Type: Package
Title: Statistical Inference in High Dimensional Regression
Version: 2.1.0
Description: The goal of SIHR is to provide inference procedures in the high-dimensional generalized linear regression setting for: (1) linear functionals <doi:10.48550/arXiv.1904.12891> <doi:10.48550/arXiv.2012.07133>, (2) conditional average treatment effects, (3) quadratic functionals <doi:10.48550/arXiv.1909.01503>, (4) inner product, (5) distance.
License: GPL-3
Encoding: UTF-8
RoxygenNote: 7.3.1
URL: https://zywang0701.github.io/SIHR/
BugReports: https://github.com/zywang0701/SIHR/issues
Imports: CVXR, glmnet, stats
Suggests: knitr, rmarkdown, R.rsp
VignetteBuilder: knitr, R.rsp
NeedsCompilation: no
Packaged: 2024-04-24 18:11:49 UTC; zw425
Author: Zhenyu Wang [aut], Prabrisha Rakshit [aut], Tony Cai [aut], Zijian Guo [aut, cre]
Maintainer: Zijian Guo <zijguo@stat.rutgers.edu>
Repository: CRAN
Date/Publication: 2024-04-24 18:40:04 UTC

Inference for difference of linear combinations of the regression vectors in high dimensional generalized linear regressions

Description

Computes the bias-corrected estimator of the difference of linear combinations of the regression vectors for the high dimensional generalized linear regressions and the corresponding standard error.

Usage

CATE(
  X1,
  y1,
  X2,
  y2,
  loading.mat,
  model = c("linear", "logistic", "logistic_alter"),
  intercept = TRUE,
  intercept.loading = FALSE,
  beta.init1 = NULL,
  beta.init2 = NULL,
  lambda = NULL,
  mu = NULL,
  prob.filter = 0.05,
  rescale = 1.1,
  verbose = FALSE
)

Arguments

X1

Design matrix for the first sample, of dimension n_1 x p

y1

Outcome vector for the first sample, of length n_1

X2

Design matrix for the second sample, of dimension n_2 x p

y2

Outcome vector for the second sample, of length n_1

loading.mat

Loading matrix, nrow=p, each column corresponds to a loading of interest

model

The high dimensional regression model, either "linear" or "logistic" or "logistic_alter"

intercept

Should intercept(s) be fitted for the initial estimators (default = TRUE)

intercept.loading

Should intercept term be included for the loading (default = FALSE)

beta.init1

The initial estimator of the regression vector for the 1st data (default = NULL)

beta.init2

The initial estimator of the regression vector for the 2nd data (default = NULL)

lambda

The tuning parameter in fitting initial model. If NULL, it will be picked by cross-validation. (default = NULL)

mu

The dual tuning parameter used in the construction of the projection direction. If NULL it will be searched automatically. (default = NULL)

prob.filter

The threshold of estimated probabilities for filtering observations in logistic regression. (default = 0.05)

rescale

The factor to enlarge the standard error to account for the finite sample bias. (default = 1.1)

verbose

Should intermediate message(s) be printed (default = FALSE)

Value

A list consists of plugin estimators, debiased estimators, and confidence intervals. For logistic regression, it also returns those items after probability transformation.

est.plugin.vec

The vector of plugin(biased) estimators for the linear combination of regression coefficients, length of ncol(loading.mat); corresponding to different column in loading.mat

est.debias.vec

The vector of bias-corrected estimators for the linear combination of regression coefficients, length of ncol(loading.mat); corresponding to different column in loading.mat

se.vec

The vector of standard errors of the bias-corrected estimators, length of ncol(loading.mat); corresponding to different column in loading.mat

prob.debias.vec

The vector of bias-corrected estimators after probability transformation, length of ncol(loading.mat); corresponding to different column in loading.mat.

prob.se.vec

The vector of standard errors of the bias-corrected estimators after probability transformation, length of ncol(loading.mat); corresponding to different column in loading.mat.

Examples

X1 <- matrix(rnorm(100 * 5), nrow = 100, ncol = 5)
y1 <- -0.5 + X1[, 1] * 0.5 + X1[, 2] * 1 + rnorm(100)
X2 <- matrix(rnorm(90 * 5), nrow = 90, ncol = 5)
y2 <- -0.4 + X2[, 1] * 0.48 + X2[, 2] * 1.1 + rnorm(90)
loading1 <- c(1, 1, rep(0, 3))
loading2 <- c(-0.5, -1, rep(0, 3))
loading.mat <- cbind(loading1, loading2)
Est <- CATE(X1, y1, X2, y2, loading.mat, model = "linear")

## compute confidence intervals
ci(Est, alpha = 0.05, alternative = "two.sided")

## summary statistics
summary(Est)

Inference for weighted quadratic functional of difference of the regression vectors (excluding the intercept term) in high dimensional generalized linear regressions.

