pnt: Non-central t Probability Distribution - Algorithms and...
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pnt R Documentation
Non-central t Probability Distribution - Algorithms and Approximations
Description
Compute different approximations for the non-central t-Distribution
cumulative probability distribution function.
Usage
pntR (t, df, ncp, lower.tail = TRUE, log.p = FALSE,
use.pnorm = (df > 4e5 ||
ncp^2 > 2*log(2)*1021), # .Machine$double.min.exp = -1022
itrmax = 1000, errmax = 1e-12, verbose = TRUE)
pntR1 (t, df, ncp, lower.tail = TRUE, log.p = FALSE,
use.pnorm = (df > 4e5 ||
ncp^2 > 2*log(2)*1021),
itrmax = 1000, errmax = 1e-12, verbose = TRUE)
pntP94 (t, df, ncp, lower.tail = TRUE, log.p = FALSE,
itrmax = 1000, errmax = 1e-12, verbose = TRUE)
pntP94.1 (t, df, ncp, lower.tail = TRUE, log.p = FALSE,
itrmax = 1000, errmax = 1e-12, verbose = TRUE)
pnt3150 (t, df, ncp, lower.tail = TRUE, log.p = FALSE, M = 1000, verbose = TRUE)
pnt3150.1 (t, df, ncp, lower.tail = TRUE, log.p = FALSE, M = 1000, verbose = TRUE)
pntLrg (t, df, ncp, lower.tail = TRUE, log.p = FALSE)
pntJW39 (t, df, ncp, lower.tail = TRUE, log.p = FALSE)
pntJW39.0 (t, df, ncp, lower.tail = TRUE, log.p = FALSE)
pntVW13 (t, df, ncp, lower.tail = TRUE, log.p = FALSE,
keepS = FALSE, verbose = FALSE)
pntGST23_T6 (t, df, ncp, lower.tail = TRUE, log.p = FALSE,
y1.tol = 1e-8, Mterms = 20, alt = FALSE, verbose = TRUE)
pntGST23_T6.1(t, df, ncp, lower.tail = TRUE, log.p = FALSE,
y1.tol = 1e-8, Mterms = 20, alt = FALSE, verbose = TRUE)
## *Non*-asymptotic, (at least partly much) better version of R's Lenth(1998) algorithm
pntGST23_1(t, df, ncp, lower.tail = TRUE, log.p = FALSE,
j0max = 1e4, # for now
IxpqFUN = Ixpq,
alt = FALSE, verbose = TRUE, ...)
Arguments
t
vector of quantiles (called q in pt(..)).
df
degrees of freedom (> 0, maybe non-integer). df
= Inf is allowed.
ncp
non-centrality parameter \delta \ge 0;
If omitted, use the central t distribution.
log, log.p
logical; if TRUE, probabilities p are given as log(p).
lower.tail
logical; if TRUE (default), probabilities are
P[X \le x], otherwise, P[X > x].
use.pnorm
logical indicating if the
pnorm() approximation of Abramowitz and Stegun (26.7.10)
should be used, which is available as pntLrg().
The default corresponds to R pt()'s own behaviour (which
is suboptimal).
itrmax
number of iterations / terms.
errmax
convergence bound for the iterations.
verbose
logical or integer determining the amount of
diagnostic print out to the console.
M
positive integer specifying the number of terms to use in the series.
keepS
logical indicating if the function should
return a list with component cdf and other
informational elements, or just the CDF values directly (by default).
y1.tol
positive tolerance for warning if y:= t^2/(t^2 + df)
is too close to 1 (as the formulas use 1/(1-y)).
Mterms
number of summation terms for pntGST23_T6().
j0max
experimental: large integer limiting the summation
terms in pntGST23_1() .
IxpqFUN
the (scaled) incomplete beta function I_x(p,q) to be
used; currently, it defaults to the Ixpq function derived from
Nico Temme's Maple code for “Table 1” in Gil et al. (2023).
alt
logical specifying if and how log-scale should
be used. Experimental and not-yet-tested.
...
further arguments passed to IxpqFUN().
Details
pntR1():a pure R version of the (C level)
code of R's own pt(), additionally giving more
flexibility (via arguments use.pnorm, itrmax, errmax
whose defaults here have been hard-coded in R's C code called by pt()).
