#### Examples for non-central t #### ========================== ### ### Johnson, Kotz, ...; 2nd ed, Vol 2. Chapter 31, around p.520 c({}## This stems from the two files , "/u/maechler/R/MM/NUMERICS/dpq-functions/pnt-ex.R" , "/u/maechler/R/MM/NUMERICS/dpq-functions/t-nonc-approx.R" , "SEE ALSO ./pnt-prec.R (still current problems!)" , " ==========" ) ## and originally, had ## source("/u/maechler/R/MM/NUMERICS/dpq-functions/t-nonc-fn.R") ## - - - - - - - - - - - - - - - - - - - - - - - - - - library(DPQ) ## source("/u/maechler/R/Pkgs/DPQ/R/t-nonc-fn.R") stopifnot(exprs = { require(graphics) require(sfsmisc) }) source(system.file(package="DPQ", "test-tools.R", mustWork=TRUE)) ## => showProc.time(), ... list_() , loadList() , readRDS_() , save2RDS() relErrV <- sfsmisc::relErrV if(!dev.interactive(orNone=TRUE)) pdf("t-nonc_P-1.pdf") ### Part I -- former pnt-ex.R ### --------------------------- ### Find value of delta after which pt(*,*, ncp=delta) === 0 : ### MM{2014}: This is no longer true: pt(ncp -> Inf) now "goes all the way to underflow" ### and pt(*, log.p=TRUE) "is ok" ## This would be from theory : (dmax <- sqrt(-2*log(2* .Machine$double.xmin))) # 37.62189 .5*exp(-dmax^2/2) # 2.225074e-308 pt.pos <- function(delta,q,df) (pt(q=q,df=df,ncp=delta) > 0) - 1/2 ## see 'MM{2014}' above: this no longer works ## d1 <- uniroot(pt.pos, q=10,df=2, lower=30, upper= 40, tol=1e-10)$root ## d2 <- uniroot(pt.pos, q= 1,df=2, lower=30, upper= 40, tol=1e-10)$root ## d1 == d2 # TRUE d3 <- uniroot(pt.pos, q=.1, df=20, lower=30, upper= 40, tol=1e-10)$root ## d4 <- uniroot(pt.pos, q=100,df=20, lower=30, upper= 40, tol=1e-10)$root ## d3 != d4 # T !! ## d1 == d4 # T ## c(d1,d3) ## 38.57550 38.56226 d5 <- uniroot(pt.pos, q=.01,df= 2, lower=30, upper= 40, tol=1e-10)$root d6 <- uniroot(pt.pos, q=.01,df=20, lower=30, upper= 40, tol=1e-10)$root c(d3, d5, d6) ## unique(sort(c(d1,d2,d3,d4,d5,d6))) ##[1] 38.45745 38.46036 38.56226 38.57550 ###--- When DEBUG_pnt is compiled in: pt(20,20,20) pt(20,20,25) pt( 1,20,25) pt(30, 1,25) pt(100, 1,25) ## p #{iter} result pt(30,30,30) # 1.84694e-196 600 .4672569 pt(35,35,35) # 4.93855e-267 786 .4694869 pt(36,36,36) # 1.88862e-282 827 .4698806 d <- 37 ;pt(d,d,d)# 2.65703e-298 868 .4702595 d <- 38 ;pt(d,d,d)# 1.37516e-314 902 .4706245 -- no longer d <- 38.5;pt(d,d,d)# 6.91692e-323 (799) s<0 => error-bound < 0 (!) wrong convergence => .4702131 ##--- pt(39,39,39)# p = 0 ==> result=0 --- no longer ## FIXME: pntR1(40,40, 38.5, verbose=2) ## and then edit ---> gives the *.out file ##--------- FIXME --- work via sink() or capture.output -- to get rr data.frame() if(FALSE) { t.file <- "/u/maechler/R/MM/NUMERICS/dpq-functions/pnt-40-40-38.5.out" nr <- names(rr <- read.table(t.file, header= TRUE)) summary(rr) matplot(rr[,"it"], rr[,-1], type='l', log="y") ##--> Graphic axis "buglet" par(mfrow=c(4,2)); f <- FF ~ it for(i in 2:ncol(rr)) {f[[2]] <- as.name(nr[i]); plot(f, data=rr, type='l')} par(mfrow=c(4,2)); f <- FF ~ it for(i in 2:ncol(rr)) {f[[2]] <- as.name(nr[i]); plot(f, data=rr, type='l', log='xy')} } ##----------- end{FIXME}--------------------------------------------- ###------------------------ dcdflib ptnc() ---- computations and plots ## ======= ~~~~~ ##==> ~/R/Pkgs/dcdflib/tests/ --------- ## How does density look in these extremes ? ## ------- curve(dt(x, df=10,ncp=10000), 0, 85000, col=2, log="y") # log-density curve(dt(x, df=10,ncp=10000, log=TRUE), 1e3, 1e9, col=2, ylim = c(-30,-8),log="x") ## is it *not* log concave -- or is that a numeric error in the tails? ## ditto ? curve(dt(x, df=10,ncp= 1e7, log=TRUE), 1e5, 1e11, col=2, ylim = c(-35,-15),log="x") curve(dt(x, df=10,ncp= 1e7), 5e6, 3e7, col=2) integrate(function(x) dt(x, df=10,ncp= 1e7), 5e6, 3e7) ## 1.004937 with absolute error < 8e-08 hm... --> seems error in tails yl <- c(-80,0) curve(dt(x, df=10,ncp= 10, log=TRUE), 0, 600, col=2, n=5001, ylim=yl) curve(dt(x, df=10,ncp= 15, log=TRUE), 0, 800, col=2, n=5001, ylim=yl) ## Log-log curve(exp(dt(x, df=10,ncp= 15, log=TRUE)), 1, 1e7, log="xy", col=2, n=5001, axes=FALSE, ylab = "", main="dt(x, df=10,ncp= 15) -- Log-Log scale") eaxis(1) op <- par(las=2) eaxis(2, at = 10^pretty(log10(axTicks(2, log=TRUE)), 10), at.small =FALSE) par(op) mtext(R.version.string, cex = .6, adj=1) ## NB: Mathematica (e.g) ./pnt-ex.nb -- shows linear {in log-log} tail ## From: Jerry Lewis ## To: Martin Maechler ## Subject: R pnt.c function ## Date: Wed, 6 Nov 2013 21:36:16 +0000 ## I noticed your comment in the code ## /*----------- DEBUGGING ------------- ## * ## * make CFLAGS='-DDEBUG_pnt -g' ## * -- Feb.3, 1999; M.Maechler: ## - For 't > ncp > 20' (or so) the result is completely WRONG! ## */ ## while chasing down accuracy problems for calculating 1-sided normal ## tolerance limits. For example trying to reproduce values in Odeh & ## Owen's Table 1.1 by n <- c(102,104); round(qt(.995,n-1,-qnorm(.0001)*sqrt(n))/sqrt(n),3) ## the value R calculates for n==104 is clearly wrong, since the ## k-factor fails to be decreasing in n (inaccurate for all larger n ## as well). ## MM: graphically convincing is e.g. n <- seq(80,120, by=0.5) summary(ncp.n <- -qnorm(.0001)*sqrt(n)) qt995 <- qt(.995, n-1, ncp=ncp.n) / sqrt(n) summary(warnings())# 45 identical warnings: ## full precision may not have been achieved in 'pnt{final}' plot(n, qt995, type = "l", col=2, lwd=2) ## => visual kink in {102, 103}: abline(v = 102:103, lty=3, col="gray") qtSc <- function(n, p = 0.995, delta = .0001) { stopifnot(length(p) == 1, length(delta) == 1, n > 1, delta > 0, 0 < p, p < 1) ncp.n <- -qnorm(delta) * sqrt(n) qt(p, n-1, ncp=ncp.n) / sqrt(n) } n <- seq(80,120, by=.5); plot(n, qtSc(n), type = "l", col=2, lwd=1.5) summary(warnings()) # --> "full precision may not have been achieved in 'pnt{final}'" ## It's the *smaller* n's which give the warning: assign("last.warning", NULL, envir=as.environment(find("last.warning"))) # (non-API) n <- seq(80,102, by=.5); plot(n, qtSc(n), type = "l", col=2, lwd=1.5) try(# works "locally" but nut with the whole script stopifnot(length(n) == length(warnings())) ## ) ## These give *no* warnings [and we know they are inaccurate!]: n <- seq(104,120, by=.5); plot(n, qtSc(n), type = "l", col=2, lwd=1.5) ## MM: the bug really is in pt(*,df, ncp), i.e., pnt.c: n <- seq(80,120, by=0.25) summary(ncp.n <- -qnorm(.0001)*sqrt(n)) qq <- 4.6*sqrt(n) plot(qq, pt(qq, n-1, ncp=ncp.n), type = "l") ## - yes! cbind(n, qq, ncp.n, pt = pt(qq, n-1, ncp=ncp.n))## it is clearly at ## ncp = ## even simpler -- curve(pt(37,37,x), 37, 38) ## When x>0 and ncp>0, a cheap fix would substitute pf(x^2,1,df,ncp^2) ## for pt(x,df,ncp) whenever pnorm(-ncp) is negligible ## (as it should be when ncp>20). ## MM: According to ## https://en.wikipedia.org/wiki/Noncentral_t-distribution#Related_distributions ## If T is noncentral t-distributed with ν degrees of freedom and ## noncentrality parameter μ and F = T^2, then F has a noncentral ## F-distribution with 1 numerator degree of freedom, ν denominator degrees of freedom, ## and noncentrality parameter μ^2. ## MM: If t >= 0, ## pf(t^2, df1=1, df2=nu, ncp=mu^2) = P(T^2 <= t^2) = P(|T| <= t) = ## ================================ ## = P(-t <= T <= t) = pt(t, df, ncp) - pt(-t, df, ncp) ## ================================ ## and now, indeed, if pt(-t, df,ncp) << pt(t, df, ncp) is negligble, we can use pf(...) ## HOWEVER, 1) this not only depends on ncp !! ## 2) Code below suggests that noncentral pf() is not quite ok for small t^2 ## MM: let's see: ptq <- pt(qq, df= n-1, ncp = ncp.n) pfq2 <- pf(qq^2, df1=1, df2= n-1, ncp = ncp.n^2) cbind(n, qq, ncp.n, pt = ptq, "pf.q^2" = pfq2, rel.E = 1 - ptq/pfq2) lines(qq, pfq2, col=adjustcolor("blue", 0.4), lwd=3) ##==> he is right that pf() is good *here* chk.t.F <- function(x, df, ncp, tolerance = 1e-9) { pt. <- pt(x, df= df, ncp = ncp) pf. <- pf(x^2, df1=1, df2= df, ncp = ncp^2) re <- relErrV(pf., pt.) if(any(abs(re) > tolerance)) warning("rel.error too large: ", max(abs(re))) invisible(re) } chk.t.F(x=qq, df=n-1, ncp=ncp.n) ## too large: 0.00894 ## many are not so good.. ## these look good: cols <- c(palette()[3], adjustcolor(2, 0.4)) curve(pt(x, df=11.282, ncp=30), 0, 50, n=512, col=3) curve(pf(x^2, df1=1, df2=11.282, ncp=30^2), n=512, col=cols[2], lwd=4, add=TRUE) legend("top", c("pt()", "pf()"), col=cols, lwd=c(1,4), bty="n") ## but not if we look at the left tail in log scale: ## ========== curve(pt(x, df=11.282, ncp=30), 0, 30, n=512, col=3, log="y",yaxt="n"); eaxis(2) curve(pf(x^2, df1=1, df2=11.282, ncp=30^2), n=512, col=adjustcolor(2, 