post selection estimation of Sharpe ratios." Arxiv e-print (2026). https://arxiv.org/abs/2606.01650

See Also

Useful links:

• https://github.com/shabbychef/epsiwal

• Report bugs at https://github.com/shabbychef/epsiwal/issues

---

News for package 'epsiwal':

Description

News for package ‘epsiwal’

epsiwal Initial Version 0.2.0 (2026-06-08)

• fix numerical stability issues in ptruncnorm and downstream utilities (CIs).

• add MLE estimator of Reid, Taylor, Tibshirani.

• add helper functions for the case of conditioning on the max of a vector.

epsiwal Initial Version 0.1.0 (2019-06-28)

• first CRAN release.

---

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, ...)

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.

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 $\eta^{\top}\mu$.

Value

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

Author(s)

Steven E. Pav [email protected]

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)

---

mle_connorm_max .

Description

Maximum likelihood estimate of normal mean, conditioning on the max.

Usage

mle_connorm_max(yk, yk1, sigma = 1, rho = 0, ...)

Arguments

yk	the observed maximum value, $y_k$.
yk1	a vector of the other observed values, $y_{k1}$, or just the scalar second largest value.
sigma	the common standard deviation.
rho	the common correlation.
...	dots are passed to uniroot.

Details

Computes the maximum likelihood estimate of unknown mean of a normal vector conditional on the one element being the maximum.

Let $y$ be multivariate normal with unknown mean $\mu$ and known covariance $\Sigma$. We assume that $\Sigma$ is compound symmetric with common variance $\sigma^2$ and common correlation $\rho$.

Conditional on $y_k \ge y_i$ for all $i$, we compute the maximum likelihood estimate of $\mu_k$.

Value

The maximum likelihood estimate of $\mu_k$.

Author(s)

Steven E. Pav [email protected]

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_max, the CDF function, pconnorm, the more general version, mle_connorm.

---

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 <= x] otherwise, P[X > x].
log.p	logical; if TRUE, probabilities p are returned 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, the MLE function, mle_connorm.

---

pconnorm_max .

Description

CDF of the conditional normal variate, conditioning on the max.

Usage

pconnorm_max(
  yk,
  yk1,
  mu_k,
  sigma = 1,
  rho = 0,
  lower.tail = TRUE,
  log.p = FALSE
)

Arguments

yk	the observed maximum value, $y_k$.
yk1	a vector of the other observed values, $y_{k1}$, or just the scalar second largest value.
mu_k	the scalar mean of the maximal element $\mu_k$.
sigma	the common standard deviation.
rho	the common correlation.
lower.tail	logical; if TRUE (default), probabilities are P[X <= x] otherwise, P[X > x].
log.p	logical; if TRUE, probabilities p are returned as log(p).

Details

Computes the CDF of the conditional maximum of a normal vector using the truncated normal from the polyhedral lemma. Let $y$ be multivariate normal where the maximal observed element is known to have mean $\mu_k$, and the vector has known covariance $\Sigma$. We assume that $\Sigma$ is compound symmetric with common variance $\sigma^2$ and common correlation $\rho$.

Conditional on $y_k \ge y_i$ for all $i$, we compute the CDF of $y_k$

Value

The CDF.

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 general CDF function, pconnorm, the MLE function, mle_connorm_max, the confidence interval function, ci_connorm_max.

---

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 <= x] otherwise, P[X > x].
log.p	logical; if TRUE, probabilities p are returned 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)

---