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R/AllClasses.R
In melt: Multiple Empirical Likelihood Tests
#' \linkS4class{ControlEL} class
#'
#' S4 class for computational details of empirical likelihood.
#'
#' @slot maxit A single integer for the maximum number of iterations for the
#' optimization with respect to \eqn{\theta}.
#' @slot maxit_l A single integer for the maximum number of iterations for the
#' optimization with respect to \eqn{\lambda}.
#' @slot tol A single numeric for the convergence tolerance denoted by
#' \eqn{\epsilon}. The iteration stops when
#' \deqn{\|P \nabla l(\theta^{(k)})\| < \epsilon.}
#' @slot tol_l A single numeric for the relative convergence tolerance denoted
#' by \eqn{\delta}. The iteration stops when
#' \deqn{\|\lambda^{(k)} - \lambda^{(k - 1)}\| <
#' \delta\|\lambda^{(k - 1)}\| + \delta^2.}
#' @slot step A single numeric for the step size \eqn{\gamma} for the projected
#' gradient descent method.
#' @slot th A single numeric for the threshold for the negative empirical
#' log-likelihood ratio.
#' @slot verbose A single logical for whether to print a message on the
#' convergence status.
#' @slot keep_data A single logical for whether to keep the data used for
#' fitting model objects.
#' @slot nthreads A single integer for the number of threads for parallel
#' computation via OpenMP (if available).
#' @slot seed A single integer for the seed for random number generation.
#' @slot b A single integer for the number of bootstrap replicates.
#' @slot m A single integer for the number of Monte Carlo samples.
#' @aliases ControlEL
#' @examples
#' showClass("ControlEL")
setClass("ControlEL",
slots = c(
maxit = "integer", maxit_l = "integer", tol = "numeric", tol_l = "numeric",
step = "ANY", th = "ANY", verbose = "logical", keep_data = "logical",
nthreads = "integer", seed = "ANY", b = "integer", m = "integer"
)
)
#' \linkS4class{EL} class
#'
#' S4 class for empirical likelihood.
#'
#' @details Let \eqn{X_i} be independent and identically distributed
#' \eqn{p}-dimensional random variable from an unknown distribution \eqn{P}
#' for \eqn{i = 1, \dots, n}. We assume that \eqn{P} has a positive definite
#' covariance matrix. For a parameter of interest
#' \eqn{\theta(F) \in {\rm{I\!R}}^p}, consider a \eqn{p}-dimensional smooth
#' estimating function \eqn{g(X_i, \theta)} with a moment condition
#' \deqn{\textrm{E}[g(X_i, \theta)] = 0.}
#' We assume that there exists an unique \eqn{\theta_0} that solves the above
#' equation. Given a value of \eqn{\theta}, the (profile) empirical likelihood
#' ratio is defined by
#' \deqn{R(\theta) =
#' \max_{p_i}\left\{\prod_{i = 1}^n np_i :
#' \sum_{i = 1}^n p_i g(X_i, \theta) = 0, p_i \geq 0, \sum_{i = 1}^n p_i = 1
#' \right\}.}
#' The Lagrange multiplier \eqn{\lambda \equiv \lambda(\theta)} of the dual
#' problem leads to
#' \deqn{p_i = \frac{1}{n}\frac{1}{1 + \lambda^\top g(X_i, \theta)},}
#' where \eqn{\lambda} solves
#' \deqn{\frac{1}{n}\sum_{i = 1}^n \frac{g(X_i, \theta)}
#' {1 + \lambda^\top g(X_i, \theta)} = 0.}
#' Then the empirical log-likelihood ratio is given by
#' \deqn{\log R(\theta) = -\sum_{i = 1}^n
#' \log(1 + \lambda^\top g(X_i, \theta)).}
#' This problem can be efficiently solved by the Newton-Raphson method when
#' the zero vector is contained in the interior of the convex hull of
#' \eqn{\{g(X_i, \theta)\}_{i = 1}^n}.
#'
#' It is known that \eqn{-2\log R(\theta_0)} converges in
#' distribution to \eqn{\chi^2_p}, where \eqn{\chi^2_p} has a chi-square
#' distribution with \eqn{p} degrees of freedom. See the references below for
#' more details.
#' @slot optim A list of the following optimization results:
#' * `par` A numeric vector of the specified parameters.
#' * `lambda` A numeric vector of the Lagrange multipliers of the dual
#' problem corresponding to `par`.
#' * `iterations` A single integer for the number of iterations performed.
#' * `convergence` A single logical for the convergence status.
#' * `cstr` A single logical for whether constrained EL optimization is
#' performed or not.
#' @slot logp A numeric vector of the log probabilities of the empirical
#' likelihood.
#' @slot logl A single numeric of the empirical log-likelihood.
#' @slot loglr A single numeric of the empirical log-likelihood ratio.
#' @slot statistic A single numeric of minus twice the empirical log-likelihood
#' ratio with an asymptotic chi-square distribution.
#' @slot df A single integer for the degrees of freedom of the statistic.
#' @slot pval A single numeric for the \eqn{p}-value of the statistic.
#' @slot nobs A single integer for the number of observations.
#' @slot npar A single integer for the number of parameters.
#' @slot weights A numeric vector of the re-scaled weights used for the model
#' fitting.
#' @slot coefficients A numeric vector of the maximum empirical likelihood
#' estimates of the parameters.
#' @slot method A single character for the method dispatch in internal
#' functions.
#' @slot data A numeric matrix of the data for the model fitting.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases EL
#' @references Owen A (2001). \emph{Empirical Likelihood}. Chapman & Hall/CRC.
#' \doi{10.1201/9781420036152}.
#' @references Qin J, Lawless J (1994).
#' “Empirical Likelihood and General Estimating Equations.”
#' \emph{The Annals of Statistics}, 22(1), 300--325.
#' \doi{10.1214/aos/1176325370}.
#' @examples
#' showClass("EL")
setClass("EL",
slots = c(
optim = "list", logp = "numeric", logl = "numeric", loglr = "numeric",
statistic = "numeric", df = "integer", pval = "numeric", nobs = "integer",
npar = "integer", weights = "numeric", coefficients = "numeric",
method = "character", data = "ANY", control = "ControlEL"
)
)
#' \linkS4class{CEL} class
#'
#' S4 class for constrained empirical likelihood. It inherits from
#' \linkS4class{EL} class. Note that the `optim` slot has constrained
#' optimization results with respect to the parameters, not the Lagrange
#' multiplier.
#'
#' @details Let \eqn{l(\theta)} denote minus twice the empirical log-likelihood
#' ratio function. We consider a linear hypothesis of the form
#' \deqn{L\theta = r,} where the left-hand-side \eqn{L} is a \eqn{q} by
#' \eqn{p} matrix and the right-hand-side \eqn{r} is a \eqn{q}-dimensional
#' vector. Under some regularity conditions, \eqn{l(\theta)} converges in
#' distribution to \eqn{\chi^2_q} under the constraint of hypothesis, i.e.,
#' \deqn{\min_{\theta: L\theta = r} l(\theta) \to_d \chi^2_q .}
#'
#' Minimization of \eqn{l(\theta)} with respect to \eqn{\theta} is
#' computationally expensive since it implicitly involves the
#' evaluation step as described in \linkS4class{EL}. Further, depending on the
#' form of \eqn{g(X_i, \theta)} and the constraint, the optimization problem
#' can be nonconvex and have multiple local minima. For this reason, the
#' package \pkg{melt} only considers linear hypotheses and performs local
#' minimization of \eqn{l(\theta)} using projected gradient descent method.
#' With the orthogonal projection matrix \eqn{P} and a step size \eqn{\gamma},
#' the algorithm updates \eqn{\theta} as
#' \deqn{\theta^{(k + 1)} \leftarrow \theta^{(k)} -
#' \gamma P \nabla l(\theta^{(k)}),}
#' where \eqn{\nabla l(\theta^{(k)})} denotes the gradient of \eqn{l} at
#' \eqn{\theta^{(k)}}. The first order optimality condition is
#' \eqn{P \nabla l(\theta) = 0}, which is used as the stopping criterion.
#' @slot optim A list of the following optimization results:
#' * `par` A numeric vector of the solution to the constrained optimization
#' problem.
#' * `lambda` A numeric vector of the Lagrange multipliers of the dual
#' problem corresponding to `par`.
#' * `iterations` A single integer for the number of iterations performed.
#' * `convergence` A single logical for the convergence status.
#' * `cstr` A single logical for whether constrained EL optimization is
#' performed or not.