Description

Inference for weighted quadratic functional of difference of the regression vectors (excluding the intercept term) in high dimensional generalized linear regressions.

Usage

Dist(
  X1,
  y1,
  X2,
  y2,
  G,
  A = NULL,
  model = c("linear", "logistic", "logistic_alter"),
  intercept = TRUE,
  beta.init1 = NULL,
  beta.init2 = NULL,
  split = TRUE,
  lambda = NULL,
  mu = NULL,
  prob.filter = 0.05,
  rescale = 1.1,
  tau = c(0.25, 0.5, 1),
  verbose = FALSE
)

Arguments

X1

Design matrix for the first sample, of dimension n_1 x p

y1

Outcome vector for the first sample, of length n_1

X2

Design matrix for the second sample, of dimension n_2 x p

y2

Outcome vector for the second sample, of length n_1

G

The set of indices, G in the quadratic form

A

The matrix A in the quadratic form, of dimension |G|\times|G|. If NULL A would be set as the |G|\times|G| submatrix of the population covariance matrix corresponding to the index set G (default = NULL)

model

The high dimensional regression model, either "linear" or "logistic" or "logistic_alter"

intercept

Should intercept(s) be fitted for the initial estimators (default = TRUE)

beta.init1

The initial estimator of the regression vector for the 1st data (default = NULL)

beta.init2

The initial estimator of the regression vector for the 2nd data (default = NULL)

split

Sampling splitting or not for computing the initial estimators. It take effects only when beta.init1 = NULL or beta.init2 = NULL. (default = TRUE)

lambda

The tuning parameter in fitting initial model. If NULL, it will be picked by cross-validation. (default = NULL)

mu

The dual tuning parameter used in the construction of the projection direction. If NULL it will be searched automatically. (default = NULL)

prob.filter

The threshold of estimated probabilities for filtering observations in logistic regression. (default = 0.05)

rescale

The factor to enlarge the standard error to account for the finite sample bias. (default = 1.1)

tau

The enlargement factor for asymptotic variance of the bias-corrected estimator to handle super-efficiency. It allows for a scalar or vector. (default = c(0.25,0.5, 1))

verbose

Should intermediate message(s) be printed. (default = FALSE)

Value

est.plugin

The plugin(biased) estimator for the quadratic form of the regression vectors restricted to G

est.debias

The bias-corrected estimator of the quadratic form of the regression vectors

se

Standard errors of the bias-corrected estimator, length of tau; corrsponding to different values of tau

Examples

X1 <- matrix(rnorm(100 * 5), nrow = 100, ncol = 5)
y1 <- -0.5 + X1[, 1] * 0.5 + X1[, 2] * 1 + rnorm(100)
X2 <- matrix(rnorm(90 * 5), nrow = 90, ncol = 5)
y2 <- -0.4 + X2[, 1] * 0.48 + X2[, 2] * 1.1 + rnorm(90)
G <- c(1, 2)
A <- matrix(c(1.5, 0.8, 0.8, 1.5), nrow = 2, ncol = 2)
Est <- Dist(X1, y1, X2, y2, G, A, model = "linear")

## compute confidence intervals
ci(Est, alpha = 0.05, alternative = "two.sided")

## summary statistics
summary(Est)

Inference for weighted inner product of the regression vectors in high dimensional generalized linear regressions

Description

Inference for weighted inner product of the regression vectors in high dimensional generalized linear regressions

Usage

InnProd(
  X1,
  y1,
  X2,
  y2,
  G,
  A = NULL,
  model = c("linear", "logistic", "logistic_alter"),
  intercept = TRUE,
  beta.init1 = NULL,
  beta.init2 = NULL,
  split = TRUE,
  lambda = NULL,
  mu = NULL,
  prob.filter = 0.05,
  rescale = 1.1,
  tau = c(0.25, 0.5, 1),
  verbose = FALSE
)

