R/rlm.test.R

In lawstat: Tools for Biostatistics, Public Policy, and Law

#' Robust L1 Moment-Based (RLM) Goodness-of-Fit Test for the Laplace Distribution
#' 
#' Robust test for the Laplace distribution. Two options for calculating critical 
#' values, namely, approximated with Chi-square distribution and empirical, 
#' are available.
#' 
#' @details The test is based on a joint statistic using skewness and kurtosis 
#' coefficients. In particular, RLM uses the Average Absolute Deviation from the Median 
#' (MAAD), a robust estimate of standard deviation. See 
#' \insertCite{Gel_2010;textual}{lawstat}.
#'
#'
#' @inheritParams rjb.test 
#'
#'
#' @return A list of class \code{"htest"} with the following components:
#' \item{statistic}{the value of the test statistic.}
#' \item{parameter}{the degrees of freedom.}
#' \item{p.value}{the \eqn{p}-value of the test.}
#' \item{method}{type of test was performed.}
#' \item{data.name}{a character string giving the name of the data.}
#' 
#' @references
#' \insertAllCited{}
#' 
#' @seealso \code{\link{sj.test}}, \code{\link{rjb.test}}, \code{\link{rqq}}, 
#' \code{\link[tseries]{jarque.bera.test}}
#' 
#' @keywords distribution robust htest
#' 
#' @author Kimihiro Noguchi, W. Wallace Hui, Yulia R. Gel
#' 
#' @export
#' @examples
#' ## Laplace distributed data
#' x = rexp(100) - rexp(100)
#' rlm.test(x)
rlm.test <-
    function (x,
              crit.values = c("chisq.approximation", "empirical"),
              N = 0)
    {
        ### set the default option to be chi-square approximation ###
        
        crit.values = match.arg(crit.values)
        
        ### stop the function if it is not a vector ###
        
        if (NCOL(x) > 1) {
            stop("x is not a vector or univariate time series")
        }
        
        ### stop the function if the data has missing values ###
        
        if (any(is.na(x))) {
            stop("NAs in x")
        }
        
        ### stop the function if the number of Monte Carlo simulations is not provided ###
        
        if ((crit.values == "empirical") & (N == 0)) {
            stop(
                "number of Monte Carlo simulations N should be provided for the empirical critical values"
            )
        }
        
        ### initialize variables ###
        
        DNAME <- deparse(substitute(x))
        n <- length(x)
        m1 <- sum(x) / n
        m3 <- sum((x - m1) ^ 3) / n
        m4 <- sum((x - m1) ^ 4) / n
        
        ### construct the test statistic ###
        
        J <- sqrt(2) * mean(abs(x - median(x)))
        ek <- 6
        b1 <- (m3 / (J ^ 3)) ^ 2
        b2 <- (m4 / (J ^ 4) - ek) ^ 2
        vk <- 1200 / n
        vs <- 60 / n
        statistic <- b1 / vs + b2 / vk
        METHOD <- "Robust L1 moment-based goodness-of-fit test"
        
        ### calculate empirical critical values ###
        
        if (crit.values == "empirical")
        {
            METHOD <-
                paste(METHOD, "using empirical critical values with N =", N)
            
            ### create a vector "jb" to store statistics ###
            
            jb <- double(N)
            
            ### generate random Laplace variables to calculate critical values ###
            
            for (k in 1:N)
            {
                e <- rexp(n) - rexp(n)
                m1 <- sum(e) / n
                m3 <- sum((e - m1) ^ 3) / n
                m4 <- sum((e - m1) ^ 4) / n
                J <- sqrt(2) * mean(abs(e - median(e)))
                ek <- 6
                b1 <- (m3 / (J ^ 3)) ^ 2
                b2 <- (m4 / (J ^ 4) - ek) ^ 2
                vk <- 1200 / n
                vs <- 60 / n
                jb[k] <- b1 / vs + b2 / vk
            }
            
            ### sort the generated statistics ###
            
            y <- sort(jb)
            
            ### set the p-value to zero if the statistic is greater than maximum of generated statistics ###
            
            if (statistic >= max(y))
            {
                p.value = 0
            }
            
            ### set the p-value to one if the statistic is smaller than minimum of generated statistics ###
            
            else if (statistic <= min(y))
            {
                p.value = 1
            }
            
            ### calculate the p-value in the case the statistic is between min and max of generated statistics ###
            
            else
            {
                an <- which(y == max(y[I(y < statistic)]))
                bn <- which(y == min(y[I(y >= statistic)]))
                a <- max(y[I(y < statistic)])
                b <- min(y[I(y >= statistic)])
                pa <- (an - 1) / (N - 1)
                pb <- (bn - 1) / (N - 1)
                alpha <- (statistic - a) / (b - a)
                p.value = 1 - alpha * pb - (1 - alpha) * pa
            }
            
        }
        
        ### calculate the p-value using a Chi-squared approximation ###
        
        else
        {
            METHOD <-
                paste(METHOD,
                      "using a Chi-squared approximated critical values")
            p.value <- pchisq(statistic, df = 2, lower.tail = FALSE)
        }
        
        ### display output ###
        
        STATISTIC = statistic
        names(STATISTIC) <- paste("Chi-squared statistic")
        PARAMETER <- 2
        names(PARAMETER) <- "df"
        structure(
            list(
                statistic = STATISTIC,
                parameter = PARAMETER,
                p.value = p.value,
                method = METHOD,
                data.name = DNAME
            ),
            class = "htest"
        )
    }