#' @slot logp A numeric vector of the log probabilities of the constrained
#' empirical likelihood.
#' @slot logl A single numeric of the constrained empirical log-likelihood.
#' @slot loglr A single numeric of the constrained empirical log-likelihood
#' ratio.
#' @slot statistic A single numeric of minus twice the constrained empirical
#' log-likelihood ratio with an asymptotic chi-square distribution.
#' @slot df A single integer for the degrees of freedom of the statistic.
#' @slot pval A single numeric for the \eqn{p}-value of the statistic.
#' @slot nobs A single integer for the number of observations.
#' @slot npar A single integer for the number of parameters.
#' @slot weights A numeric vector of the re-scaled weights used for the model
#' fitting.
#' @slot coefficients A numeric vector of the maximum empirical likelihood
#' estimates of the parameters.
#' @slot method A single character for the method dispatch in internal
#' functions.
#' @slot data A numeric matrix of the data for the model fitting.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases CEL
#' @references Adimari G, Guolo A (2010).
#' “A Note on the Asymptotic Behaviour of Empirical Likelihood Statistics.”
#' \emph{Statistical Methods & Applications}, 19(4), 463--476.
#' \doi{10.1007/s10260-010-0137-9}.
#' @references Qin J, Lawless J (1995).
#' “Estimating Equations, Empirical Likelihood and Constraints on Parameters.”
#' \emph{Canadian Journal of Statistics}, 23(2), 145--159.
#' \doi{10.2307/3315441}.
#' @examples
#' showClass("CEL")
setClass("CEL", contains = "EL")
setOldClass("terms")
#' \linkS4class{LM} class
#'
#' S4 class for linear models with empirical likelihood. It inherits from
#' \linkS4class{CEL} class.
#'
#' @details The overall test involves a constrained optimization problem. All
#' the parameters except for the intercept are constrained to zero. The
#' `optim` slot contains the results. When there is no intercept, all
#' parameters are set to zero, and the results need to be understood in terms
#' of \linkS4class{EL} class since no constrained optimization is involved.
#' Once the solution is found, the log probabilities (`logp`) and the
#' (constrained) empirical likelihood values (`logl`, `loglr`, `statistic`)
#' readily follow, along with the degrees of freedom (`df`) and the
#' \eqn{p}-value (`pval`). The significance tests for each parameter also
#' involve constrained optimization problems where only one parameter is
#' constrained to zero. The `sigTests` slot contains the results.
#' @slot sigTests A list of the following results of significance tests:
#' * `statistic` A numeric vector of minus twice the (constrained) empirical
#' log-likelihood ratios with asymptotic chi-square distributions.
#' * `iterations` An integer vector for the number of iterations performed for
#' each parameter.
#' * `convergence` A logical vector for the convergence status of each
#' parameter.
#' @slot call A matched call.
#' @slot terms A [`terms`] object used.
#' @slot misc A list of various outputs obtained from the model fitting process.
#' They are used in other generics and methods.
#' @slot optim A list of the following optimization results:
#' * `par` A numeric vector of the solution to the (constrained) optimization
#' problem.
#' * `lambda` A numeric vector of the Lagrange multipliers of the dual
#' problem corresponding to `par`.
#' * `iterations` A single integer for the number of iterations performed.
#' * `convergence` A single logical for the convergence status.
#' @slot logp A numeric vector of the log probabilities of the (constrained)
#' empirical likelihood.
#' @slot logl A single numeric of the (constrained) empirical log-likelihood.
#' @slot loglr A single numeric of the (constrained) empirical log-likelihood
#' ratio.
#' @slot statistic A single numeric of minus twice the (constrained) empirical
#' log-likelihood ratio with an asymptotic chi-square distribution.
#' @slot df A single integer for the degrees of freedom of the statistic.
#' @slot pval A single numeric for the \eqn{p}-value of the statistic.
#' @slot nobs A single integer for the number of observations.
#' @slot npar A single integer for the number of parameters.
#' @slot weights A numeric vector of the re-scaled weights used for the model
#' fitting.
#' @slot coefficients A numeric vector of the maximum empirical likelihood
#' estimates of the parameters.
#' @slot method A single character for the method dispatch in internal
#' functions.
#' @slot data A numeric matrix of the data for the model fitting.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases LM
#' @examples
#' showClass("LM")
setClass("LM",
slots = c(sigTests = "ANY", call = "call", terms = "terms", misc = "list"),
contains = "CEL"
)
setOldClass("family")
#' \linkS4class{GLM} class
#'
#' S4 class for generalized linear models. It inherits from \linkS4class{LM}
#' class.
#'
#' @details The overall test involves a constrained optimization problem. All
#' the parameters except for the intercept are constrained to zero. The
#' `optim` slot contains the results. When there is no intercept, all
#' parameters are set to zero, and the results need to be understood in terms
#' of \linkS4class{EL} class since no constrained optimization is involved.
#' Once the solution is found, the log probabilities (`logp`) and the
#' (constrained) empirical likelihood values (`logl`, `loglr`, `statistic`)
#' readily follow, along with the degrees of freedom (`df`) and the
#' \eqn{p}-value (`pval`). The significance tests for each parameter also
#' involve constrained optimization problems where only one parameter is
#' constrained to zero. The `sigTests` slot contains the results.
#' @slot family A [`family`] object used.
#' @slot dispersion A single numeric for the estimated dispersion parameter.
#' @slot sigTests A list of the following results of significance tests:
#' * `statistic` A numeric vector of minus twice the (constrained) empirical
#' log-likelihood ratios with asymptotic chi-square distributions.
#' * `iterations` An integer vector for the number of iterations performed for
#' each parameter.
#' * `convergence` A logical vector for the convergence status of each
#' parameter.
#' * `cstr` A single logical for whether constrained EL optimization is
#' performed or not.
#' @slot call A matched call.
#' @slot terms A [`terms`] object used.
#' @slot misc A list of various outputs obtained from the model fitting process.
#' They are used in other generics and methods.
#' @slot optim A list of the following optimization results:
#' * `par` A numeric vector of the solution to the (constrained) optimization
#' problem.
#' * `lambda` A numeric vector of the Lagrange multipliers of the dual
#' problem corresponding to `par`.
#' * `iterations` A single integer for the number of iterations performed.
#' * `convergence` A single logical for the convergence status.
#' @slot logp A numeric vector of the log probabilities of the (constrained)
#' empirical likelihood.
#' @slot logl A single numeric of the (constrained) empirical log-likelihood.
#' @slot loglr A single numeric of the (constrained) empirical log-likelihood
#' ratio.
#' @slot statistic A single numeric of minus twice the (constrained) empirical
#' log-likelihood ratio with an asymptotic chi-square distribution.
#' @slot df A single integer for the degrees of freedom of the statistic.
#' @slot pval A single numeric for the \eqn{p}-value of the statistic.
#' @slot nobs A single integer for the number of observations.
#' @slot npar A single integer for the number of parameters.
#' @slot weights A numeric vector of the re-scaled weights used for the model
#' fitting.
#' @slot coefficients A numeric vector of the maximum empirical likelihood
#' estimates of the parameters.
#' @slot method A single character for the method dispatch in internal
#' functions.
#' @slot data A numeric matrix of the data for the model fitting.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases GLM
#' @examples
#' showClass("GLM")
setClass("GLM",
slots = c(family = "family", dispersion = "numeric"),
contains = "LM"
)
#' \linkS4class{ConfregEL} class
#'
#' S4 class for confidence region. It inherits from `"matrix"`.
#'
#' @slot estimates A numeric vector of length two for the parameter estimates.
#' @slot level A single numeric for the confidence level required.
#' @slot cv A single numeric for the critical value for calibration of empirical
#' likelihood ratio statistic.
#' @slot pnames A character vector of length two for the name of parameters.
#' @aliases ConfregEL
#' @examples
#' showClass("ConfregEL")
setClass("ConfregEL",
slots = c(
estimates = "numeric", level = "numeric", cv = "numeric",
pnames = "character"
),
contains = "matrix"
)
#' \linkS4class{ELD} class
#'
#' S4 class for empirical likelihood displacement. It inherits from `"numeric"`.
#'
#' @aliases ELD
#' @examples
#' showClass("ELD")
setClass("ELD", contains = "numeric")
#' \linkS4class{ELMT} class
#'
#' S4 class for empirical likelihood multiple tests.
#'
#' @slot estimates A list of numeric vectors of the estimates of the linear
#' hypotheses.
#' @slot statistic A numeric vector of minus twice the (constrained) empirical
#' log-likelihood ratios with asymptotic chi-square distributions.
