R/BMTfit.mme.R

In BMT: The BMT Distribution

#'@title Moment Matching Fit of the BMT Distribution to Non-censored Data.
#'  
#'@description Fit of the BMT distribution to non-censored data by moment 
#'  matching (mme).
#'  
#'@rdname BMTfit.mme
#'@name BMTfit.mme
#'  
#'@details This function is not intended to be called directly but is internally
#'  called in \code{\link{BMTfit}} when used with the moment matching method.
#'  
#'  \code{BMTfit.mme} is based on the function \code{\link{mmedist}} but it 
#'  focuses on the moment matching parameter estimation for the BMT distribution
#'  (see \code{\link{BMT}} for details about the BMT distribution and 
#'  \code{\link{mmedist}} for details about moment matching fit of univariate 
#'  distributions).
#'  
#'  For each parameter of the BMT distribution we choose a moment or measure. 
#'  Mean for \code{p1}, standard deviation for \code{p2}, Pearson_s skewness for
#'  \code{p3}, and Pearson's kurtosis for \code{p4}.
#'  
#'@param data A numeric vector with the observed values for non-censored data.
#'@param start A named list giving the initial values of parameters of the BMT 
#'  distribution or a function of data computing initial values and returning a 
#'  named list. (see the 'details' section of 
#'  \code{\link{mledist}}).
#'@param fix.arg An optional named list giving the values of fixed parameters of
#'  the BMT distribution or a function of data computing (fixed) parameter 
#'  values and returning a named list. Parameters with fixed value are thus NOT 
#'  estimated. (see the 'details' section of 
#'  \code{\link{mledist}}).
#'@param type.p.3.4 Type of parametrization asociated to p3 and p4. "t w" means 
#'  tails weights parametrization (default) and "a-s" means asymmetry-steepness 
#'  parametrization.
#'@param type.p.1.2 Type of parametrization asociated to p1 and p2. "c-d" means 
#'  domain parametrization (default) and "l-s" means location-scale 
#'  parametrization.
#'@param optim.method \code{"default"} (see the 'details' section of 
#'  \code{\link{mledist}}) or optimization method to pass to 
#'  \code{\link{optim}}.
#'@param custom.optim A function carrying the optimization (see the 'details' 
#'  section of \code{\link{mledist}}).
#'@param silent A logical to remove or show warnings when bootstraping.
#'@param \dots Further arguments to be passed to generic functions or to the 
#'  function \code{"mmedist"}. See \code{\link{mmedist}} for details.
#'  
#'@return \code{BMTfit.mme} returns a list with following components,
#'  
#'  \item{estimate}{ the parameter estimates.}
#'  
#'  \item{convergence}{ an integer code for the convergence of 
#'  \code{\link{optim}}/\code{\link{constrOptim}} defined as below or defined by
#'  the user in the user-supplied optimization function.
#'  
#'  \code{0} indicates successful convergence.
#'  
#'  \code{1} indicates that the iteration limit of \code{\link{optim}} has been 
#'  reached.
#'  
#'  \code{10} indicates degeneracy of the Nealder-Mead simplex.
#'  
#'  \code{100} indicates that \code{\link{optim}} encountered an internal error.
#'  }
#'  
#'  \item{value}{the value of the corresponding objective function of the 
#'  estimation method at the estimate.}
#'  
#'  \item{hessian}{a symmetric matrix computed by \code{\link{optim}} as an 
#'  estimate of the Hessian at the solution found or computed in the 
#'  user-supplied optimization function.}
#'  
#'  \item{loglik}{the log-likelihood value.}
#'  
#'  \item{order}{the vector of moment(s) matched: mean (1), standard deviation 
#'  (2), Pearson's skewness (3), Pearson's kurtosis (4).}
#'  
#'  \item{memp}{the empirical moment function. }
#'  
#'  \item{optim.function}{the name of the optimization function used for maximum
#'  product of spacing.}
#'  
#'  \item{optim.method}{when \code{\link{optim}} is used, the name of the 
#'  algorithm used, \code{NULL} otherwise.}
#'  
#'  \item{fix.arg}{the named list giving the values of parameters of the named 
#'  distribution that must kept fixed rather than estimated or \code{NULL} if 
#'  there are no such parameters. }
#'  
#'  \item{fix.arg.fun}{the function used to set the value of \code{fix.arg} or 
#'  \code{NULL}.}
#'  
#'  \item{weights}{the vector of weigths used in the estimation process or 
#'  \code{NULL}.}
#'  
#'  \item{counts}{A two-element integer vector giving the number of calls to the
#'  log-likelihood function and its gradient respectively. This excludes those 
#'  calls needed to compute the Hessian, if requested, and any calls to 
#'  log-likelihood function to compute a finite-difference approximation to the 
#'  gradient. \code{counts} is returned by \code{\link{optim}} or the 
#'  user-supplied function or set to \code{NULL}.}
#'  
#'  \item{optim.message}{A character string giving any additional information 
#'  returned by the optimizer, or \code{NULL}. To understand exactly the 
#'  message, see the source code.}
#'  
#'@references Torres-Jimenez, C. J. (2017, September), \emph{Comparison of estimation
#'  methods for the BMT distribution}. ArXiv e-prints.
