#' Anderson-Darling Test of Goodness-of-Fit
#'
#' @description Performs the Anderson-Darling test of goodness-of-fit to a specified continuous univariate proba
#' bility distribution
#'
#' @usage ad.test(x, null = "punif", ..., nullname)
#'
#' @param x Numeric vector of data values.
#' @param null A function, or a character string giving the name of a function, to compute the cumulative
#' distribution function for the null distribution
#' @param ... Additional arguments for the cumulative distribution function.
#' @param nullname Optional character string describing the null distribution. The default is "uniform distribution".
#'
#' @details This command performs the Anderson-Darling test of goodness-of-fit to the distribution specified
#' by the argument null. It is assumed that the values in x are independent and identically distributed
#' random values, with some cumulative distribution function F. The null hypothesis is that F is the
#' function specified by the argument null, while the alternative hypothesis is that F is some other
#' function.
#'
#' @return An object of class "htest" representing the result of the hypothesis test
#'
#' @author Original C code by George Marsaglia and John Marsaglia. R interface by Adrian Baddeley
#'
#' @references
#'
#' Anderson, T.W. and Darling, D.A. (1952) Asymptotic theory of certain ’goodness-of-fit’ criteria
#' based on stochastic processes. Annals of Mathematical Statistics 23, 193–212.
#'
#' Anderson, T.W. and Darling, D.A. (1954) A test of goodness of fit. Journal of the American Statistical Association 49, 765–769.
#'
#' Marsaglia, G. and Marsaglia, J. (2004) Evaluating the Anderson-Darling Distribution. Journal of Statistical Software 9 (2), 1–5.
#' February 2004. http://www.jstatsoft.org/v09/i02
#'
#' @examples
#'
#' x <- rnorm(10, mean=2, sd=1)
#' ad.test(x, "pnorm", mean=2, sd=1)
#'
#' @export
ad.test <- function(x, null="punif", ..., nullname) {
xname <- deparse(substitute(x))
nulltext <- deparse(substitute(null))
if(is.character(null)) nulltext <- null
if(missing(nullname) || is.null(nullname)) {
reco <- recogniseCdf(nulltext)
nullname <- if(!is.null(reco)) reco else
paste("distribution", sQuote(nulltext))
}
stopifnot(is.numeric(x))
x <- as.vector(x)
n <- length(x)
F0 <- if(is.function(null)) null else
if(is.character(null)) get(null, mode="function") else
stop("Argument 'null' should be a function, or the name of a function")
U <- F0(x, ...)
if(any(U < 0 | U > 1))
stop("null distribution function returned values outside [0,1]")
U <- sort(U)
k <- seq_len(n)
## call Marsaglia C code
z <- .C("ADtestR",
x = as.double(U),
n = as.integer(n),
adstat = as.double(numeric(1)),
pvalue = as.double(numeric(1)),
PACKAGE = 'ECTools'
)
STATISTIC <- z$adstat
names(STATISTIC) <- "An"
PVAL <- z$pvalue
METHOD <- c("Anderson-Darling test of goodness-of-fit",
paste("Null hypothesis:", nullname))
extras <- list(...)
parnames <- intersect(names(extras), names(formals(F0)))
if(length(parnames) > 0) {
pars <- extras[parnames]
pard <- character(0)
for(i in seq_along(parnames))
pard[i] <- paste(parnames[i], "=", paste(pars[[i]], collapse=" "))
pard <- paste("with",
ngettext(length(pard), "parameter", "parameters"),
" ",
paste(pard, collapse=", "))
METHOD <- c(METHOD, pard)
}
out <- list(statistic = STATISTIC,
p.value = PVAL,
method = METHOD,
data.name = xname)
class(out) <- "htest"
return(out)
}
pAD <- function(q, n=Inf, lower.tail=TRUE, fast=TRUE) {
q <- as.numeric(q)
p <- rep(NA_real_, length(q))
if(any(ones <- is.infinite(q) & (q == Inf)))
p[ones] <- 1
if(any(zeroes <- (is.finite(q) & q <= 0) | (is.infinite(q) & (q == -Inf))))
p[zeroes] <- 0
ok <- is.finite(q) & (q > 0)
nok <- sum(ok)
if(nok > 0) {
if(is.finite(n)) {
z <- .C("ADprobN",
a = as.double(q[ok]),
na = as.integer(nok),
nsample = as.integer(n),
prob = as.double(numeric(nok)),
PACKAGE="ECTools")
p[ok] <- z$prob
} else if(fast) {
## fast version adinf()
z <- .C("ADprobApproxInf",
a = as.double(q[ok]),
na = as.integer(nok),
prob = as.double(numeric(nok)),
PACKAGE="ECTools")
p[ok] <- z$prob
} else {
## slow, accurate version ADinf()
z <- .C("ADprobExactInf",
a = as.double(q[ok]),
na = as.integer(nok),
prob = as.double(numeric(nok)),
PACKAGE="ECTools")
p[ok] <- z$prob
}
}
if(!lower.tail)
p <- 1 - p
return(p)
}
qAD <- local({
f <- function(x, N, P, Fast) {
pAD(x, N, fast=Fast) - P
}
qAD <- function(p, n=Inf, lower.tail=TRUE, fast=TRUE) {
## quantiles of null distribution of Anderson-Darling test statistic
stopifnot(all(p >= 0))
stopifnot(all(p <= 1))
if(!lower.tail) p <- 1-p
ans <- rep(NA_real_, length(p))
for(i in which(p >= 0 & p < 1))
ans[i] <- uniroot(f, c(0, 1), N=n, P=p[i], Fast=fast, extendInt="up")$root
return(ans)
}
qAD
})
recogniseCdf <- function(s="punif") {
if(!is.character(s) || length(s) != 1) return(NULL)
if(nchar(s) <= 1 || substr(s,1,1) != "p") return(NULL)
root <- substr(s, 2, nchar(s))
a <- switch(root,
beta = "beta",
binom = "binomial",
birthday = "birthday coincidence",
cauchy = "Cauchy",
chisq = "chi-squared",
exp = "exponential",
f = "F",
gamma = "Gamma",
geom = "geometric",
hyper = "hypergeometric",
lnorm = "log-normal",
logis = "logistic",
nbinom = "negative binomial",
norm = "Normal",
pois = "Poisson",
t = "Student's t",
tukey = "Tukey (Studentized range)",
unif = "uniform",
weibull = "Weibull",
NULL)
if(!is.null(a))
return(paste(a, "distribution"))
b <- switch(root,
AD = "Anderson-Darling",
CvM = "Cramer-von Mises",
wilcox = "Wilcoxon Rank Sum",
NULL)
if(!is.null(b))
return(paste("null distribution of", b, "Test Statistic"))
return(NULL)
}
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