#' @slot df An integer vector of the marginal degrees of freedom of the
#' statistic.
#' @slot pval A numeric vector for the multiplicity adjusted \eqn{p}-values.
#' @slot cv A single numeric for the multiplicity adjusted critical value.
#' @slot rhs A numeric vector for the right-hand sides of the hypotheses.
#' @slot lhs A numeric matrix for the left-hand side of the hypotheses.
#' @slot alpha A single numeric for the overall significance level.
#' @slot calibrate A single character for the calibration method used.
#' @slot weights A numeric vector of the re-scaled weights used for the model
#' fitting.
#' @slot coefficients A numeric vector of the maximum empirical likelihood
#' estimates of the parameters.
#' @slot method A single character for the method dispatch in internal
#' functions.
#' @slot data A numeric matrix of the data for the model fitting.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases ELMT
#' @examples
#' showClass("ELMT")
setClass("ELMT",
slots = c(
estimates = "list", statistic = "numeric", df = "integer", pval = "numeric",
cv = "numeric", rhs = "numeric", lhs = "matrix", alpha = "numeric",
calibrate = "character", weights = "numeric", coefficients = "numeric",
method = "character", data = "ANY", control = "ControlEL"
)
)
#' \linkS4class{ELT} class
#'
#' S4 class for empirical likelihood test.
#'
#' @slot optim A list of the following optimization results:
#' * `par` A numeric vector of the solution to the (constrained) optimization
#' problem.
#' * `lambda` A numeric vector of the Lagrange multipliers of the dual
#' problem corresponding to `par`.
#' * `iterations` A single integer for the number of iterations performed.
#' * `convergence` A single logical for the convergence status.
#' * `cstr` A single logical for whether constrained EL optimization is
#' performed or not.
#' @slot logp A numeric vector of the log probabilities of the (constrained)
#' empirical likelihood.
#' @slot logl A single numeric of the (constrained) empirical log-likelihood.
#' @slot loglr A single numeric of the (constrained) empirical log-likelihood
#' ratio.
#' @slot statistic A single numeric of minus twice the (constrained) empirical
#' log-likelihood ratio with an asymptotic chi-square distribution.
#' @slot df A single integer for the chi-square degrees of freedom of the
#' statistic.
#' @slot pval A single numeric for the (calibrated) \eqn{p}-value of the
#' statistic.
#' @slot cv A single numeric for the critical value.
#' @slot rhs A numeric vector for the right-hand side of the hypothesis.
#' @slot lhs A numeric matrix for the left-hand side of the hypothesis.
#' @slot alpha A single numeric for the significance level.
#' @slot calibrate A single character for the calibration method used.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases ELT
#' @examples
#' showClass("ELT")
setClass("ELT",
slots = c(
optim = "list", logp = "numeric", logl = "numeric", loglr = "numeric",
statistic = "numeric", df = "integer", pval = "numeric", cv = "numeric",
rhs = "numeric", lhs = "matrix", alpha = "numeric", calibrate = "character",
control = "ControlEL"
)
)
#' \linkS4class{QGLM} class
#'
#' S4 class for generalized linear models with quasi-likelihood methods. It
#' inherits from \linkS4class{GLM} class.
#'
#' @details The overall test involves a constrained optimization problem. All
#' the parameters except for the intercept are constrained to zero. The
#' `optim` slot contains the results. When there is no intercept, all
#' parameters are set to zero, and the results need to be understood in terms
#' of \linkS4class{EL} class since no constrained optimization is involved.
#' Once the solution is found, the log probabilities (`logp`) and the
#' (constrained) empirical likelihood values (`logl`, `loglr`, `statistic`)
#' readily follow, along with the degrees of freedom (`df`) and the
#' \eqn{p}-value (`pval`). The significance tests for each parameter also
#' involve constrained optimization problems where only one parameter is
#' constrained to zero. The `sigTests` slot contains the results.
#' @slot family A [`family`] object used.
#' @slot dispersion A single numeric for the estimated dispersion parameter.
#' @slot sigTests A list of the following results of significance tests:
#' * `statistic` A numeric vector of minus twice the (constrained) empirical
#' log-likelihood ratios with asymptotic chi-square distributions.
#' * `iterations` An integer vector for the number of iterations performed for
#' each parameter.
#' * `convergence` A logical vector for the convergence status of each
#' parameter.
#' * `cstr` A single logical for whether constrained EL optimization is
#' performed or not.
#' @slot call A matched call.
#' @slot terms A [`terms`] object used.
#' @slot misc A list of various outputs obtained from the model fitting process.
#' They are used in other generics and methods.
#' @slot optim A list of the following optimization results:
#' * `par` A numeric vector of the solution to the (constrained) optimization
#' problem.
#' * `lambda` A numeric vector of the Lagrange multipliers of the dual
#' problem corresponding to `par`.
#' * `iterations` A single integer for the number of iterations performed.
#' * `convergence` A single logical for the convergence status.
#' @slot logp A numeric vector of the log probabilities of the (constrained)
#' empirical likelihood.
#' @slot logl A single numeric of the (constrained) empirical log-likelihood.
#' @slot loglr A single numeric of the (constrained) empirical log-likelihood
#' ratio.
#' @slot statistic A single numeric of minus twice the (constrained) empirical
#' log-likelihood ratio with an asymptotic chi-square distribution.
#' @slot df A single integer for the degrees of freedom of the statistic.
#' @slot pval A single numeric for the \eqn{p}-value of the statistic.
#' @slot nobs A single integer for the number of observations.
#' @slot npar A single integer for the number of parameters.
#' @slot weights A numeric vector of the re-scaled weights used for the model
#' fitting.
#' @slot coefficients A numeric vector of the maximum empirical likelihood
#' estimates of the parameters.
#' @slot method A single character for the method dispatch in internal
#' functions.
#' @slot data A numeric matrix of the data for the model fitting.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases QGLM
#' @examples
#' showClass("QGLM")
setClass("QGLM", contains = "GLM")
#' \linkS4class{SD} class
#'
#' S4 class for standard deviation. It inherits from \linkS4class{EL} class.
#'
#' @slot optim A list of the following optimization results:
#' * `par` A numeric vector of the specified parameters.
#' * `lambda` A numeric vector of the Lagrange multipliers of the dual
#' problem corresponding to `par`.
#' * `iterations` A single integer for the number of iterations performed.
#' * `convergence` A single logical for the convergence status.
#' * `cstr` A single logical for whether constrained EL optimization is
#' performed or not.
#' @slot logp A numeric vector of the log probabilities of the empirical
#' likelihood.
#' @slot logl A single numeric of the empirical log-likelihood.
#' @slot loglr A single numeric of the empirical log-likelihood ratio.
#' @slot statistic A single numeric of minus twice the empirical log-likelihood
#' ratio with an asymptotic chi-square distribution.
#' @slot df A single integer for the degrees of freedom of the statistic.
#' @slot pval A single numeric for the \eqn{p}-value of the statistic.
#' @slot nobs A single integer for the number of observations.
#' @slot npar A single integer for the number of parameters.
#' @slot weights A numeric vector of the re-scaled weights used for the model
#' fitting.
#' @slot coefficients A numeric vector of the maximum empirical likelihood
#' estimates of the parameters.
#' @slot method A single character for the method dispatch in internal
#' functions.
#' @slot data A numeric matrix of the data for the model fitting.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases SD
#' @examples
#' showClass("SD")
setClass("SD", contains = "EL")
#' \linkS4class{SummaryEL} class
#'
#' S4 class for a summary of \linkS4class{EL} objects.
#'
#' @slot optim A list of the following optimization results:
#' * `par` A numeric vector of the specified parameters.
#' * `lambda` A numeric vector of the Lagrange multipliers of the dual
#' problem corresponding to `par`.
#' * `iterations` A single integer for the number of iterations performed.
#' * `convergence` A single logical for the convergence status.
#' * `cstr` A single logical for whether constrained EL optimization is
#' performed or not.
#' @slot logl A single numeric of the empirical log-likelihood.
#' @slot loglr A single numeric of the empirical log-likelihood ratio.
#' @slot statistic A single numeric of minus twice the empirical log-likelihood
#' ratio with an asymptotic chi-square distribution.
#' @slot df A single integer for the degrees of freedom of the statistic.
#' @slot pval A single numeric for the \eqn{p}-value of the statistic.
#' @slot nobs A single integer for the number of observations.
#' @slot npar A single integer for the number of parameters.
#' @slot weighted A single logical for whether the data are weighted or not.