Arguments

X1

Design matrix for the first sample, of dimension n_1 x p

y1

Outcome vector for the first sample, of length n_1

X2

Design matrix for the second sample, of dimension n_2 x p

y2

Outcome vector for the second sample, of length n_1

G

The set of indices, G in the quadratic form

A

The matrix A in the quadratic form, of dimension |G|\times|G|. If NULL A would be set as the |G|\times|G| submatrix of the population covariance matrix corresponding to the index set G (default = NULL)

model

The high dimensional regression model, either "linear" or "logistic" or "logistic_alter"

intercept

Should intercept(s) be fitted for the initial estimators (default = TRUE)

beta.init1

The initial estimator of the regression vector for the 1st data (default = NULL)

beta.init2

The initial estimator of the regression vector for the 2nd data (default = NULL)

split

Sampling splitting or not for computing the initial estimators. It take effects only when beta.init1 = NULL or beta.init2 = NULL. (default = TRUE)

lambda

The tuning parameter in fitting initial model. If NULL, it will be picked by cross-validation. (default = NULL)

mu

The dual tuning parameter used in the construction of the projection direction. If NULL it will be searched automatically. (default = NULL)

prob.filter

The threshold of estimated probabilities for filtering observations in logistic regression. (default = 0.05)

rescale

The factor to enlarge the standard error to account for the finite sample bias. (default = 1.1)

tau

The enlargement factor for asymptotic variance of the bias-corrected estimator to handle super-efficiency. It allows for a scalar or vector. (default = c(0.25,0.5, 1))

verbose

Should intermediate message(s) be printed. (default = FALSE)

Value

est.plugin

The plugin(biased) estimator for the inner product form of the regression vectors restricted to G

est.debias

The bias-corrected estimator of the inner product form of the regression vectors

se

Standard errors of the bias-corrected estimator, length of tau; corrsponding to different values of tau

Examples

X1 <- matrix(rnorm(100 * 5), nrow = 100, ncol = 5)
y1 <- -0.5 + X1[, 1] * 0.5 + X1[, 2] * 1 + rnorm(100)
X2 <- matrix(rnorm(90 * 5), nrow = 90, ncol = 5)
y2 <- -0.4 + X2[, 1] * 0.48 + X2[, 2] * 1.1 + rnorm(90)
G <- c(1, 2)
A <- matrix(c(1.5, 0.8, 0.8, 1.5), nrow = 2, ncol = 2)
Est <- InnProd(X1, y1, X2, y2, G, A, model = "linear")

## compute confidence intervals
ci(Est, alpha = 0.05, alternative = "two.sided")

## summary statistics
summary(Est)

Inference for linear combination of the regression vector in high dimensional generalized linear regression

Description

Inference for linear combination of the regression vector in high dimensional generalized linear regression

Usage

LF(
  X,
  y,
  loading.mat,
  model = c("linear", "logistic", "logistic_alter"),
  intercept = TRUE,
  intercept.loading = FALSE,
  beta.init = NULL,
  lambda = NULL,
  mu = NULL,
  prob.filter = 0.05,
  rescale = 1.1,
  verbose = FALSE
)

Arguments

X

Design matrix, of dimension n x p

y

Outcome vector, of length n

loading.mat

Loading matrix, nrow=p, each column corresponds to a loading of interest

model

The high dimensional regression model, either "linear" or "logistic" or "logistic_alter"

intercept

Should intercept be fitted for the initial estimator (default = TRUE)

intercept.loading

Should intercept term be included for the loading (default = FALSE)

beta.init

The initial estimator of the regression vector (default = NULL)

lambda

The tuning parameter in fitting initial model. If NULL, it will be picked by cross-validation. (default = NULL)

mu

The dual tuning parameter used in the construction of the projection direction. If NULL it will be searched automatically. (default = NULL)

prob.filter

The threshold of estimated probabilities for filtering observations in logistic regression. (default = 0.05)

rescale

The factor to enlarge the standard error to account for the finite sample bias. (default = 1.1)

verbose

Should intermediate message(s) be printed. (default = FALSE)

Value

est.plugin.vec

The vector of plugin(biased) estimators for the linear combination of regression coefficients, length of ncol(loading.mat); each corresponding to a loading of interest

est.debias.vec

The vector of bias-corrected estimators for the linear combination of regression coefficients, length of ncol(loading.mat); each corresponding to a loading of interest

se.vec

The vector of standard errors of the bias-corrected estimators, length of ncol(loading.mat); each corresponding to a loading of interest

proj.mat

The matrix of projection directions; each column corresponding to a loading of interest.

