skedastic (version 2.0.2)

dDtrend: Probability mass function of nonparametric trend statistic \(D\)

Description

This function computes \(\Pr(D = k)\), i.e. the probability mass function for \(D=\sum_{i=1}^{n} (R_i - i)^2\), the nonparametric trend statistic proposed by Lehmann75;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.

Usage

dDtrend(k = "all", n, override = FALSE)

Value

A double vector containing the probabilities corresponding to the integers in its names attribute.

Arguments

k

An integer of length \(\ge 1\) or a character "all" (the default) indicating that the probability mass function should be applied to the entire support of \(D\).

n

A positive integer representing the number of observations in the series. Note that computation time increases rapidly with \(n\) and is infeasible for \(n>11\).

override

A logical. By default, the 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.

Details

The function is used within horn in computing \(p\)-values for Horn's nonparametric test for heteroskedasticity in a linear regression model Horn81skedastic. The support of \(D\) consists of consecutive even numbers from 0 to \(\frac{n(n-1)(n+1)}{3}\), with the exception of the case \(n=3\), when the value 4 is excluded from the support. Note that computation speed for k = "all" is about the same as when k is set to an individual integer value, because the entire distribution is still computed in the latter case.

References

See Also

horn

Examples

prob <- dDtrend(k = "all", n = 6)
values <- as.integer(names(prob))
plot(c(values[1], values[1]), c(0, prob[1]), type = "l",
  axes = FALSE, xlab = expression(k), ylab = expression(Pr(D == k)),
  xlim = c(0, 70), yaxs = "i", ylim = c(0, 1.05 * max(prob)))
  axis(side = 1, at = seq(0, 70, 10), las = 2)
for (i in seq_along(values)) {
 lines(c(values[i], values[i]), c(0, prob[i]))
}