#'  
#'  Torres-Jimenez, C. J. (2018), \emph{The BMT Item Response Theory model: A 
#'  new skewed distribution family with bounded domain and an IRT model based on
#'  it}, PhD thesis, Doctorado en ciencias - Estadistica, Universidad Nacional 
#'  de Colombia, Sede Bogota.
#'  
#'@seealso See \code{\link{BMT}} for the BMT density, distribution, quantile 
#'  function and random deviates. See \code{\link{BMTfit.qme}}, 
#'  \code{\link{BMTfit.mle}}, \code{\link{BMTfit.mge}}, 
#'  \code{\link{BMTfit.mpse}} and \code{\link{BMTfit.mqde}} for other estimation
#'  methods. See \code{\link{optim}} and \code{\link{constrOptim}} for 
#'  optimization routines. See \code{\link{BMTfit}} and \code{\link{fitdist}} 
#'  for functions that return an objetc of class \code{"fitdist"}.
#'  
#'@author Camilo Jose Torres-Jimenez [aut,cre] \email{cjtorresj@unal.edu.co}
#'  
#'@source Based on the function \code{\link{mmedist}} of the R package: 
#'  \code{\link{fitdistrplus}}
#'  
#'  Delignette-Muller ML and Dutang C (2015), \emph{fitdistrplus: An R Package 
#'  for Fitting Distributions}. Journal of Statistical Software, 64(4), 1-34.
#'  
#' @examples
#' # (1) basic fit by moment matching estimation
#' set.seed(1234)
#' x1 <- rBMT(n=100, p3=0.25, p4=0.75)
#' BMTfit.mme(x1)
#' 
#' # (3) how to change the optimisation method?
#' BMTfit.mme(x1, optim.method="L-BFGS-B") 
#' BMTfit.mme(x1, custom.optim="nlminb")
#' 
#' # (4) estimation of the tails weights parameters of the BMT 
#' # distribution with domain fixed at [0,1]
#' BMTfit.mme(x1, start=list(p3=0.5, p4=0.5), fix.arg=list(p1=0, p2=1))
#' 
#' # (5) estimation of the asymmetry-steepness parameters of the BMT 
#' # distribution with domain fixed at [0,1]
#' BMTfit.mme(x1, start=list(p3=0, p4=0.5), type.p.3.4 = "a-s", 
#'            fix.arg=list(p1=0, p2=1))
#' 
#'@keywords distribution

#####################
#' @rdname BMTfit.mme
#' @export BMTfit.mme
BMTfit.mme <- function(data,
                       start = list(p3 = 0.5, p4 = 0.5, p1 = min(data) - 0.1, p2 = max(data) + 0.1),
                       fix.arg = NULL, type.p.3.4 = "t w", type.p.1.2 = "c-d", 
                       optim.method = "Nelder-Mead", custom.optim = NULL, silent = TRUE, ...){
  # Control data
  if (!(is.vector(data) & is.numeric(data) & length(data) > 1))
    stop("data must be a numeric vector of length greater than 1")
  # Further arguments to be passed
  my3dots <- list(...)    