Examples

X <- matrix(rnorm(100 * 5), nrow = 100, ncol = 5)
y <- -0.5 + X[, 1] * 0.5 + X[, 2] * 1 + rnorm(100)
loading1 <- c(1, 1, rep(0, 3))
loading2 <- c(-0.5, -1, rep(0, 3))
loading.mat <- cbind(loading1, loading2)
Est <- LF(X, y, loading.mat, model = "linear")

## compute confidence intervals
ci(Est, alpha = 0.05, alternative = "two.sided")

## summary statistics
summary(Est)

Inference for quadratic forms of the regression vector in high dimensional generalized linear regressions

Description

Inference for quadratic forms of the regression vector in high dimensional generalized linear regressions

Usage

QF(
  X,
  y,
  G,
  A = NULL,
  model = c("linear", "logistic", "logistic_alter"),
  intercept = TRUE,
  beta.init = NULL,
  split = TRUE,
  lambda = NULL,
  mu = NULL,
  prob.filter = 0.05,
  rescale = 1.1,
  tau = c(0.25, 0.5, 1),
  verbose = FALSE
)

Arguments

X

Design matrix, of dimension n x p

y

Outcome vector, of length n

G

The set of indices, G in the quadratic form

A

The matrix A in the quadratic form, of dimension |G|\times|G|. If NULL A would be set as the |G|\times|G| submatrix of the population covariance matrix corresponding to the index set G (default = NULL)

model

The high dimensional regression model, either "linear" or "logistic" or "logistic_alter"

intercept

Should intercept be fitted for the initial estimator (default = TRUE)

beta.init

The initial estimator of the regression vector (default = NULL)

split

Sampling splitting or not for computing the initial estimator. It take effects only when beta.init = NULL. (default = TRUE)

lambda

The tuning parameter in fitting initial model. If NULL, it will be picked by cross-validation. (default = NULL)

mu

The dual tuning parameter used in the construction of the projection direction. If NULL it will be searched automatically. (default = NULL)

prob.filter

The threshold of estimated probabilities for filtering observations in logistic regression. (default = 0.05)

rescale

The factor to enlarge the standard error to account for the finite sample bias. (default = 1.1)

tau

The enlargement factor for asymptotic variance of the bias-corrected estimator to handle super-efficiency. It allows for a scalar or vector. (default = c(0.25,0.5,1))

verbose

Should intermediate message(s) be printed. (default = FALSE)

Value

est.plugin

The plugin(biased) estimator for the quadratic form of the regression vector restricted to G

est.debias

The bias-corrected estimator of the quadratic form of the regression vector

se

Standard errors of the bias-corrected estimator, length of tau; corrsponding to different values of tau

Examples

X <- matrix(rnorm(100 * 5), nrow = 100, ncol = 5)
y <- X[, 1] * 0.5 + X[, 2] * 1 + rnorm(100)
G <- c(1, 2)
A <- matrix(c(1.5, 0.8, 0.8, 1.5), nrow = 2, ncol = 2)
Est <- QF(X, y, G, A, model = "linear")
## compute confidence intervals
ci(Est, alpha = 0.05, alternative = "two.sided")

## summary statistics
summary(Est)

Confidence Intervals for Bias-corrected Estimators

Description

generic function

Usage

ci(
  object,
  probability = FALSE,
  alpha = 0.05,
  alternative = c("two.sided", "less", "greater"),
  ...
)

Arguments

object

An object of class

probability

Whether returns CI with probability transformation or not (default=FALSE)

alpha

Level of significance to construct confidence interval (default=0.05)

alternative

Indicates the alternative hypothesis to construct confidence interval and must be one of "two.sided" (default), "less", or "greater".

...

arguments to pass down

Confidence Intervals for Bias-corrected CATE Estimators

Description

Computes confidence intervals for bias-corrected estimators; Each row corresponds to a loading.

Usage

## S3 method for class 'CATE'
ci(
  object,
  probability = FALSE,
  alpha = 0.05,
  alternative = c("two.sided", "less", "greater"),
  ...
)

Arguments

object

An object of class 'CATE', a result of a call to 'CATE'

probability

Whether returns CI with probability transformation or not (default=FALSE)

alpha

Level of significance to construct confidence interval (default=0.05)

alternative

Indicates the alternative hypothesis to construct confidence interval and must be one of "two.sided" (default), "less", or "greater".

