qqtest (version 1.0)

dkay: dkay The density function of the K distribution

Description

The K density function on df degrees of freedom and non-centrality parameter ncp.

A K distribution is the square root of a chi-square divided by its degrees of freedom. That is, if x is chi-squared on m degrees of freedom, then y = sqrt(x/m) is K on m degrees of freedom. Under standard normal theory, K is the distribution of the pivotal quantity s/sigma where s is the sample standard deviation and sigma is the standard deviation parameter of the normal density. K is the natural distribution for tests and confidence intervals about sigma. K densities are more nearly symmetric than are chi-squared and concentrate near 1. As the degrees of freedom increase, they become more symmetric, more concentrated, and more nearly normally distributed.

Usage

dkay(x, df, ncp = 0, log.p = FALSE)

Value

dkay gives the density evaluated at the values of x.

Invalid arguments will result in return value NaN, with a warning.

The length of the result is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments are recycled to the length of the result. Only the first elements of the logical arguments are used.

Examples

dkay(1, 20)
#
# compare K density to that of chi as degrees of freedom increase
op <-par(mfrow=c(1,2))
p <- seq(0.001, .999, 0.001)
#
# First get all the chi-square densities and plot them
xchi5 <- qchisq(p,5)
dchi5 <- dchisq(xchi5,5)
xchi10 <- qchisq(p,10)
dchi10 <- dchisq(xchi10,10)
xchi20 <- qchisq(p,20)
dchi20 <- dchisq(xchi20,20)
xchi30 <- qchisq(p,30)
dchi30 <- dchisq(xchi20,30)
xlim <- range(xchi5, xchi10, xchi20, xchi30)
ylim <- range(dchi5, dchi10, dchi20, dchi30)
plot(xchi5, dchi5, type="l", xlab="x", ylab="density",
     xlim=xlim, ylim=ylim,
     main="chi-squared densities")
lines(xchi10, dchi10, lty=2)
lines(xchi20, dchi20, lty=3)
lines(xchi20, dchi30, lty=4)
legend("topright",
       legend=c("df = 5", "df = 10", "df = 20", "df = 30"),
       lty=c(1,2,3,4),
       title="degrees of freedom",
       cex=0.75, bty="n")
#
# Now get all the K densities and plot them
xkay5 <- qkay(p,5)
dkay5 <- dkay(xkay5,5)
xkay10 <- qkay(p,10)
dkay10 <- dkay(xkay10,10)
xkay20 <- qkay(p,20)
dkay20 <- dkay(xkay20,20)
xkay30 <- qkay(p,30)
dkay30 <- dkay(xkay20,30)
xlim <- range(xkay5, xkay10, xkay20, xkay30)
ylim <- range(dkay5, dkay10, dkay20, dkay30)
plot(xkay5, dkay5, type="l",
     xlab="x", ylab="density",
     xlim=xlim, ylim=ylim,
     main="K densities")
lines(xkay10, dkay10, lty=2)
lines(xkay20, dkay20, lty=3)
lines(xkay20, dkay30, lty=4)
legend("topright",
       legend=c("df = 5", "df = 10", "df = 20", "df = 30"),
       lty=c(1,2,3,4),
       title="degrees of freedom",
       cex=0.75, bty="n")
par(op)
#
# Note how K densities are more nearly symmetric and concentrate near 1.
# As the degrees of freedom increase, they become more symmetric,
# more concentrated, and more nearly normally distributed.