#' @slot coefficients A numeric vector of the maximum empirical likelihood
#' estimates of the parameters.
#' @slot method A single character for the method dispatch in internal
#' functions.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases SummaryEL
#' @examples
#' showClass("SummaryEL")
setClass("SummaryEL", slots = c(
optim = "list", logl = "numeric", loglr = "numeric", statistic = "numeric",
df = "integer", pval = "numeric", nobs = "integer", npar = "integer",
weighted = "logical", coefficients = "numeric", method = "character",
control = "ControlEL"
))
#' \linkS4class{SummaryELMT} class
#'
#' S4 class for a summary of \linkS4class{ELMT} objects.
#'
#' @slot aliased A named logical vector showing if the original coefficients are
#' aliased.
#' @aliases SummaryELMT
#' @examples
#' showClass("SummaryELMT")
setClass("SummaryELMT", slots = c(
estimates = "list", statistic = "numeric", df = "integer", pval = "numeric",
cv = "numeric", rhs = "numeric", lhs = "matrix", alpha = "numeric",
calibrate = "character"
))
#' \linkS4class{SummaryELT} class
#'
#' S4 class for a summary of \linkS4class{ELT} objects.
#'
#' @slot optim A list of the following optimization results:
#' * `par` A numeric vector of the solution to the (constrained) optimization
#' problem.
#' * `lambda` A numeric vector of the Lagrange multipliers of the dual
#' problem corresponding to `par`.
#' * `iterations` A single integer for the number of iterations performed.
#' * `convergence` A single logical for the convergence status.
#' * `cstr` A single logical for whether constrained EL optimization is
#' performed or not.
#' @slot logl A single numeric of the (constrained) empirical log-likelihood.
#' @slot loglr A single numeric of the (constrained) empirical log-likelihood
#' ratio.
#' @slot statistic A single numeric of minus twice the (constrained) empirical
#' log-likelihood ratio with an asymptotic chi-square distribution.
#' @slot df A single integer for the chi-square degrees of freedom of the
#' statistic.
#' @slot pval A single numeric for the (calibrated) \eqn{p}-value of the
#' statistic.
#' @slot cv A single numeric for the critical value.
#' @slot rhs A numeric vector for the right-hand side of the hypothesis.
#' @slot lhs A numeric matrix for the left-hand side of the hypothesis.
#' @slot alpha A single numeric for the significance level.
#' @slot calibrate A single character for the calibration method used.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases SummaryELT
#' @examples
#' showClass("SummaryELT")
setClass("SummaryELT", slots = c(
optim = "list", logl = "numeric", loglr = "numeric", statistic = "numeric",
df = "integer", pval = "numeric", cv = "numeric", rhs = "numeric",
lhs = "matrix", alpha = "numeric", calibrate = "character",
control = "ControlEL"
))
#' \linkS4class{SummaryLM} class
#'
#' S4 class for a summary of \linkS4class{LM} objects.
#'
#' @slot coefficients A numeric matrix of the results of significance tests.
#' @slot intercept A single logical for whether the given model has an intercept
#' term or not.
#' @slot na.action Information returned by [`model.frame`] on the special
#' handling of `NA`s.
#' @slot call A matched call.
#' @slot terms A [`terms`] object used.
#' @slot aliased A named logical vector showing if the original coefficients are
#' aliased.
#' @slot optim A list of the following optimization results:
#' * `par` A numeric vector of the solution to the (constrained) optimization
#' problem.
#' * `lambda` A numeric vector of the Lagrange multipliers of the dual
#' problem corresponding to `par`.
#' * `iterations` A single integer for the number of iterations performed.
#' * `convergence` A single logical for the convergence status.
#' * `cstr` A single logical for whether constrained EL optimization is
#' performed or not.
#' @slot logl A single numeric of the empirical log-likelihood.
#' @slot loglr A single numeric of the empirical log-likelihood ratio.
#' @slot statistic A single numeric of minus twice the (constrained) empirical
#' log-likelihood ratio for the overall test.
#' @slot df A single integer for the degrees of freedom of the statistic.
#' @slot pval A single numeric for the \eqn{p}-value of the statistic.
#' @slot nobs A single integer for the number of observations.
#' @slot npar A single integer for the number of parameters.
#' @slot weighted A single logical for whether the data are weighted or not.
#' @slot method A single character for the method dispatch in internal
#' functions.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases SummaryLM
#' @examples
#' showClass("SummaryLM")
setClass("SummaryLM", slots = c(
coefficients = "matrix", intercept = "logical", na.action = "ANY",
call = "call", terms = "terms", aliased = "logical", optim = "list",
logl = "numeric", loglr = "numeric", statistic = "numeric", df = "integer",
pval = "numeric", nobs = "integer", npar = "integer", weighted = "logical",
method = "character", control = "ControlEL"
))
#' \linkS4class{SummaryGLM} class
#'
#' S4 class for a summary of \linkS4class{GLM} objects. It inherits from
#' \linkS4class{SummaryLM} class.
#'
#' @slot family A [`family`] object used.
#' @slot dispersion A single numeric for the estimated dispersion parameter.
#' @slot coefficients A numeric matrix of the results of significance tests.
#' @slot intercept A single logical for whether the given model has an intercept
#' term or not.
#' @slot na.action Information returned by [`model.frame`] on the special
#' handling of `NA`s.
#' @slot call A matched call.
#' @slot terms A [`terms`] object used.
#' @slot aliased A named logical vector showing if the original coefficients are
#' aliased.
#' @slot optim A list of the following optimization results:
#' * `par` A numeric vector of the solution to the (constrained) optimization
#' problem.
#' * `lambda` A numeric vector of the Lagrange multipliers of the dual
#' problem corresponding to `par`.
#' * `iterations` A single integer for the number of iterations performed.
#' * `convergence` A single logical for the convergence status.
#' * `cstr` A single logical for whether constrained EL optimization is
#' performed or not.
#' @slot logl A single numeric of the empirical log-likelihood.
#' @slot loglr A single numeric of the empirical log-likelihood ratio.
#' @slot statistic A single numeric of minus twice the (constrained) empirical
#' log-likelihood ratio for the overall test.
#' @slot df A single integer for the degrees of freedom of the statistic.
#' @slot pval A single numeric for the \eqn{p}-value of the statistic.
#' @slot nobs A single integer for the number of observations.
#' @slot npar A single integer for the number of parameters.
#' @slot weighted A single logical for whether the data are weighted or not.
#' @slot method A single character for the method dispatch in internal
#' functions.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases SummaryGLM
#' @examples
#' showClass("SummaryGLM")
setClass("SummaryGLM",
slots = c(family = "family", dispersion = "numeric"), contains = "SummaryLM"
)
#' \linkS4class{SummaryQGLM} class
#'
#' S4 class for a summary of \linkS4class{QGLM} objects. It inherits from
#' \linkS4class{SummaryGLM} class.
#'
#' @slot family A [`family`] object used.
#' @slot dispersion A single numeric for the estimated dispersion parameter.
#' @slot coefficients A numeric matrix of the results of significance tests.
#' @slot intercept A single logical for whether the given model has an intercept
#' term or not.
#' @slot na.action Information returned by [`model.frame`] on the special
#' handling of `NA`s.
#' @slot call A matched call.
#' @slot terms A [`terms`] object used.
#' @slot aliased A named logical vector showing if the original coefficients are
#' aliased.
#' @slot optim A list of the following optimization results:
#' * `par` A numeric vector of the solution to the (constrained) optimization
#' problem.
#' * `lambda` A numeric vector of the Lagrange multipliers of the dual
#' problem corresponding to `par`.
#' * `iterations` A single integer for the number of iterations performed.
#' * `convergence` A single logical for the convergence status.
#' * `cstr` A single logical for whether constrained EL optimization is
#' performed or not.
#' @slot logl A single numeric of the empirical log-likelihood.
#' @slot loglr A single numeric of the empirical log-likelihood ratio.
#' @slot statistic A single numeric of minus twice the (constrained) empirical
#' log-likelihood ratio for the overall test.
#' @slot df A single integer for the degrees of freedom of the statistic.
#' @slot pval A single numeric for the \eqn{p}-value of the statistic.
#' @slot nobs A single integer for the number of observations.
#' @slot npar A single integer for the number of parameters.
#' @slot weighted A single logical for whether the data are weighted or not.
#' @slot method A single character for the method dispatch in internal
#' functions.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases SummaryQGLM
#' @examples
#' showClass("SummaryQGLM")
setClass("SummaryQGLM", contains = "SummaryGLM")
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#' \linkS4class{ControlEL} class
#'
#' S4 class for computational details of empirical likelihood.