...

arguments to pass down

Value

A matrix with columns giving lower and upper confidence limits for bias-corrected estimators.

Examples

## Not run: 
## -- Continuing the CATE(.) example:
out <- ci(Est)
out

## End(Not run)

Confidence Intervals for Bias-corrected Dist Estimators

Description

Computes confidence intervals for bias-corrected estimator

Usage

## S3 method for class 'Dist'
ci(
  object,
  probability = FALSE,
  alpha = 0.05,
  alternative = c("two.sided", "less", "greater"),
  ...
)

Arguments

object

An object of class 'Dist', a result of a call to 'Dist'

alpha

Level of significance to construct confidence interval (default=0.05)

alternative

Indicates the alternative hypothesis to construct confidence interval and must be one of "two.sided" (default), "less", or "greater".

...

arguments to pass down

Value

A vector giving lower and upper confidence limits for bias-corrected estimator

Examples

## Not run: 
## -- Continuing the Dist(.) example:
out <- ci(Est)
out

## End(Not run)

Confidence Intervals for Bias-corrected InnProd Estimators

Description

Computes confidence intervals for bias-corrected estimator

Usage

## S3 method for class 'InnProd'
ci(
  object,
  probability = FALSE,
  alpha = 0.05,
  alternative = c("two.sided", "less", "greater"),
  ...
)

Arguments

object

An object of class 'InnProd', a result of a call to 'InnProd'

alpha

Level of significance to construct confidence interval (default=0.05)

alternative

Indicates the alternative hypothesis to construct confidence interval and must be one of "two.sided" (default), "less", or "greater".

...

arguments to pass down

Value

A vector giving lower and upper confidence limits for bias-corrected estimator

Examples

## Not run: 
## -- Continuing the InnProd(.) example:
out <- ci(Est)
out

## End(Not run)

Confidence Intervals for Bias-corrected LF Estimators

Description

Computes confidence intervals for bias-corrected estimators; Each row corresponds to a loading.

Usage

## S3 method for class 'LF'
ci(
  object,
  probability = FALSE,
  alpha = 0.05,
  alternative = c("two.sided", "less", "greater"),
  ...
)

Arguments

object

An object of class 'LF', a result of a call to 'LF'

probability

Whether returns CI with probability transformation or not (default=FALSE)

alpha

Level of significance to construct confidence interval (default=0.05)

alternative

Indicates the alternative hypothesis to construct confidence interval and must be one of "two.sided" (default), "less", or "greater".

...

arguments to pass down

Value

A matrix with columns giving lower and upper confidence limits for bias-corrected estimators.

Examples

## Not run: 
## -- Continuing the LF(.) example:
out <- ci(Est)
out

## End(Not run)

Confidence Intervals for Bias-corrected QF Estimators

Description

Computes confidence intervals for bias-corrected estimators; Each row corresponds to a tau value.

Usage

## S3 method for class 'QF'
ci(
  object,
  probability = FALSE,
  alpha = 0.05,
  alternative = c("two.sided", "less", "greater"),
  ...
)

Arguments

object

An object of class 'QF', a result of a call to 'QF'

probability

Whether returns CI with probability transformation or not (default=FALSE)

alpha

Level of significance to construct confidence interval (default=0.05)

alternative

Indicates the alternative hypothesis to construct confidence interval and must be one of "two.sided" (default), "less", or "greater".

...

arguments to pass down

Value

A matrix with columns giving lower and upper confidence limits for bias-corrected estimators, with rows corresponding to different tau.

Examples

## Not run: 
## -- Continuing the QF(.) example:
out <- ci(Est)
out

## End(Not run)

Printing Summarizing CATE

Description

'print' method for class 'summary.CATE'

Usage

## S3 method for class 'summary.CATE'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

x

An object of class 'summary.CATE', a result of a call to 'summary.CATE'

digits

The number of digits to use when printing

...

arguments to pass down

Examples

## Not run: 
#' ##-- Continuing the CATE(.) example:
summary(Est)

## End(Not run)

Printing summarizing Dist

Description

'print' method for class 'summary.Dist'

Usage

## S3 method for class 'summary.Dist'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

x

An object of class 'summary.Dist', a result of a call to 'summary.Dist'

digits

The number of digits to use when printing

...