This implements an improved version of the AS 243 algorithm from
Lenth(1989);
- R's help on non-central
pt() says: -
This computes the lower tail only, so the upper tail suffers from
cancellation and a warning will be given when this is likely to be
significant.
- and (in ‘Note:’)
The code for non-zero
ncp is principally intended to be used for moderate
values of ncp: it will not be highly accurate,
especially in the tails, for large values.
pntR():the Vectorize()d version of pntR1().
pntP94(), pntP94.1():New versions of
pntR1(), pntR(); using the Posten (1994) algorithm.
pntP94() is the Vectorize()d version of
pntP94.1().
pnt3150(), pnt3150.1():-
Simple inefficient but hopefully correct version of pntP94..()
This is really a direct implementation of formula
(31.50), p.532 of Johnson, Kotz and Balakrishnan (1995)
pntLrg():provides the pnorm()
approximation (to the non-central t) from
Abramowitz and Stegun (26.7.10), p.949; which should be employed only for
large df and/or ncp.
pntJW39.0():use the Jennett & Welch (1939) approximation
see Johnson et al. (1995), p. 520, after (31.26a). This is still
fast for huge ncp but has wrong asymptotic tail
for |t| \to \infty. Crucially needs b=b_chi(df).
pntJW39():is an improved version of pntJW39.0(),
using 1-b =b_chi(df, one.minus=TRUE) to avoid
cancellation when computing 1 - b^2.
pntGST23_T6():(and pntGST23_T6.1() for
informational purposes only) use the Gil et al.(2023)'s
approximation of their Theorem 6.
pntGST23_1():implements Gil et al.(2023)'s direct
pbeta() based formula (1), which is very close to
Lenth's algorithm.
pntVW13():use MM's R translation of Viktor
Witkowský (2013)'s matlab implementation.
Value
a number for pntJKBf1() and .pntJKBch1().
a numeric vector of the same length as the maximum of the lengths of
x, df, ncp for pntJKBf() and .pntJKBch().
Author(s)
Martin Maechler
References
Johnson, N.L., Kotz, S. and Balakrishnan, N. (1995)
Continuous Univariate Distributions Vol~2, 2nd ed.; Wiley;
chapter 31, Section 5 Distribution Function, p.514 ff
Lenth, R. V. (1989). Algorithm AS 243 —
Cumulative distribution function of the non-central t distribution,
JRSS C (Applied Statistics) 38, 185–189.
Abramowitz, M. and Stegun, I. A. (1972)
Handbook of Mathematical Functions. New York: Dover;
formula (26.7.10), p.949
Posten, Harry O. (1994)
A new algorithm for the noncentral t distribution function,
Journal of Statistical Computation and Simulation 51, 79–87;
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/00949659408811623")}.
– not yet implemented –
Chattamvelli, R. and Shanmugam, R. (1994)
An enhanced algorithm for noncentral t-distribution,
Journal of Statistical Computation and Simulation 49, 77–83.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/00949659408811561")}
– not yet implemented –
Akahira, Masafumi. (1995).
A higher order approximation to a percentage point of the noncentral t distribution,
Communications in Statistics - Simulation and Computation 24:3, 595–605;
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/03610919508813261")}
Michael Perakis and Evdokia Xekalaki (2003)
On a Comparison of the Efficacy of Various Approximations of the Critical Values for Tests on the Process Capability Indices CPL, CPU, and Cpk,
Communications in Statistics - Simulation and Computation 32, 1249–1264;
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1081/SAC-120023888")}
Witkovský, Viktor (2013)
A Note on Computing Extreme Tail Probabilities of the Noncentral T Distribution with Large
Noncentrality Parameter,
Acta Universitatis Palackianae Olomucensis, Facultas Rerum Naturalium, Mathematica
52(2), 131–143.
Gil A., Segura J., and Temme N.M. (2023)
New asymptotic representations of the noncentral t-distribution,
Stud Appl Math. 151, 857–882; \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/sapm.12609")} ;
acronym “GST23”.
See Also
pt, for R's version of non-central t probabilities.
