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#' WP method background function WP_unknown
#'
#' This is a background/internal function called by \code{WP}. It computes the maximum likelihood estimator
#' of R0 assuming that the serial distribution is unknown but comes from a discretized gamma distribution.
#' The function then implements a simple grid search algorithm to obtain the maximum likelihood estimator
#' of R0 as well as the gamma parameters.
#'
#' @param NT Vector of case counts.
#' @param B Length of grid for shape and scale (grid search parameter).
#' @param shape.max Maximum shape value (grid \code{search} parameter).
#' @param scale.max Maximum scale value (grid \code{search} parameter).
#' @param tol cutoff value for cumulative distribution function of the serial distribution (defaults to 0.999).
#'
#' @return The function returns \code{Rhat}, the maximum likelihood estimator of R0, as well as the maximum
#' likelihood estimator of the discretized serial distribution given by \code{p} (the probability mass
#' function) and \code{range.max} (the distribution has support on the integers one to \code{range.max}).
#' The function also returns \code{resLL} (all values of the log-likelihood) at \code{shape} (grid for
#' shape parameter) and at \code{scale} (grid for scale parameter), as well as \code{resR0} (the full
#' vector of maximum likelihood estimators), \code{JJ} (the locations for the likelihood for these), and
#' \code{J0} (the location for the maximum likelihood estimator \code{Rhat}). If \code{JJ} and \code{J0}
#' are not the same, this means that the maximum likelihood estimator is not unique.
#'
#' @keywords internal
WP_unknown <- function(NT, B=100, shape.max=10, scale.max=10, tol=0.999) {
shape <- seq(0, shape.max, length.out=B+1)
scale <- seq(0, scale.max, length.out=B+1)
shape <- shape[-1]
scale <- scale[-1]
resLL <- matrix(0,B,B)
resR0 <- matrix(0,B,B)
for (i in 1:B) {
for (j in 1:B) {
range.max <- ceiling(qgamma(tol, shape=shape[i], scale=scale[j]))
p <- diff(pgamma(0:range.max, shape=shape[i], scale=scale[j]))
p <- p / sum(p)
mle <- WP_known(NT, p)
resLL[i,j] <- computeLL(p, NT, mle)
resR0[i,j] <- mle
}
}
J0 <- which.max(resLL)
R0hat <- resR0[J0]
JJ <- which(resLL == resLL[J0], arr.ind=TRUE)
range.max <- ceiling(qgamma(tol, shape=shape[JJ[1]], scale=scale[JJ[2]]))
p <- diff(pgamma(0:range.max, shape=shape[JJ[1]], scale=scale[JJ[2]]))
p <- p / sum(p)
return(list(Rhat=R0hat, J0=J0, ll=resLL, Rs=resR0, scale=scale, shape=shape, JJ=JJ, p=p, range.max=range.max))
}
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