#' Modified Moment bound
#'
#' Calculates the Modified Moment confidence bound for the maximum error in an
#' audit population according to the methodology described by Dworin &
#' Grimlund (1986).
#'
#' @usage modified.moment(bookValues, auditValues, pop.type = "inventory", confidence = 0.95)
#'
#' @param bookValues A vector of book values from sample.
#' @param auditValues A vector of corresponding audit values from the sample.
#' @param pop.type A character defining the type of population audited.
#' \emph{inventory} for inventory populations. \emph{accounts} for populations
#' of accounts receivable.
#' @param confidence The amount of confidence desired from the bound
#' (on a scale from 0 to 1), defaults to 95\% confidence.
#'
#' @return An estimate of the mean taint per dollar unit in the population.
#'
#' @section Details: EMPTY FOR NOW
#'
#' @author Koen Derks, \email{k.derks@nyenrode.nl}
#'
#' @seealso
#'
#' @references Dworin, L., & Grimlund, R. A. (1986). Dollar-unit sampling: A
#' comparison of the quasi-Bayesian and moment bounds. Accounting Review, 36-57.
#'
#' @examples
#' # Create an imaginary data set
#' bookValues <- rgamma(n = 2400, shape = 1, rate = 0.001)
#' error.rate <- 0.1
#' error <- sample(0:1, 2400, TRUE, c(1-error.rate, error.rate))
#' taint <- rchisq(n = 2400, df = 1) / 10
#' auditValues <- bookValues - (error * taint * bookValues)
#' frame <- data.frame( bookValues = round(bookValues,2),
#' auditValues = round(auditValues,2))
#' # Draw a sample
#' samp.probs <- frame$bookValues/sum(frame$bookValues)
#' sample.no <- sample(1:nrow(frame), 100, FALSE, samp.probs)
#' sample <- frame[sample.no, ]
#' # Calculate bound
#' modified.moment(bookValues = sample$bookValues,
#' auditValues = sample$auditValues,
#' pop.type = "inventory",
#' confidence = 0.95)
#'
#' @keywords bound
#'
#' @export
modified.moment <- function(bookValues,
auditValues,
pop.type = "inventory",
confidence = 0.95){
if(!(length(bookValues) == length(auditValues)))
stop("bookValues must be the same length as auditValues")
n <- length(auditValues)
t <- bookValues - auditValues
z <- t / bookValues
z <- subset(z, z != 0)
M <- length(z)
if(pop.type == "inventory" & length(z) > 0){
zstar <- 0.81 * (1-0.667 * tanh(10*abs(mean(z))))
} else if(pop.type == "inventory" & length(z) == 0){
zstar <- 0.81 * (1-0.667 * tanh(10*0))
}
if(pop.type == "accounts" & length(z) > 0){
zstar <- 0.81 * (1-0.667 * tanh(10 * mean(z))) * (1+0.667 * tanh(M/10))
} else if(pop.type == "accounts" & length(z) == 0){
zstar <- 0.81 * (1-0.667 * tanh(10 * 0)) * (1+0.667 * tanh(0/10))
}
ncm1_z <- (zstar^1 + sum(z^1)) / (M + 1)
ncm2_z <- (zstar^2 + sum(z^2)) / (M + 1)
ncm3_z <- (zstar^3 + sum(z^3)) / (M + 1)
ncm1_e <- (M+1)/(n+2)
ncm2_e <- ((M+2)/(n+3)) * ncm1_e
ncm3_e <- ((M+3)/(n+4)) * ncm2_e
ncm1_t <- ncm1_e * ncm1_z
ncm2_t <- (ncm1_e * ncm2_z + ((n - 1) * ncm2_e * ncm1_z^2)) / n
ncm3_t <- ((ncm1_e * ncm3_z + (3 * (n - 1) * ncm2_e * ncm1_z *
ncm2_z)) / n^2) + (((n - 1) * (n - 2) * ncm3_e *
ncm1_z^3)/(n^2))
cm2_t <- ncm2_t - ncm1_t^2
cm3_t <- ncm3_t - (3 * ncm1_t * ncm2_t) + (2 * ncm1_t^3)
A <- (4 * cm2_t^3)/(cm3_t^2)
B <- cm3_t / (2 * cm2_t)
G <- ncm1_t - ((2 * cm2_t^2)/cm3_t)
bound <- G + (A * B * (1 + (qnorm(confidence, mean = 0, sd = 1)/
sqrt(9*A)) - (1/(9*A)))^3)
return(bound)
}
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