#'
#' @slot maxit A single integer for the maximum number of iterations for the
#' optimization with respect to \eqn{\theta}.
#' @slot maxit_l A single integer for the maximum number of iterations for the
#' optimization with respect to \eqn{\lambda}.
#' @slot tol A single numeric for the convergence tolerance denoted by
#' \eqn{\epsilon}. The iteration stops when
#' \deqn{\|P \nabla l(\theta^{(k)})\| < \epsilon.}
#' @slot tol_l A single numeric for the relative convergence tolerance denoted
#' by \eqn{\delta}. The iteration stops when
#' \deqn{\|\lambda^{(k)} - \lambda^{(k - 1)}\| <
#' \delta\|\lambda^{(k - 1)}\| + \delta^2.}
#' @slot step A single numeric for the step size \eqn{\gamma} for the projected
#' gradient descent method.
#' @slot th A single numeric for the threshold for the negative empirical
#' log-likelihood ratio.
#' @slot verbose A single logical for whether to print a message on the
#' convergence status.
#' @slot keep_data A single logical for whether to keep the data used for
#' fitting model objects.
#' @slot nthreads A single integer for the number of threads for parallel
#' computation via OpenMP (if available).
#' @slot seed A single integer for the seed for random number generation.
#' @slot b A single integer for the number of bootstrap replicates.
#' @slot m A single integer for the number of Monte Carlo samples.
#' @aliases ControlEL
#' @examples
#' showClass("ControlEL")
setClass("ControlEL",
slots = c(
maxit = "integer", maxit_l = "integer", tol = "numeric", tol_l = "numeric",
step = "ANY", th = "ANY", verbose = "logical", keep_data = "logical",
nthreads = "integer", seed = "ANY", b = "integer", m = "integer"
)
)
#' \linkS4class{EL} class
#'
#' S4 class for empirical likelihood.
#'
#' @details Let \eqn{X_i} be independent and identically distributed
#' \eqn{p}-dimensional random variable from an unknown distribution \eqn{P}
#' for \eqn{i = 1, \dots, n}. We assume that \eqn{P} has a positive definite
#' covariance matrix. For a parameter of interest
#' \eqn{\theta(F) \in {\rm{I\!R}}^p}, consider a \eqn{p}-dimensional smooth
#' estimating function \eqn{g(X_i, \theta)} with a moment condition
#' \deqn{\textrm{E}[g(X_i, \theta)] = 0.}
#' We assume that there exists an unique \eqn{\theta_0} that solves the above
#' equation. Given a value of \eqn{\theta}, the (profile) empirical likelihood
#' ratio is defined by
#' \deqn{R(\theta) =
#' \max_{p_i}\left\{\prod_{i = 1}^n np_i :
#' \sum_{i = 1}^n p_i g(X_i, \theta) = 0, p_i \geq 0, \sum_{i = 1}^n p_i = 1
#' \right\}.}
#' The Lagrange multiplier \eqn{\lambda \equiv \lambda(\theta)} of the dual
#' problem leads to
#' \deqn{p_i = \frac{1}{n}\frac{1}{1 + \lambda^\top g(X_i, \theta)},}
#' where \eqn{\lambda} solves
#' \deqn{\frac{1}{n}\sum_{i = 1}^n \frac{g(X_i, \theta)}
#' {1 + \lambda^\top g(X_i, \theta)} = 0.}
#' Then the empirical log-likelihood ratio is given by
#' \deqn{\log R(\theta) = -\sum_{i = 1}^n
#' \log(1 + \lambda^\top g(X_i, \theta)).}
#' This problem can be efficiently solved by the Newton-Raphson method when
#' the zero vector is contained in the interior of the convex hull of
#' \eqn{\{g(X_i, \theta)\}_{i = 1}^n}.
#'
#' It is known that \eqn{-2\log R(\theta_0)} converges in
#' distribution to \eqn{\chi^2_p}, where \eqn{\chi^2_p} has a chi-square
#' distribution with \eqn{p} degrees of freedom. See the references below for
#' more details.
#' @slot optim A list of the following optimization results:
#' * `par` A numeric vector of the specified parameters.
#' * `lambda` A numeric vector of the Lagrange multipliers of the dual
#' problem corresponding to `par`.
#' * `iterations` A single integer for the number of iterations performed.
#' * `convergence` A single logical for the convergence status.
#' * `cstr` A single logical for whether constrained EL optimization is
#' performed or not.
#' @slot logp A numeric vector of the log probabilities of the empirical
#' likelihood.
#' @slot logl A single numeric of the empirical log-likelihood.
#' @slot loglr A single numeric of the empirical log-likelihood ratio.
#' @slot statistic A single numeric of minus twice the empirical log-likelihood
#' ratio with an asymptotic chi-square distribution.
#' @slot df A single integer for the degrees of freedom of the statistic.
#' @slot pval A single numeric for the \eqn{p}-value of the statistic.
#' @slot nobs A single integer for the number of observations.
#' @slot npar A single integer for the number of parameters.
#' @slot weights A numeric vector of the re-scaled weights used for the model
#' fitting.
#' @slot coefficients A numeric vector of the maximum empirical likelihood
#' estimates of the parameters.
#' @slot method A single character for the method dispatch in internal
#' functions.
#' @slot data A numeric matrix of the data for the model fitting.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases EL
#' @references Owen A (2001). \emph{Empirical Likelihood}. Chapman & Hall/CRC.
#' \doi{10.1201/9781420036152}.
#' @references Qin J, Lawless J (1994).
#' “Empirical Likelihood and General Estimating Equations.”
#' \emph{The Annals of Statistics}, 22(1), 300--325.
#' \doi{10.1214/aos/1176325370}.
#' @examples
#' showClass("EL")
setClass("EL",
slots = c(
optim = "list", logp = "numeric", logl = "numeric", loglr = "numeric",
statistic = "numeric", df = "integer", pval = "numeric", nobs = "integer",
npar = "integer", weights = "numeric", coefficients = "numeric",
method = "character", data = "ANY", control = "ControlEL"
)
)
#' \linkS4class{CEL} class
#'
#' S4 class for constrained empirical likelihood. It inherits from
#' \linkS4class{EL} class. Note that the `optim` slot has constrained
#' optimization results with respect to the parameters, not the Lagrange
#' multiplier.
#'
#' @details Let \eqn{l(\theta)} denote minus twice the empirical log-likelihood
#' ratio function. We consider a linear hypothesis of the form
#' \deqn{L\theta = r,} where the left-hand-side \eqn{L} is a \eqn{q} by
#' \eqn{p} matrix and the right-hand-side \eqn{r} is a \eqn{q}-dimensional
#' vector. Under some regularity conditions, \eqn{l(\theta)} converges in
#' distribution to \eqn{\chi^2_q} under the constraint of hypothesis, i.e.,
#' \deqn{\min_{\theta: L\theta = r} l(\theta) \to_d \chi^2_q .}
#'
#' Minimization of \eqn{l(\theta)} with respect to \eqn{\theta} is
#' computationally expensive since it implicitly involves the
#' evaluation step as described in \linkS4class{EL}. Further, depending on the
#' form of \eqn{g(X_i, \theta)} and the constraint, the optimization problem
#' can be nonconvex and have multiple local minima. For this reason, the
#' package \pkg{melt} only considers linear hypotheses and performs local
#' minimization of \eqn{l(\theta)} using projected gradient descent method.
#' With the orthogonal projection matrix \eqn{P} and a step size \eqn{\gamma},
#' the algorithm updates \eqn{\theta} as
#' \deqn{\theta^{(k + 1)} \leftarrow \theta^{(k)} -
#' \gamma P \nabla l(\theta^{(k)}),}
#' where \eqn{\nabla l(\theta^{(k)})} denotes the gradient of \eqn{l} at
#' \eqn{\theta^{(k)}}. The first order optimality condition is
#' \eqn{P \nabla l(\theta) = 0}, which is used as the stopping criterion.
#' @slot optim A list of the following optimization results:
#' * `par` A numeric vector of the solution to the constrained optimization
#' problem.
#' * `lambda` A numeric vector of the Lagrange multipliers of the dual
#' problem corresponding to `par`.
#' * `iterations` A single integer for the number of iterations performed.
#' * `convergence` A single logical for the convergence status.
#' * `cstr` A single logical for whether constrained EL optimization is
#' performed or not.
#' @slot logp A numeric vector of the log probabilities of the constrained
#' empirical likelihood.
