#' Data Driven Smooth Test for Exponentiality
#'
#' Performs data driven smooth test for composite hypothesis of exponentiality.
#' Null density is given by \eqn{f(z;gamma) = exp(-z/gamma)} for z >= 0 and 0 otherwise.
#' Modelling alternatives similarly as in Kallenberg and Ledwina (1997 a,b).
#'
#' @param x a (non-empty) numeric vector of data values
#' @param base a function which returns an orthonormal system, possible choice: \code{ddst.base.legendre} for the Legendre polynomials and \code{ddst.base.cos} for the cosine system
#' @param d.n an integer specifying the maximum dimension considered, only for advanced users
#' @param c a calibrating parameter in the penalty in the model selection rule
#' @param nr an integer specifying the number of runs for a p-value and a critical value computation if any
#' @param compute.p a logical value indicating whether to compute a p-value or not
#' @param alpha a significance level
#' @param compute.cv a logical value indicating whether to compute a critical value corresponding to the significance level alpha or not
#' @param ... further arguments
#'
#' @return An object of class \code{htest}
#' \item{statistic }{the value of the test statistic.}
#' \item{parameter }{the number of choosen coordinates (k).}
#' \item{method }{a character string indicating the parameters of performed test. }
#' \item{data.name }{a character string giving the name(s) of the data. }
#' \item{p.value }{the p-value for the test, computed only if \code{compute.p=T}.}
#'
#' @export
#'
#' @references
#' Kallenberg, W.C.M., Ledwina, T. (1997 a). Data driven smooth tests for composite hypotheses: Comparison of powers. \eqn{ J. Statist. Comput. Simul.} \bold{ 59}, 101--121.
#'
#' Kallenberg, W.C.M., Ledwina, T. (1997 b). Data driven smooth tests when the hypothesis is composite. \eqn{ J. Amer. Statist. Assoc.} \bold{ 92}, 1094--1104.
#'
#' @examples
#' set.seed(7)
#' # H0 is true
#' z <- rexp(80,4)
#' \dontrun{
#' t <- ddst.exp.test (z, compute.p = TRUE, d.n = 10)
#' t
#' plot(t)
#'
#' # H0 is false
#' z = rchisq(80,4)
#' (t = ddst.exp.test (z, compute.p = TRUE, d.n = 10))
#' t$p.value
#' plot(t)
#' }
#' @keywords htest
#' @aliases ddst.exp.Nk
#'
`ddst.exp.test` <-
function(x,
base = ddst.base.legendre,
d.n = 10,
c = 100,
nr = 100000,
compute.p = TRUE,
alpha = 0.05,
compute.cv = TRUE,
...) {
# method.name = as.character(substitute(base))
base = ddst.base.legendre
method.name = "ddst.base.legendre"
n = length(x)
if (n < 5)
stop("length(x) should be at least 5")
er = mean(x)
maxN = max(min(d.n, n - 2, 20), 1)
coord = numeric(maxN)
u = numeric(maxN)
for (j in 1:maxN) {
u[j] = ddst.phi(pexp(x, 1 / er), j, base)
coord[j] = t(u[1:j]) %*% MMexp[[j]] %*% u[1:j] * n
}
l = ddst.IIC(coord, n, c)
attr(l, "names") = "T*"
t = coord[l]
attr(t, "names") = "W*T*"
result = list(statistic = t,
parameter = l,
coordinates = coord - c(0, coord[-d.n]),
method = "Data Driven Smooth Test for Expotentiality")
result$data.name = paste(paste(as.character(substitute(x)), collapse = ""),
", base: ",
method.name,
", c: ",
c,
", d.n: ",
d.n,
sep = "")
class(result) = c("htest", "ddst.test")
if (compute.p | compute.cv) {
tmp = numeric(nr)
for (i in 1:nr) {
y = rexp(length(x))
tmpC = ddst.exp.Nk(y, base, Dmax = d.n, n = length(y))
l = ddst.IIC(tmpC, n, c)
tmp[i] = tmpC[l]
}
p.val = mean(tmp > t)
if (compute.p) {
result$p.value = mean(tmp > t)
}
if (compute.cv) {
result$cv = quantile(tmp, alpha)
}
}
result
}
`MMexp` <-
list(
structure(4, .Dim = c(1L, 1L)),
structure(
c(7.75, 2.90473750965556,
2.90473750965556, 2.25),
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),
.Dim = c(20L, 20L)
)
)
`ddst.exp.Nk` <-
function(x,
base = ddst.base.legendre,
Dmax = 5,
n = length(x)) {
er = mean(x)
maxN = max(min(Dmax, n - 2, 20), 1)
coord = numeric(maxN)
u = numeric(maxN)
for (j in 1:maxN) {
u[j] = ddst.phi(pexp(x, 1 / er), j, base)
coord[j] = t(u[1:j]) %*% MMexp[[j]] %*% u[1:j] * n
}
coord
}
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