Examples
tt <- seq(0, 10, len = 21)
ncp <- seq(0, 6, len = 31)
pt3R <- outer(tt, ncp, pt, , df = 3)
pt3JKB <- outer(tt, ncp, pntR, df = 3)# currently verbose
stopifnot(all.equal(pt3R, pt3JKB, tolerance = 4e-15))# 64-bit Lnx: 2.78e-16
## Gil et al.(2023) -- Table 1 p.869
str(GST23_tab1 <- read.table(header=TRUE, text = "
x pnt_x_delta Rel.accuracy l_y j_max
5 0.7890745035061528e-20 0.20e-13 0.29178 254
8 0.1902963697413609e-07 0.40e-12 0.13863 294
11 0.4649258368179092e-03 0.12e-09 0.07845 310
14 0.2912746016055676e-01 0.11e-07 0.04993 317
17 0.1858422833307925e-00 0.41e-06 0.03441 321
20 0.4434882973203470e-00 0.82e-05 0.02510 323"))
x1 <- c(5,8,11,14,17,20)
(p1 <- pt (x1, df=10.3, ncp=20))
(p1R <- pntR(x1, df=10.3, ncp=20)) # verbose=TRUE is default
all.equal(p1, p1R, tolerance=0) # 4.355452e-15 {on x86_64} as have *no* LDOUBLE on R level
stopifnot(all.equal(p1, p1R))
## NB: According to Gil et al., the first value (x=5) is really wrong
## p1.23 <- .. Gil et al., Table 1:
p1.23.11 <- pntGST23_T6(x1, df=10.3, ncp=20, Mterms = 11)
p1.23.20 <- pntGST23_T6(x1, df=10.3, ncp=20, Mterms = 20, verbose=TRUE)
# ==> Mterms = 11 is good only for x=5
p1.23.50 <- pntGST23_T6(x1, df=10.3, ncp=20, Mterms = 50, verbose=TRUE)
x <- 4:40 ; df <- 10.3
ncp <- 20
p1 <- pt (x, df=df, ncp=ncp)
pG1 <- pntGST23_1(x, df=df, ncp=ncp)
pG1.bR <- pntGST23_1(x, df=df, ncp=ncp,
IxpqFUN = \(x, l_x=.5-x+.5, p, q) Ixpq(x,l_x, p,q))
pG1.BR <- pntGST23_1(x, df=df, ncp=ncp,
IxpqFUN = \(x, l_x, p, q) pbeta(x, p,q))
cbind(x, p1, pG1, pG1.bR, pG1.BR)
all.equal(pG1, p1, tolerance=0) # 1.034 e-12
all.equal(pG1, pG1.bR, tolerance=0) # 2.497031 e-13
all.equal(pG1, pG1.BR, tolerance=0) # 2.924698 e-13
all.equal(pG1.BR,pG1.bR,tolerance=0)# 1.68644 e-13
stopifnot(exprs = {
all.equal(pG1, p1, tolerance = 4e-12)
all.equal(pG1, pG1.bR, tolerance = 1e-12)
all.equal(pG1, pG1.BR, tolerance = 1e-12)
})
ncp <- 40 ## is > 37.62 = "critical" for Lenth' algorithm
### --------- pntVW13() --------------------------------------------------
## length 1 arguments:
str(rr <- pntVW13(t = 1, df = 2, ncp = 3, verbose=TRUE, keepS=TRUE))
all.equal(rr$cdf, pt(1,2,3), tol = 0)# "Mean relative difference: 4.956769e-12"
stopifnot( all.equal(rr$cdf, pt(1,2,3)) )
str(rr <- pntVW13(t = 1:19, df = 2, ncp = 3, verbose=TRUE, keepS=TRUE))
str(r2 <- pntVW13(t = 1, df = 2:20, ncp = 3, verbose=TRUE, keepS=TRUE))
str(r3 <- pntVW13(t = 1, df = 2:20, ncp = 3:21, verbose=TRUE, keepS=TRUE))
pt1.10.5_T <- 4.34725285650591657e-5 # Ex. 7 of Witkovsky(2013)
pt1.10.5 <- pntVW13(1, 10, 5)
all.equal(pt1.10.5_T, pt1.10.5, tol = 0)# TRUE! (Lnx Fedora 40; 2024-07-04);
# 3.117e-16 (Macbuilder R 4.4.0, macOS Ventura 13.3.1)
stopifnot(exprs = {
identical(rr$cdf, r1 <- pntVW13(t = 1:19, df = 2, ncp = 3))
identical(r1[1], pntVW13(1, 2, 3))
identical(r1[7], pntVW13(7, 2, 3))
all.equal(pt1.10.5_T, pt1.10.5, tol = 9e-16)# NB even tol=0 (64 Lnx)
})
## However, R' pt() is only equal for the very first
cbind(t = 1:19, pntVW = r1, pt = pt(1:19, 2,3))
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| pnt | R Documentation |
Non-central t Probability Distribution - Algorithms and Approximations
Description
Compute different approximations for the non-central t-Distribution cumulative probability distribution function.