#' @slot logl A single numeric of the constrained empirical log-likelihood.
#' @slot loglr A single numeric of the constrained empirical log-likelihood
#' ratio.
#' @slot statistic A single numeric of minus twice the constrained empirical
#' log-likelihood ratio with an asymptotic chi-square distribution.
#' @slot df A single integer for the degrees of freedom of the statistic.
#' @slot pval A single numeric for the \eqn{p}-value of the statistic.
#' @slot nobs A single integer for the number of observations.
#' @slot npar A single integer for the number of parameters.
#' @slot weights A numeric vector of the re-scaled weights used for the model
#' fitting.
#' @slot coefficients A numeric vector of the maximum empirical likelihood
#' estimates of the parameters.
#' @slot method A single character for the method dispatch in internal
#' functions.
#' @slot data A numeric matrix of the data for the model fitting.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases CEL
#' @references Adimari G, Guolo A (2010).
#' “A Note on the Asymptotic Behaviour of Empirical Likelihood Statistics.”
#' \emph{Statistical Methods & Applications}, 19(4), 463--476.
#' \doi{10.1007/s10260-010-0137-9}.
#' @references Qin J, Lawless J (1995).
#' “Estimating Equations, Empirical Likelihood and Constraints on Parameters.”
#' \emph{Canadian Journal of Statistics}, 23(2), 145--159.
#' \doi{10.2307/3315441}.
#' @examples
#' showClass("CEL")
setClass("CEL", contains = "EL")
setOldClass("terms")
#' \linkS4class{LM} class
#'
#' S4 class for linear models with empirical likelihood. It inherits from
#' \linkS4class{CEL} class.
#'
#' @details The overall test involves a constrained optimization problem. All
#' the parameters except for the intercept are constrained to zero. The
#' `optim` slot contains the results. When there is no intercept, all
#' parameters are set to zero, and the results need to be understood in terms
#' of \linkS4class{EL} class since no constrained optimization is involved.
#' Once the solution is found, the log probabilities (`logp`) and the
#' (constrained) empirical likelihood values (`logl`, `loglr`, `statistic`)
#' readily follow, along with the degrees of freedom (`df`) and the
#' \eqn{p}-value (`pval`). The significance tests for each parameter also
#' involve constrained optimization problems where only one parameter is
#' constrained to zero. The `sigTests` slot contains the results.
#' @slot sigTests A list of the following results of significance tests:
#' * `statistic` A numeric vector of minus twice the (constrained) empirical
#' log-likelihood ratios with asymptotic chi-square distributions.
#' * `iterations` An integer vector for the number of iterations performed for
#' each parameter.
#' * `convergence` A logical vector for the convergence status of each
#' parameter.
#' @slot call A matched call.
#' @slot terms A [`terms`] object used.
#' @slot misc A list of various outputs obtained from the model fitting process.
#' They are used in other generics and methods.
#' @slot optim A list of the following optimization results:
#' * `par` A numeric vector of the solution to the (constrained) optimization
#' problem.
#' * `lambda` A numeric vector of the Lagrange multipliers of the dual
#' problem corresponding to `par`.
#' * `iterations` A single integer for the number of iterations performed.
#' * `convergence` A single logical for the convergence status.
#' @slot logp A numeric vector of the log probabilities of the (constrained)
#' empirical likelihood.
#' @slot logl A single numeric of the (constrained) empirical log-likelihood.
#' @slot loglr A single numeric of the (constrained) empirical log-likelihood
#' ratio.
#' @slot statistic A single numeric of minus twice the (constrained) empirical
#' log-likelihood ratio with an asymptotic chi-square distribution.
#' @slot df A single integer for the degrees of freedom of the statistic.
#' @slot pval A single numeric for the \eqn{p}-value of the statistic.
#' @slot nobs A single integer for the number of observations.
#' @slot npar A single integer for the number of parameters.
#' @slot weights A numeric vector of the re-scaled weights used for the model
#' fitting.
#' @slot coefficients A numeric vector of the maximum empirical likelihood
#' estimates of the parameters.
#' @slot method A single character for the method dispatch in internal
#' functions.
#' @slot data A numeric matrix of the data for the model fitting.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases LM
#' @examples
#' showClass("LM")
setClass("LM",
slots = c(sigTests = "ANY", call = "call", terms = "terms", misc = "list"),
contains = "CEL"
)
setOldClass("family")
#' \linkS4class{GLM} class
#'
#' S4 class for generalized linear models. It inherits from \linkS4class{LM}
#' class.
#'
#' @details The overall test involves a constrained optimization problem. All
#' the parameters except for the intercept are constrained to zero. The
#' `optim` slot contains the results. When there is no intercept, all
#' parameters are set to zero, and the results need to be understood in terms
#' of \linkS4class{EL} class since no constrained optimization is involved.
#' Once the solution is found, the log probabilities (`logp`) and the
#' (constrained) empirical likelihood values (`logl`, `loglr`, `statistic`)
#' readily follow, along with the degrees of freedom (`df`) and the
#' \eqn{p}-value (`pval`). The significance tests for each parameter also
#' involve constrained optimization problems where only one parameter is
#' constrained to zero. The `sigTests` slot contains the results.
#' @slot family A [`family`] object used.
#' @slot dispersion A single numeric for the estimated dispersion parameter.
#' @slot sigTests A list of the following results of significance tests:
#' * `statistic` A numeric vector of minus twice the (constrained) empirical
#' log-likelihood ratios with asymptotic chi-square distributions.
#' * `iterations` An integer vector for the number of iterations performed for
#' each parameter.
#' * `convergence` A logical vector for the convergence status of each
#' parameter.
#' * `cstr` A single logical for whether constrained EL optimization is
#' performed or not.
#' @slot call A matched call.
#' @slot terms A [`terms`] object used.
#' @slot misc A list of various outputs obtained from the model fitting process.
#' They are used in other generics and methods.
#' @slot optim A list of the following optimization results:
#' * `par` A numeric vector of the solution to the (constrained) optimization
#' problem.
#' * `lambda` A numeric vector of the Lagrange multipliers of the dual
#' problem corresponding to `par`.
#' * `iterations` A single integer for the number of iterations performed.
#' * `convergence` A single logical for the convergence status.
#' @slot logp A numeric vector of the log probabilities of the (constrained)
#' empirical likelihood.
#' @slot logl A single numeric of the (constrained) empirical log-likelihood.
#' @slot loglr A single numeric of the (constrained) empirical log-likelihood
#' ratio.
#' @slot statistic A single numeric of minus twice the (constrained) empirical
#' log-likelihood ratio with an asymptotic chi-square distribution.
#' @slot df A single integer for the degrees of freedom of the statistic.
#' @slot pval A single numeric for the \eqn{p}-value of the statistic.
#' @slot nobs A single integer for the number of observations.
#' @slot npar A single integer for the number of parameters.
#' @slot weights A numeric vector of the re-scaled weights used for the model
#' fitting.
#' @slot coefficients A numeric vector of the maximum empirical likelihood
#' estimates of the parameters.
#' @slot method A single character for the method dispatch in internal
#' functions.
#' @slot data A numeric matrix of the data for the model fitting.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases GLM
#' @examples
#' showClass("GLM")
setClass("GLM",
slots = c(family = "family", dispersion = "numeric"),
contains = "LM"
)
#' \linkS4class{ConfregEL} class
#'
#' S4 class for confidence region. It inherits from `"matrix"`.
#'
#' @slot estimates A numeric vector of length two for the parameter estimates.
#' @slot level A single numeric for the confidence level required.
#' @slot cv A single numeric for the critical value for calibration of empirical
#' likelihood ratio statistic.
#' @slot pnames A character vector of length two for the name of parameters.
#' @aliases ConfregEL
#' @examples
#' showClass("ConfregEL")
setClass("ConfregEL",
slots = c(
estimates = "numeric", level = "numeric", cv = "numeric",
pnames = "character"
),
contains = "matrix"
)
#' \linkS4class{ELD} class
#'
#' S4 class for empirical likelihood displacement. It inherits from `"numeric"`.
#'
#' @aliases ELD
#' @examples
#' showClass("ELD")
setClass("ELD", contains = "numeric")
#' \linkS4class{ELMT} class
#'
#' S4 class for empirical likelihood multiple tests.
#'
#' @slot estimates A list of numeric vectors of the estimates of the linear
#' hypotheses.
#' @slot statistic A numeric vector of minus twice the (constrained) empirical
#' log-likelihood ratios with asymptotic chi-square distributions.
#' @slot df An integer vector of the marginal degrees of freedom of the
#' statistic.
