qtAppr {DPQ} | R Documentation |
Compute Approximate Quantiles of the (Non-Central) t-Distribution
Description
Compute quantiles (inverse distribution values) for the non-central t distribution. using Johnson,Kotz,.. p.521, formula (31.26 a) (31.26 b) & (31.26 c)
Note that qt(.., ncp=*)
did not exist yet in 1999, when MM
implemented qtAppr()
.
qtNappr()
approximates t-quantiles for large df
, i.e., when
close to the Gaussian / normal distribution, using up to 4 asymptotic
terms from Abramowitz & Stegun 26.7.5 (p.949).
Usage
qtAppr (p, df, ncp, lower.tail = TRUE, log.p = FALSE, method = c("a", "b", "c"))
qtNappr(p, df, lower.tail = TRUE, log.p=FALSE, k)
Arguments
p |
vector of probabilities. |
df |
degrees of freedom |
ncp |
non-centrality parameter |
lower.tail , log.p |
logical, see, e.g., |
method |
a string specifying the approximation method to be used. |
k |
an integer in {0,1,2,3,4}, choosing the number of terms in |
Value
numeric vector of length length(p + df + ncp)
with approximate t-quantiles.
References
Johnson, N.L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions Vol~2, 2nd ed.; Wiley; chapter 31, Section 6 Approximation, p.519 ff
Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. New York: Dover; formula (26.7.5), p.949; https://en.wikipedia.org/wiki/Abramowitz_and_Stegun provides links to the full text which is in public domain.
See Also
Our qtU()
; several non-central density and probability
approximations in dntJKBf
, and e.g., pntR
.
Further, R's qt
.
Examples
qts <- function(p, df) {
cbind(qt = qt(p, df=df)
, qtN0 = qtNappr(p, df=df, k=0)
, qtN1 = qtNappr(p, df=df, k=1)
, qtN2 = qtNappr(p, df=df, k=2)
, qtN3 = qtNappr(p, df=df, k=3)
, qtN4 = qtNappr(p, df=df, k=4)
)
}
p <- (0:100)/100
ii <- 2:100 # drop p=0 & p=1 where q*(p, .) == +/- Inf
df <- 100 # <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
qsp1c <- qts(p, df = df)
matplot(p, qsp1c, type="l") # "all on top"
(dq <- (qsp1c[,-1] - qsp1c[,1])[ii,])
matplot(p[ii], dq, type="l", col=2:6,
main = paste0("difference qtNappr(p,df) - qt(p,df), df=",df), xlab=quote(p))
matplot(p[ii], pmax(abs(dq), 1e-17), log="y", type="l", col=2:6,
main = paste0("abs. difference |qtNappr(p,df) - qt(p,df)|, df=",df), xlab=quote(p))
legend("bottomright", paste0("k=",0:4), col=2:6, lty=1:5, bty="n")
matplot(p[ii], abs(dq/qsp1c[ii,"qt"]), log="y", type="l", col=2:6,
main = sprintf("rel.error qtNappr(p, df=%g, k=*)",df), xlab=quote(p))
legend("left", paste0("k=",0:4), col=2:6, lty=1:5, bty="n")
df <- 2000 # <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
qsp1c <- qts(p, df=df)
(dq <- (qsp1c[,-1] - qsp1c[,1])[ii,])
matplot(p[ii], dq, type="l", col=2:6,
main = paste0("difference qtNappr(p,df) - qt(p,df), df=",df), xlab=quote(p))
legend("top", paste0("k=",0:4), col=2:6, lty=1:5)
matplot(p[ii], pmax(abs(dq), 1e-17), log="y", type="l", col=2:6,
main = paste0("abs.diff. |qtNappr(p,df) - qt(p,df)|, df=",df), xlab=quote(p))
legend("right", paste0("k=",0:4), col=2:6, lty=1:5, bty="n")
matplot(p[ii], abs(dq/qsp1c[ii,"qt"]), log="y", type="l", col=2:6,
main = sprintf("rel.error qtNappr(p, df=%g, k=*)",df), xlab=quote(p))
legend("left", paste0("k=",0:4), col=2:6, lty=1:5, bty="n")