Usage
pntR (t, df, ncp, lower.tail = TRUE, log.p = FALSE,
use.pnorm = (df > 4e5 ||
ncp^2 > 2*log(2)*1021), # .Machine$double.min.exp = -1022
itrmax = 1000, errmax = 1e-12, verbose = TRUE)
pntR1 (t, df, ncp, lower.tail = TRUE, log.p = FALSE,
use.pnorm = (df > 4e5 ||
ncp^2 > 2*log(2)*1021),
itrmax = 1000, errmax = 1e-12, verbose = TRUE)
pntP94 (t, df, ncp, lower.tail = TRUE, log.p = FALSE,
itrmax = 1000, errmax = 1e-12, verbose = TRUE)
pntP94.1 (t, df, ncp, lower.tail = TRUE, log.p = FALSE,
itrmax = 1000, errmax = 1e-12, verbose = TRUE)
pnt3150 (t, df, ncp, lower.tail = TRUE, log.p = FALSE, M = 1000, verbose = TRUE)
pnt3150.1 (t, df, ncp, lower.tail = TRUE, log.p = FALSE, M = 1000, verbose = TRUE)
pntLrg (t, df, ncp, lower.tail = TRUE, log.p = FALSE)
pntJW39 (t, df, ncp, lower.tail = TRUE, log.p = FALSE)
pntJW39.0 (t, df, ncp, lower.tail = TRUE, log.p = FALSE)
pntVW13 (t, df, ncp, lower.tail = TRUE, log.p = FALSE,
keepS = FALSE, verbose = FALSE)
pntGST23_T6 (t, df, ncp, lower.tail = TRUE, log.p = FALSE,
y1.tol = 1e-8, Mterms = 20, alt = FALSE, verbose = TRUE)
pntGST23_T6.1(t, df, ncp, lower.tail = TRUE, log.p = FALSE,
y1.tol = 1e-8, Mterms = 20, alt = FALSE, verbose = TRUE)
## *Non*-asymptotic, (at least partly much) better version of R's Lenth(1998) algorithm
pntGST23_1(t, df, ncp, lower.tail = TRUE, log.p = FALSE,
j0max = 1e4, # for now
IxpqFUN = Ixpq,
alt = FALSE, verbose = TRUE, ...)
Arguments
t |
vector of quantiles (called |
df |
degrees of freedom ( |
ncp |
non-centrality parameter |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are
|
use.pnorm |
The default corresponds to R |
itrmax |
number of iterations / terms. |
errmax |
convergence bound for the iterations. |
verbose |
|
M |
positive integer specifying the number of terms to use in the series. |
keepS |
|
y1.tol |
positive tolerance for warning if |
Mterms |
number of summation terms for |
j0max |
experimental: large integer limiting the summation
terms in |
IxpqFUN |
the (scaled) incomplete beta function |
alt |
|
... |
further arguments passed to |
Details
pntR1():a pure R version of the (C level) code of R's own
pt(), additionally giving more flexibility (via argumentsuse.pnorm,itrmax,errmaxwhose defaults here have been hard-coded in R's C code called bypt()).This implements an improved version of the AS 243 algorithm from Lenth(1989);
- R's help on non-central
pt()says: -
This computes the lower tail only, so the upper tail suffers from cancellation and a warning will be given when this is likely to be significant.
- and (in ‘Note:’)
The code for non-zero
ncpis principally intended to be used for moderate values ofncp: it will not be highly accurate, especially in the tails, for large values.