#' @slot pval A numeric vector for the multiplicity adjusted \eqn{p}-values.
#' @slot cv A single numeric for the multiplicity adjusted critical value.
#' @slot rhs A numeric vector for the right-hand sides of the hypotheses.
#' @slot lhs A numeric matrix for the left-hand side of the hypotheses.
#' @slot alpha A single numeric for the overall significance level.
#' @slot calibrate A single character for the calibration method used.
#' @slot weights A numeric vector of the re-scaled weights used for the model
#' fitting.
#' @slot coefficients A numeric vector of the maximum empirical likelihood
#' estimates of the parameters.
#' @slot method A single character for the method dispatch in internal
#' functions.
#' @slot data A numeric matrix of the data for the model fitting.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases ELMT
#' @examples
#' showClass("ELMT")
setClass("ELMT",
slots = c(
estimates = "list", statistic = "numeric", df = "integer", pval = "numeric",
cv = "numeric", rhs = "numeric", lhs = "matrix", alpha = "numeric",
calibrate = "character", weights = "numeric", coefficients = "numeric",
method = "character", data = "ANY", control = "ControlEL"
)
)
#' \linkS4class{ELT} class
#'
#' S4 class for empirical likelihood test.
#'
#' @slot optim A list of the following optimization results:
#' * `par` A numeric vector of the solution to the (constrained) optimization
#' problem.
#' * `lambda` A numeric vector of the Lagrange multipliers of the dual
#' problem corresponding to `par`.
#' * `iterations` A single integer for the number of iterations performed.
#' * `convergence` A single logical for the convergence status.
#' * `cstr` A single logical for whether constrained EL optimization is
#' performed or not.
#' @slot logp A numeric vector of the log probabilities of the (constrained)
#' empirical likelihood.
#' @slot logl A single numeric of the (constrained) empirical log-likelihood.
#' @slot loglr A single numeric of the (constrained) empirical log-likelihood
#' ratio.
#' @slot statistic A single numeric of minus twice the (constrained) empirical
#' log-likelihood ratio with an asymptotic chi-square distribution.
#' @slot df A single integer for the chi-square degrees of freedom of the
#' statistic.
#' @slot pval A single numeric for the (calibrated) \eqn{p}-value of the
#' statistic.
#' @slot cv A single numeric for the critical value.
#' @slot rhs A numeric vector for the right-hand side of the hypothesis.
#' @slot lhs A numeric matrix for the left-hand side of the hypothesis.
#' @slot alpha A single numeric for the significance level.
#' @slot calibrate A single character for the calibration method used.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases ELT
#' @examples
#' showClass("ELT")
setClass("ELT",
slots = c(
optim = "list", logp = "numeric", logl = "numeric", loglr = "numeric",
statistic = "numeric", df = "integer", pval = "numeric", cv = "numeric",
rhs = "numeric", lhs = "matrix", alpha = "numeric", calibrate = "character",
control = "ControlEL"
)
)
#' \linkS4class{QGLM} class
#'
#' S4 class for generalized linear models with quasi-likelihood methods. It
#' inherits from \linkS4class{GLM} class.
#'
#' @details The overall test involves a constrained optimization problem. All
#' the parameters except for the intercept are constrained to zero. The
#' `optim` slot contains the results. When there is no intercept, all
#' parameters are set to zero, and the results need to be understood in terms
#' of \linkS4class{EL} class since no constrained optimization is involved.
#' Once the solution is found, the log probabilities (`logp`) and the
#' (constrained) empirical likelihood values (`logl`, `loglr`, `statistic`)
#' readily follow, along with the degrees of freedom (`df`) and the
#' \eqn{p}-value (`pval`). The significance tests for each parameter also
#' involve constrained optimization problems where only one parameter is
#' constrained to zero. The `sigTests` slot contains the results.
#' @slot family A [`family`] object used.
#' @slot dispersion A single numeric for the estimated dispersion parameter.
#' @slot sigTests A list of the following results of significance tests:
#' * `statistic` A numeric vector of minus twice the (constrained) empirical
#' log-likelihood ratios with asymptotic chi-square distributions.
#' * `iterations` An integer vector for the number of iterations performed for
#' each parameter.
#' * `convergence` A logical vector for the convergence status of each
#' parameter.
#' * `cstr` A single logical for whether constrained EL optimization is
#' performed or not.
#' @slot call A matched call.
#' @slot terms A [`terms`] object used.
#' @slot misc A list of various outputs obtained from the model fitting process.
#' They are used in other generics and methods.
#' @slot optim A list of the following optimization results:
#' * `par` A numeric vector of the solution to the (constrained) optimization
#' problem.
#' * `lambda` A numeric vector of the Lagrange multipliers of the dual
#' problem corresponding to `par`.
#' * `iterations` A single integer for the number of iterations performed.
#' * `convergence` A single logical for the convergence status.
#' @slot logp A numeric vector of the log probabilities of the (constrained)
#' empirical likelihood.
#' @slot logl A single numeric of the (constrained) empirical log-likelihood.
#' @slot loglr A single numeric of the (constrained) empirical log-likelihood
#' ratio.
#' @slot statistic A single numeric of minus twice the (constrained) empirical
#' log-likelihood ratio with an asymptotic chi-square distribution.
#' @slot df A single integer for the degrees of freedom of the statistic.
#' @slot pval A single numeric for the \eqn{p}-value of the statistic.
#' @slot nobs A single integer for the number of observations.
#' @slot npar A single integer for the number of parameters.
#' @slot weights A numeric vector of the re-scaled weights used for the model
#' fitting.
#' @slot coefficients A numeric vector of the maximum empirical likelihood
#' estimates of the parameters.
#' @slot method A single character for the method dispatch in internal
#' functions.
#' @slot data A numeric matrix of the data for the model fitting.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases QGLM
#' @examples
#' showClass("QGLM")
setClass("QGLM", contains = "GLM")
#' \linkS4class{SD} class
#'
#' S4 class for standard deviation. It inherits from \linkS4class{EL} class.
#'
#' @slot optim A list of the following optimization results:
#' * `par` A numeric vector of the specified parameters.
#' * `lambda` A numeric vector of the Lagrange multipliers of the dual
#' problem corresponding to `par`.
#' * `iterations` A single integer for the number of iterations performed.
#' * `convergence` A single logical for the convergence status.
#' * `cstr` A single logical for whether constrained EL optimization is
#' performed or not.
#' @slot logp A numeric vector of the log probabilities of the empirical
#' likelihood.
#' @slot logl A single numeric of the empirical log-likelihood.
#' @slot loglr A single numeric of the empirical log-likelihood ratio.
#' @slot statistic A single numeric of minus twice the empirical log-likelihood
#' ratio with an asymptotic chi-square distribution.
#' @slot df A single integer for the degrees of freedom of the statistic.
#' @slot pval A single numeric for the \eqn{p}-value of the statistic.
#' @slot nobs A single integer for the number of observations.
#' @slot npar A single integer for the number of parameters.
#' @slot weights A numeric vector of the re-scaled weights used for the model
#' fitting.
#' @slot coefficients A numeric vector of the maximum empirical likelihood
#' estimates of the parameters.
#' @slot method A single character for the method dispatch in internal
#' functions.
#' @slot data A numeric matrix of the data for the model fitting.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases SD
#' @examples
#' showClass("SD")
setClass("SD", contains = "EL")
#' \linkS4class{SummaryEL} class
#'
#' S4 class for a summary of \linkS4class{EL} objects.
#'
#' @slot optim A list of the following optimization results:
#' * `par` A numeric vector of the specified parameters.
#' * `lambda` A numeric vector of the Lagrange multipliers of the dual
#' problem corresponding to `par`.
#' * `iterations` A single integer for the number of iterations performed.
#' * `convergence` A single logical for the convergence status.
#' * `cstr` A single logical for whether constrained EL optimization is
#' performed or not.
#' @slot logl A single numeric of the empirical log-likelihood.
#' @slot loglr A single numeric of the empirical log-likelihood ratio.
#' @slot statistic A single numeric of minus twice the empirical log-likelihood
#' ratio with an asymptotic chi-square distribution.
#' @slot df A single integer for the degrees of freedom of the statistic.
#' @slot pval A single numeric for the \eqn{p}-value of the statistic.
#' @slot nobs A single integer for the number of observations.
#' @slot npar A single integer for the number of parameters.
#' @slot weighted A single logical for whether the data are weighted or not.
#' @slot coefficients A numeric vector of the maximum empirical likelihood
#' estimates of the parameters.