arguments to pass down

Examples

## Not run: 
#' ##-- Continuing the Dist(.) example:
summary(Est)

## End(Not run)

Printing summarizing InnProd

Description

'print' method for class 'summary.InnProd'

Usage

## S3 method for class 'summary.InnProd'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

x

An object of class 'summary.InnProd', a result of a call to 'summary.InnProd'

digits

The number of digits to use when printing

...

arguments to pass down

Examples

## Not run: 
#' ##-- Continuing the InnProd(.) example:
summary(Est)

## End(Not run)

Printing Summarizing LF

Description

'print' method for class 'summary.LF'

Usage

## S3 method for class 'summary.LF'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

x

An object of class 'summary.LF', a result of a call to 'summary.LF'

digits

The number of digits to use when printing

...

arguments to pass down

Examples

## Not run: 
#' ##-- Continuing the LF(.) example:
summary(Est)

## End(Not run)

Printing ummarizing QF

Description

'print' method for class 'summary.QF'

Usage

## S3 method for class 'summary.QF'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

x

An object of class 'summary.QF', a result of a call to 'summary.QF'

digits

The number of digits to use when printing

...

arguments to pass down

Examples

## Not run: 
#' ##-- Continuing the QF(.) example:
summary(Est)

## End(Not run)

Summarizing CATE

Description

'summary' method for class 'CATE'

Usage

## S3 method for class 'CATE'
summary(object, ...)

Arguments

object

An object of class 'CATE', a result of a call to 'CATE'

...

arguments to pass down

Value

The function 'summary.CATE' computes and returns a list of summary statistics.

output.est

a ncol(loading.mat) x 7 matrix with columns for the loading, plugin(biased) estimators, bias-corrected estimators, its standard error, z-statistic, corresponding (two-sided) p-value and significance stars; Each row corresponds to each loading.

Examples

## Not run: 
## -- Continuing the CATE(.) example:
sEst <- summary(Est)
sEst

## End(Not run)

Summarizing Dist

Description

'summary' method for class 'Dist'

Usage

## S3 method for class 'Dist'
summary(object, ...)

Arguments

object

An object of class 'Dist', a result of a call to 'Dist'

...

arguments to pass down

Value

The function 'summary.Dist' computes and returns a list of summary statistics.

output.est

A 6-dimensional vector with elements for plugin(biased) estimators, bias-corrected estimators, its standard error, z-statistic, corresponding (two-sided) p-value and significance stars

Examples

## Not run: 
## -- Continuing the Dist(.) example:
sEst <- summary(Est)
sEst

## End(Not run)

Summarizing InnProd

Description

'summary' method for class 'InnProd'

Usage

## S3 method for class 'InnProd'
summary(object, ...)

Arguments

object

An object of class 'InnProd', a result of a call to 'InnProd'

...

arguments to pass down

Value

The function 'summary.InnProd' computes and returns a list of summary statistics.

output.est

A 6-dimensional vector with elements for plugin(biased) estimators, bias-corrected estimators, its standard error, z-statistic, corresponding (two-sided) p-value and significance stars

Examples

## Not run: 
## -- Continuing the InnProd(.) example:
sEst <- summary(Est)
sEst

## End(Not run)

Summarizing LF

Description

'summary' method for class 'LF'

Usage

## S3 method for class 'LF'
summary(object, ...)

Arguments

object

An object of class 'LF', a result of a call to 'LF'

...

arguments to pass down

Value

The function 'summary.LF' computes and returns a list of summary statistics.

output.est

a ncol(loading.mat) x 7 matrix with columns for the loading, plugin(biased) estimators, bias-corrected estimators, its standard error, z-statistic, corresponding (two-sided) p-value and significance stars; Each row corresponds to each loading.

Examples

## Not run: 
## -- Continuing the LF(.) example:
sEst <- summary(Est)
sEst

## End(Not run)

Summarizing QF

Description

'summary' method for class 'QF'

Usage

## S3 method for class 'QF'
summary(object, ...)

Arguments

object

An object of class 'QF', a result of a call to 'QF'

...

arguments to pass down

Value

The function 'summary.QF' computes and returns a list of summary statistics.

output.est

a length(tau.vec) x 7 matrix with columns for tau, plugin(biased) estimators, bias-corrected estimators, its standard error, z-statistic, corresponding (two-sided) p-value and significance stars; Each row corresponds to each tau.

Examples

## Not run: 
## -- Continuing the QF(.) example:
sEst <- summary(Est)
sEst

## End(Not run)