- R's help on non-central
pntR():the
Vectorize()d version ofpntR1().pntP94(),pntP94.1():New versions of
pntR1(),pntR(); using the Posten (1994) algorithm.pntP94()is theVectorize()d version ofpntP94.1().pnt3150(),pnt3150.1():-
Simple inefficient but hopefully correct version of pntP94..() This is really a direct implementation of formula (31.50), p.532 of Johnson, Kotz and Balakrishnan (1995)
pntLrg():provides the
pnorm()approximation (to the non-centralt) from Abramowitz and Stegun (26.7.10), p.949; which should be employed only for largedfand/orncp.pntJW39.0():use the Jennett & Welch (1939) approximation see Johnson et al. (1995), p. 520, after (31.26a). This is still fast for huge
ncpbut has wrong asymptotic tail for|t| \to \infty. Crucially needsb=b_chi(df).pntJW39():is an improved version of
pntJW39.0(), using1-b =b_chi(df, one.minus=TRUE)to avoid cancellation when computing1 - b^2.pntGST23_T6():(and
pntGST23_T6.1()for informational purposes only) use the Gil et al.(2023)'s approximation of their Theorem 6.pntGST23_1():implements Gil et al.(2023)'s direct
pbeta()based formula (1), which is very close to Lenth's algorithm.pntVW13():use MM's R translation of Viktor Witkowský (2013)'s matlab implementation.
Value
a number for pntJKBf1() and .pntJKBch1().
a numeric vector of the same length as the maximum of the lengths of
x, df, ncp for pntJKBf() and .pntJKBch().
Author(s)
Martin Maechler
References
Johnson, N.L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions Vol~2, 2nd ed.; Wiley; chapter 31, Section 5 Distribution Function, p.514 ff
Lenth, R. V. (1989). Algorithm AS 243 —
Cumulative distribution function of the non-central t distribution,
JRSS C (Applied Statistics) 38, 185–189.
Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. New York: Dover; formula (26.7.10), p.949
Posten, Harry O. (1994) A new algorithm for the noncentral t distribution function, Journal of Statistical Computation and Simulation 51, 79–87; \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/00949659408811623")}.
– not yet implemented –
Chattamvelli, R. and Shanmugam, R. (1994)
An enhanced algorithm for noncentral t-distribution,
Journal of Statistical Computation and Simulation 49, 77–83.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/00949659408811561")}
– not yet implemented –
Akahira, Masafumi. (1995).
A higher order approximation to a percentage point of the noncentral t distribution,
Communications in Statistics - Simulation and Computation 24:3, 595–605;
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/03610919508813261")}
Michael Perakis and Evdokia Xekalaki (2003) On a Comparison of the Efficacy of Various Approximations of the Critical Values for Tests on the Process Capability Indices CPL, CPU, and Cpk, Communications in Statistics - Simulation and Computation 32, 1249–1264; \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1081/SAC-120023888")}
Witkovský, Viktor (2013) A Note on Computing Extreme Tail Probabilities of the Noncentral T Distribution with Large Noncentrality Parameter, Acta Universitatis Palackianae Olomucensis, Facultas Rerum Naturalium, Mathematica 52(2), 131–143.
Gil A., Segura J., and Temme N.M. (2023) New asymptotic representations of the noncentral t-distribution, Stud Appl Math. 151, 857–882; \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/sapm.12609")} ; acronym “GST23”.
See Also
pt, for R's version of non-central t probabilities.