#' @slot method A single character for the method dispatch in internal
#' functions.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases SummaryEL
#' @examples
#' showClass("SummaryEL")
setClass("SummaryEL", slots = c(
optim = "list", logl = "numeric", loglr = "numeric", statistic = "numeric",
df = "integer", pval = "numeric", nobs = "integer", npar = "integer",
weighted = "logical", coefficients = "numeric", method = "character",
control = "ControlEL"
))
#' \linkS4class{SummaryELMT} class
#'
#' S4 class for a summary of \linkS4class{ELMT} objects.
#'
#' @slot aliased A named logical vector showing if the original coefficients are
#' aliased.
#' @aliases SummaryELMT
#' @examples
#' showClass("SummaryELMT")
setClass("SummaryELMT", slots = c(
estimates = "list", statistic = "numeric", df = "integer", pval = "numeric",
cv = "numeric", rhs = "numeric", lhs = "matrix", alpha = "numeric",
calibrate = "character"
))
#' \linkS4class{SummaryELT} class
#'
#' S4 class for a summary of \linkS4class{ELT} objects.
#'
#' @slot optim A list of the following optimization results:
#' * `par` A numeric vector of the solution to the (constrained) optimization
#' problem.
#' * `lambda` A numeric vector of the Lagrange multipliers of the dual
#' problem corresponding to `par`.
#' * `iterations` A single integer for the number of iterations performed.
#' * `convergence` A single logical for the convergence status.
#' * `cstr` A single logical for whether constrained EL optimization is
#' performed or not.
#' @slot logl A single numeric of the (constrained) empirical log-likelihood.
#' @slot loglr A single numeric of the (constrained) empirical log-likelihood
#' ratio.
#' @slot statistic A single numeric of minus twice the (constrained) empirical
#' log-likelihood ratio with an asymptotic chi-square distribution.
#' @slot df A single integer for the chi-square degrees of freedom of the
#' statistic.
#' @slot pval A single numeric for the (calibrated) \eqn{p}-value of the
#' statistic.
#' @slot cv A single numeric for the critical value.
#' @slot rhs A numeric vector for the right-hand side of the hypothesis.
#' @slot lhs A numeric matrix for the left-hand side of the hypothesis.
#' @slot alpha A single numeric for the significance level.
#' @slot calibrate A single character for the calibration method used.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases SummaryELT
#' @examples
#' showClass("SummaryELT")
setClass("SummaryELT", slots = c(
optim = "list", logl = "numeric", loglr = "numeric", statistic = "numeric",
df = "integer", pval = "numeric", cv = "numeric", rhs = "numeric",
lhs = "matrix", alpha = "numeric", calibrate = "character",
control = "ControlEL"
))
#' \linkS4class{SummaryLM} class
#'
#' S4 class for a summary of \linkS4class{LM} objects.
#'
#' @slot coefficients A numeric matrix of the results of significance tests.
#' @slot intercept A single logical for whether the given model has an intercept
#' term or not.
#' @slot na.action Information returned by [`model.frame`] on the special
#' handling of `NA`s.
#' @slot call A matched call.
#' @slot terms A [`terms`] object used.
#' @slot aliased A named logical vector showing if the original coefficients are
#' aliased.
#' @slot optim A list of the following optimization results:
#' * `par` A numeric vector of the solution to the (constrained) optimization
#' problem.
#' * `lambda` A numeric vector of the Lagrange multipliers of the dual
#' problem corresponding to `par`.
#' * `iterations` A single integer for the number of iterations performed.
#' * `convergence` A single logical for the convergence status.
#' * `cstr` A single logical for whether constrained EL optimization is
#' performed or not.
#' @slot logl A single numeric of the empirical log-likelihood.
#' @slot loglr A single numeric of the empirical log-likelihood ratio.
#' @slot statistic A single numeric of minus twice the (constrained) empirical
#' log-likelihood ratio for the overall test.
#' @slot df A single integer for the degrees of freedom of the statistic.
#' @slot pval A single numeric for the \eqn{p}-value of the statistic.
#' @slot nobs A single integer for the number of observations.
#' @slot npar A single integer for the number of parameters.
#' @slot weighted A single logical for whether the data are weighted or not.
#' @slot method A single character for the method dispatch in internal
#' functions.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases SummaryLM
#' @examples
#' showClass("SummaryLM")
setClass("SummaryLM", slots = c(
coefficients = "matrix", intercept = "logical", na.action = "ANY",
call = "call", terms = "terms", aliased = "logical", optim = "list",
logl = "numeric", loglr = "numeric", statistic = "numeric", df = "integer",
pval = "numeric", nobs = "integer", npar = "integer", weighted = "logical",
method = "character", control = "ControlEL"
))
#' \linkS4class{SummaryGLM} class
#'
#' S4 class for a summary of \linkS4class{GLM} objects. It inherits from
#' \linkS4class{SummaryLM} class.
#'
#' @slot family A [`family`] object used.
#' @slot dispersion A single numeric for the estimated dispersion parameter.
#' @slot coefficients A numeric matrix of the results of significance tests.
#' @slot intercept A single logical for whether the given model has an intercept
#' term or not.
#' @slot na.action Information returned by [`model.frame`] on the special
#' handling of `NA`s.
#' @slot call A matched call.
#' @slot terms A [`terms`] object used.
#' @slot aliased A named logical vector showing if the original coefficients are
#' aliased.
#' @slot optim A list of the following optimization results:
#' * `par` A numeric vector of the solution to the (constrained) optimization
#' problem.
#' * `lambda` A numeric vector of the Lagrange multipliers of the dual
#' problem corresponding to `par`.
#' * `iterations` A single integer for the number of iterations performed.
#' * `convergence` A single logical for the convergence status.
#' * `cstr` A single logical for whether constrained EL optimization is
#' performed or not.
#' @slot logl A single numeric of the empirical log-likelihood.
#' @slot loglr A single numeric of the empirical log-likelihood ratio.
#' @slot statistic A single numeric of minus twice the (constrained) empirical
#' log-likelihood ratio for the overall test.
#' @slot df A single integer for the degrees of freedom of the statistic.
#' @slot pval A single numeric for the \eqn{p}-value of the statistic.
#' @slot nobs A single integer for the number of observations.
#' @slot npar A single integer for the number of parameters.
#' @slot weighted A single logical for whether the data are weighted or not.
#' @slot method A single character for the method dispatch in internal
#' functions.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases SummaryGLM
#' @examples
#' showClass("SummaryGLM")
setClass("SummaryGLM",
slots = c(family = "family", dispersion = "numeric"), contains = "SummaryLM"
)
#' \linkS4class{SummaryQGLM} class
#'
#' S4 class for a summary of \linkS4class{QGLM} objects. It inherits from
#' \linkS4class{SummaryGLM} class.
#'
#' @slot family A [`family`] object used.
#' @slot dispersion A single numeric for the estimated dispersion parameter.
#' @slot coefficients A numeric matrix of the results of significance tests.
#' @slot intercept A single logical for whether the given model has an intercept
#' term or not.
#' @slot na.action Information returned by [`model.frame`] on the special
#' handling of `NA`s.
#' @slot call A matched call.
#' @slot terms A [`terms`] object used.
#' @slot aliased A named logical vector showing if the original coefficients are
#' aliased.
#' @slot optim A list of the following optimization results:
#' * `par` A numeric vector of the solution to the (constrained) optimization
#' problem.
#' * `lambda` A numeric vector of the Lagrange multipliers of the dual
#' problem corresponding to `par`.
#' * `iterations` A single integer for the number of iterations performed.
#' * `convergence` A single logical for the convergence status.
#' * `cstr` A single logical for whether constrained EL optimization is
#' performed or not.
#' @slot logl A single numeric of the empirical log-likelihood.
#' @slot loglr A single numeric of the empirical log-likelihood ratio.
#' @slot statistic A single numeric of minus twice the (constrained) empirical
#' log-likelihood ratio for the overall test.
#' @slot df A single integer for the degrees of freedom of the statistic.
#' @slot pval A single numeric for the \eqn{p}-value of the statistic.
#' @slot nobs A single integer for the number of observations.
#' @slot npar A single integer for the number of parameters.
#' @slot weighted A single logical for whether the data are weighted or not.
#' @slot method A single character for the method dispatch in internal
#' functions.
#' @slot control An object of class \linkS4class{ControlEL} constructed by
#' [el_control()].
#' @aliases SummaryQGLM
#' @examples
#' showClass("SummaryQGLM")
setClass("SummaryQGLM", contains = "SummaryGLM")
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