  if (length(my3dots) == 0) 
    my3dots <- NULL
  # Control weights
  if(!is.null(my3dots$weights))
    stop("Estimation with weights is not considered yet")
  # Control type.p.3.4. It allows partial match.
  TYPE.P.3.4 <- c("t w", "a-s") # tail weights or asymmetry-steepness
  int.type.p.3.4 <- pmatch(type.p.3.4, TYPE.P.3.4)
  if (is.na(int.type.p.3.4))
    stop("invalid type of parametrization for parameters 3 and 4")
  if (int.type.p.3.4 == -1)
    stop("ambiguous type of parametrization for parameters 3 and 4")
  # Control type.p.1.2. It allows partial match.
  TYPE.P.1.2 <- c("c-d", "l-s") # domain or location-scale
  int.type.p.1.2 <- pmatch(type.p.1.2, TYPE.P.1.2)
  if (is.na(int.type.p.1.2))
    stop("invalid type of parametrization for parameters 1 and 2")
  if (int.type.p.1.2 == -1)
    stop("ambiguous type of parametrization for parameters 1 and 2")
  # Type of parametrizations are fixed parameters
  fix.arg$type.p.3.4 <- type.p.3.4
  fix.arg$type.p.1.2 <- type.p.1.2
  # Establish box constraints according to parameters in start
  stnames <- names(start)
  m <- length(stnames)
  # Initialize all box constraints: (-Inf, Inf)
  lower <- rep(-Inf, m)
  upper <- rep(Inf, m)
  # domain parametrization
  if (int.type.p.1.2 == 1) {
    # c has to be inside (-Inf, min(data)) 
    upper[stnames == "p1"] <- min(data) - .epsilon
    # d has to be inside (max(data), Inf)
    lower[stnames == "p2"] <- max(data) + .epsilon
  }
  # location-scale parametrization
  else{
    # sigma has to be inside (0, Inf) 
    lower[stnames == "p2"] <- 0 + .epsilon
  }
  # tail weights parametrization
  if (int.type.p.3.4 == 1) {
    # Both tail weights have to be inside (0,1) 
    lower[stnames == "p3" | stnames == "p4"] <- 0 + .epsilon
    upper[stnames == "p3" | stnames == "p4"] <- 1 - .epsilon
  }
  # asymmetry-steepness parametrization
  else{
    # asymmetric has to be inside (-1, 1) 
    # steepness has to be inside (0, 1)
    lower[stnames == "p3"] <- -1 + .epsilon
    lower[stnames == "p4"] <- 0 + .epsilon
    upper[stnames == "p3" | stnames == "p4"] <- 1 - .epsilon
  }
  # nlminb optimization method
  if(!is.null(custom.optim))
    if(custom.optim=="nlminb")
      custom.optim <- .m.nlminb
  # order of moments to be used (p1-mean, p2-sd, p3-skew, p4-kurt)
  order <- as.integer(substr(stnames,2,2))
  # memp unbiased sample moments 
  memp <- function(x, order){
    n <- length(x)
    s.mean <- mean(x)
    s.sd <- sd(x)
    res <- switch(order, 
                  s.mean, 
                  s.sd, 
                  n / ((n-1) * (n-2)) * sum((x - s.mean)^3) / s.sd^3,
                  (n-1) / ((n-2) * (n-3)) * ((n+1) * n / (n-1)^2 * sum((x - s.mean)^4) / s.sd^4 - 3 * (n-1)) + 3)
    return(res)
  }
  # mmedist function of fitdistplus
  mme <- fitdistrplus::mmedist(data, "BMT", order = order, memp = memp, start = start, fix.arg = fix.arg, 
                 optim.method = optim.method, lower = lower, upper = upper,
                 custom.optim = custom.optim, silent = silent, ...)
  # Estimation with location-scale parameterization might allow data outside the estimated domain
  par <- append(mme$estimate,fix.arg)
  if (int.type.p.1.2 == 2)
    par <- BMTchangepars(par$p3, par$p4, par$type.p.3.4, par$p1, par$p2, par$type.p.1.2)
  n.obs <- sum(data < par$p1 | data > par$p2)
  if(n.obs > 0){
    text <- paste("The resultant estimated domain is [",round(par$p1,4),",",round(par$p2,4),
                  "] and there are ",n.obs," observations out of it.",sep="")
    warning(text)
  }
  return(mme)
}