Examples
tt <- seq(0, 10, len = 21)
ncp <- seq(0, 6, len = 31)
pt3R <- outer(tt, ncp, pt, , df = 3)
pt3JKB <- outer(tt, ncp, pntR, df = 3)# currently verbose
stopifnot(all.equal(pt3R, pt3JKB, tolerance = 4e-15))# 64-bit Lnx: 2.78e-16
## Gil et al.(2023) -- Table 1 p.869
str(GST23_tab1 <- read.table(header=TRUE, text = "
x pnt_x_delta Rel.accuracy l_y j_max
5 0.7890745035061528e-20 0.20e-13 0.29178 254
8 0.1902963697413609e-07 0.40e-12 0.13863 294
11 0.4649258368179092e-03 0.12e-09 0.07845 310
14 0.2912746016055676e-01 0.11e-07 0.04993 317
17 0.1858422833307925e-00 0.41e-06 0.03441 321
20 0.4434882973203470e-00 0.82e-05 0.02510 323"))
x1 <- c(5,8,11,14,17,20)
(p1 <- pt (x1, df=10.3, ncp=20))
(p1R <- pntR(x1, df=10.3, ncp=20)) # verbose=TRUE is default
all.equal(p1, p1R, tolerance=0) # 4.355452e-15 {on x86_64} as have *no* LDOUBLE on R level
stopifnot(all.equal(p1, p1R))
## NB: According to Gil et al., the first value (x=5) is really wrong
## p1.23 <- .. Gil et al., Table 1:
p1.23.11 <- pntGST23_T6(x1, df=10.3, ncp=20, Mterms = 11)
p1.23.20 <- pntGST23_T6(x1, df=10.3, ncp=20, Mterms = 20, verbose=TRUE)
# ==> Mterms = 11 is good only for x=5
p1.23.50 <- pntGST23_T6(x1, df=10.3, ncp=20, Mterms = 50, verbose=TRUE)
x <- 4:40 ; df <- 10.3
ncp <- 20
p1 <- pt (x, df=df, ncp=ncp)
pG1 <- pntGST23_1(x, df=df, ncp=ncp)
pG1.bR <- pntGST23_1(x, df=df, ncp=ncp,
IxpqFUN = \(x, l_x=.5-x+.5, p, q) Ixpq(x,l_x, p,q))
pG1.BR <- pntGST23_1(x, df=df, ncp=ncp,
IxpqFUN = \(x, l_x, p, q) pbeta(x, p,q))
cbind(x, p1, pG1, pG1.bR, pG1.BR)
all.equal(pG1, p1, tolerance=0) # 1.034 e-12
all.equal(pG1, pG1.bR, tolerance=0) # 2.497031 e-13
all.equal(pG1, pG1.BR, tolerance=0) # 2.924698 e-13
all.equal(pG1.BR,pG1.bR,tolerance=0)# 1.68644 e-13
stopifnot(exprs = {
all.equal(pG1, p1, tolerance = 4e-12)
all.equal(pG1, pG1.bR, tolerance = 1e-12)
all.equal(pG1, pG1.BR, tolerance = 1e-12)
})
ncp <- 40 ## is > 37.62 = "critical" for Lenth' algorithm
### --------- pntVW13() --------------------------------------------------
## length 1 arguments:
str(rr <- pntVW13(t = 1, df = 2, ncp = 3, verbose=TRUE, keepS=TRUE))
all.equal(rr$cdf, pt(1,2,3), tol = 0)# "Mean relative difference: 4.956769e-12"
stopifnot( all.equal(rr$cdf, pt(1,2,3)) )
str(rr <- pntVW13(t = 1:19, df = 2, ncp = 3, verbose=TRUE, keepS=TRUE))
str(r2 <- pntVW13(t = 1, df = 2:20, ncp = 3, verbose=TRUE, keepS=TRUE))
str(r3 <- pntVW13(t = 1, df = 2:20, ncp = 3:21, verbose=TRUE, keepS=TRUE))
pt1.10.5_T <- 4.34725285650591657e-5 # Ex. 7 of Witkovsky(2013)
pt1.10.5 <- pntVW13(1, 10, 5)
all.equal(pt1.10.5_T, pt1.10.5, tol = 0)# TRUE! (Lnx Fedora 40; 2024-07-04);
# 3.117e-16 (Macbuilder R 4.4.0, macOS Ventura 13.3.1)
stopifnot(exprs = {
identical(rr$cdf, r1 <- pntVW13(t = 1:19, df = 2, ncp = 3))
identical(r1[1], pntVW13(1, 2, 3))
identical(r1[7], pntVW13(7, 2, 3))
all.equal(pt1.10.5_T, pt1.10.5, tol = 9e-16)# NB even tol=0 (64 Lnx)
})
## However, R' pt() is only equal for the very first
cbind(t = 1:19, pntVW = r1, pt = pt(1:19, 2,3))
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