pDtrend: Cumulative distribution function of nonparametric trend...
In skedastic: Handling Heteroskedasticity in the Linear Regression Model

pDtrend

R Documentation

Cumulative distribution function of nonparametric trend statistic `D`

Description

This function computes \Pr(D \le k) (\Pr(D \ge k)), i.e. lower (upper) cumulative probabilities for D=\sum_{i=1}^{n} (R_i - i)^2, the nonparametric trend statistic proposed by \insertCiteLehmann75;textualskedastic, under the assumption that the ranks R_i are computed on a series of n independent and identically distributed random variables with no ties. The function may be used to compute one-sided p-values for the nonparametric test for heteroskedasticity of \insertCiteHorn81;textualskedastic. Computation time is extremely slow for n > 10 if usedata is set to FALSE; thus a normal approximation is implemented, including a continuity correction.

Usage

pDtrend(
  k,
  n,
  lower.tail = TRUE,
  exact = (n <= 10),
  tiefreq = NA,
  override = FALSE
)

Arguments

`k`	An integer of `length` `\ge 1` or a character `"all"` indicating that the cumulative distribution function should be applied to the entire support of `D`. The latter is only acceptable when `exact` is `TRUE`, since the distribution is otherwise continuous.
`n`	A positive integer representing the number of observations in the series.
`lower.tail`	A logical. Should lower tailed cumulative probability be calculated? Defaults to `TRUE`. Note that both lower- and upper- tailed cumulative probabilities are computed inclusive of `k`.
`exact`	A logical. Should exact distribution of `D` be used by calling `dDtrend`? If `FALSE`, a normal approximation is used. If `tiefreq` is not `NA` (ties are present), normal approximation is used regardless of the value of `exact`. By default, `exact` is set to `TRUE` provided that `n <= 10`. Setting `exact` to `TRUE` for `n > 11` results in an error unless `override` is set to `TRUE`.
`tiefreq`	A double vector corresponding to the value of `d_i` in \insertCiteLehmann75;textualskedastic. These are the frequencies of the various tied ranks. If ties are absent, `NA` (the default).
`override`	A logical. By default, the `dDtrend` function aborts if `n > 11` due to the prohibitively slow computation (which may cause some systems to crash). Setting this argument to `TRUE` overrides the abort. Ignored unless `exact` is `TRUE`.

Value

A double between 0 and 1 representing the probability/ies of D taking on at least (at most) the value(s) in the names attribute.

References

\insertAllCited

Examples

# For an independent sample of size 6, the probability that D is <= 50 is
# 0.8222
pDtrend(k = 50, n = 6)
# Normal approximation of the above with continuity correction is
# 0.8145
pDtrend(k = 50, n = 6, exact = FALSE)

skedastic documentation built on June 8, 2025, 11:07 a.m.