0.4), lwd=4, add=TRUE) legend("topleft", c("pt()", "pf()"), col=cols, lwd=c(1,4), bty="n") ## pf() and pt() very much differ when x gets small even for this ## large ncp ## The difference should *only* be "the other" pt() part : pt(-x, df, ncp) ## but that looks quite bad : ## Note that it should be monotone !!! pxy <- curve(pt(x, df=11.282, ncp=30, log=TRUE), -25, 30, n=512, col=3) curve(pt(x, df=5, ncp=10, log=TRUE), -20, 20, n=512, col=2, lwd=2, type="o") ## If we look at the corresponding *density* function, it's not better: curve(dt(x, df=11.282, ncp=30, log=TRUE), -25, 30, n=512, col=3) ## What about negative ncp ? [the same phenomenon} ## all fine "regular scale": curve(dt(x, df=11, ncp=-10), -25, 30, n=512, col=2, lwd=2) ## "all bad" on log scale: curve(dt(x, df=11, ncp=-10, log=TRUE), -40, 30, n=512, col=2, lwd=2) ## [Jerry, from his e-mail:] ---------------------------------------------------- ## That alone would improve the minimum accuracy from 2 to 6 significant ## figures when calculating tolerance factors for the cases that Odeh and ## Owen tabulated. ## The series expansion used by Lenth (basis of R's pnt.c function) can ## be implemented in a way that can accurately handle these cases, but I ## have not digested R's c code to see where the problem lies. The ## boundary where Lenth's p0 crosses from a normalized to a denormal binary representation is ## between the cases of n==102 and n==104, so I suspect that p0 is not ## factored out of the summation, which would accumulate reduced accuracy ## terms. ## Jerry if(!dev.interactive(orNone=TRUE)) { dev.off(); pdf("t-nonc_P-2.pdf") } ### Part II -- former t-nonc-fn.R ### ------------------------------ ### b_nu ----------------------------------- mult.fig(2, main = "b(nu) = E[ Chi_nu ] / sqrt(nu)")$old.par -> opar plot(b_chi, n=1024, col=2) ## Unfortunately, the above switch gives a small kink at nu = 300 ## 2015-01-02: b_chi() improved: no kink anymore ## ================================= ! curve(b_chi, 200,1200) curve(b_chiAsymp, add=TRUE, col='red') plot(b_chi,1,10, col='red'); curve(b_chiAsymp, add=TRUE, col='blue') ## New cutoff plot(b_chi,340.9, 341.1, n=1001)# no jump/kink visible plot(b_chi,999.9, 1000.1, n=1001)# no jump/kink visible nu <- seq(200, 1000, length=1001) plot(nu, b_chi(nu) - b_chiAsymp(nu), type='l')# b_chi(nu) > b_chiAsymp() ALWAYS ## with 'one.minus=TRUE' the reverse: b_chi(*, one.minus=TRUE) < b_chiAsymp(*, ..TRUE) plot(nu, b_chi(nu,one.minus=TRUE) - b_chiAsymp(nu,one.minus=TRUE), type='l') par(opar)# reset ### ----------- pntJW39() and pntR() ---------------------- pntR(30,30,30, verbose=2)# 600 iter pntR(30,30,ncp = 30:40) pntR(2, 10,ncp=1e5)#> C-code directly {underflow p=0, |ncp| = |delta| too large pntR(2,df=10,ncp=1e4)#> (ditto) ## t --> 0 : is it problematic ? df <- 1 df <- 10 ncp <- 1 x <- 1e-12 x <- 1e-6 (pt1 <- pntR(x * sqrt((df+2)/df), df=df+2, ncp=ncp, errmax = 1e-14)) (pt2 <- pntR(x , df=df , ncp=ncp, errmax = 1e-14)) c(pt1 - pt2, (pt1 - pt2)/x) if(FALSE) ## useful to see things debug(pntR) pntR(1e-8, df=10, ncp=1, errmax = 1e-14) pntR(1e-15, df=10, ncp=1, errmax = 1e-14) plot(function(t)pntJW39(t,30,30)) plot(function(t)pntJW39(t,100,30),-9,5, col="red", log="y") plot(function(t)pt (t,100,30),-9,5, col="blue", log="y",add=TRUE) plot(function(t)pntJW39(t,100,30),0,5, col="red", log="y")#~ lin plot(function(t)pntJW39(t,100,30),0,15, col="red", log="y") plot(function(t)pt(t,100,ncp=30),0,15, col="blue", log="y", add=TRUE) plot(function(t)pntJW39(t,100,30),0,25, col="red", log="y") plot(function(t)pntJW39(t,100,30),0,25, col="red") plot(function(t)pntJW39(t,100,30),0,45, col="red") plot(function(t)pntJW39(t,100,30),20,50, col="red", log="y") plot(function(t)pt (t,100,30),20,50, col="blue", log="y",add=TRUE) o <- par(las=1, mar=.1+c(4,4,4,4)) plot(function(t)abs(pt(t,100,30)-pntJW39(t,100,30)),20,50, col="red", log="y"); eaxis(4, at=10^-(2:8)) par(o) ## xtended x-range o <- par(las=1, mar=.1+c(4,4,4,4)) plot(function(t)abs(pt(t,100,30)-pntJW39(t,100,30)), 17, 120, n=1001, col="red", log="y"); eaxis(4) par(o) ### --------------------- qt(), qtAppr(), qtNappr(), etc --------------------- ## df=1 is pretty bad... plot(function(t)qt(t,df=1)) plot(function(t)qtAppr(t,df=1,ncp=0),col="blue",add=TRUE) ## df=2 quite a bit better.. plot(function(t)qtAppr(t,df=2,ncp=0),col="blue") plot(function(t)qt(t,df=2),add=TRUE) ## df=4 a bit better.. still only for alpha ~ in (.1, .9) plot(function(t)qtAppr(t,df=4,ncp=0),col="blue") plot(function(t)qt(t,df=4),add=TRUE) plot(function(t)abs(1-qt(t,df=4)/qtAppr(t,df=4,ncp=0)), main="rel.Error") plot(function(t)abs(1-qt(t,df=10)/qtAppr(t,df=10,ncp=0)), main="rel.Error") ## max error: 10e-5, however... still NaN's sqrt() plot(function(t)abs(1-qt(t,df=100)/qtAppr(t,df=100,ncp=0))) plot(function(t)qtAppr(t,df=100,ncp=100)) ## catastrophe : !!!! plot(function(t)qtAppr(t,df=1,ncp=100)) plot(function(t) t - pt (qtAppr(t,df=4,ncp=100),df=4,ncp=100)) ## --> pntJW39() uses MUCH better asymptotic than Abramowitz&Stegun (in pnt). plot(function(t) t - pntJW39(qtAppr(t,df=4,ncp=100),df=4,ncp=100), col='red',add=TRUE) ## Absolute Error: very small -- just proves that the two ".appr" CORRESPOND! plot(function(t) t - pntJW39(qtAppr(t,df=4,ncp=100),df=4,ncp=100)) plot(function(t)qtAppr(t,df=10,ncp=1e5)) ## Shows that pt(,, ncp=1e5) uses asymptotic form alright: plot(function(t) t - pt (qtAppr(t,df=10,ncp=1e5),df=10,ncp=1e5)) ## --> pntJW39() MUCH better ========== fitting to qtAppr !! ## ------ not necessarily better asymptotic than Abramowitz&Stegun (in pnt). plot(function(t) t - pntJW39(qtAppr(t,df=10,ncp=1e5),df=10,ncp=1e5), col='red',add=TRUE) ###--- Diverse tests, some lifted from ../R/t-nonc-fn.R ------------------------ ## [MM: the next 2 'dntJKBch(.)' were 'dntRwrong(.)' originally ] print(f. <- .dntJKBch(1:6, df=3, ncp=5, check=TRUE), digits = 4) ## now:[1] 0.0005083 0.0260607 0.1191377 0.1761045 0.1657718 0.1305411 ## was:[1] 0.0032023 0.2728377 1.5174647 2.4481320 2.4106931 1.9481189 -- wrong "but sensible" x <- seq(-1,12, by=1/16) if(FALSE) ## FIXME ?! -- internal logic error: .dntJKBch(x, df=3, ncp=5, check=TRUE) ## ## Error in (function (x, df, ncp, log = FALSE, M = 1000, check = FALSE, : ## ## exp(lterms[ii]) and terms[ii] are not equal: ## ## 'is.NA' value mismatch: 0 in current 340 in target ## -- This is now in help page examples >>> ../man/dnt.Rd <<< ## fx <- dt(x, df=3, ncp=5) ~~~~~~~~~~~~~ ## re1 <- 1 - .dntJKBch(x, df=3, ncp=5) / fx # with warnings ## re2 <- 1 - dntJKBf (x, df=3, ncp=5) / fx ## summary(warnings()) ## "In log(x * ncp * sqrt(2)/sqrt(df + x^2)) : NaNs produced" ## all.equal(re1[!is.na(re1)], re2[!is.na(re1)], tol=0)## Mean relative ...: 2.068..e-5 ## stopifnot(all.equal(re1[!is.na(re1)], re2[!is.na(re1)], tol=1e-6)) ## matplot(x, log10(abs(cbind(re1, re2))), type = "o", cex = 1/4) print(ft <- .dntJKBch(1:6, df=3, ncp=5), digits = 4) ## [1] 0.0005083 0.0260607 0.1191377 0.1761045 0.1657718 0.1305411 ## correct !! *with* the factorial ! all.equal(ft, dt(1:6, df=3, ncp=5), tol = 0) # 1.4959e-12 stopifnot(all.equal(ft, dt(1:6, df=3, ncp=5), tol = 1e-11)) ## From: Stephen Berman .. ## To: ## Subject: [R] Why does qt() return Inf with certain negative ncp values? ## Date: Mon, 13 Jun 2022 23:47:29 +0200 ## Can anyone explain why Inf appears in the following results? ## MM: options(nwarnings = 5e5) sapply(-1:-10, function(ncp) qt(1-1*(10^(-4+ncp)), 35, ncp)) ## [1] 3.6527153 3.0627759 2.4158355 1.7380812 1.0506904 0.3700821 ## [7] Inf -0.9279783 -1.5341759 -2.1085213 ## There were 1060 warnings (use warnings() to see them) summary(warnings()) ## 1060 identical warnings: ## In qt(1 - 1 * (10^(-4 + ncp)), 35, ncp) : ## full precision may not have been achieved in 'pnt{final}' nc1 <- seq(-12, -1, by=1/32) r1 <- vapply(nc1, function(ncp) qt(1 - 10^(-4+ncp), df=35, ncp=ncp), 1.23) ## There were 8264 warnings (use warnings() to see them) summary(warnings()) ## 8264 identical warnings: ## In qt(1 - 10^(-4 + ncp), df = 35, ncp = ncp) : ## full precision may not have been achieved in 'pnt{final}' ## MM: better than "1 - {lower.tail=FALSE} did not help much i <- nc1 <= -10 matplot(nc1[i], cbind(r1,r1.)[i,], type="l") ## Zoom in around -7 r2 <- sapply(seq(-6.9, -7.9, -0.1), function(ncp) qt(1-1*(10^(-4+ncp)), 35, ncp)) ## [1] -0.2268386 Inf Inf Inf -0.4857400 -0.5497784 ## [7] -0.6135402 -0.6770143 -0.7401974 -0.8030853 -0.8656810 summary(warnings()) ## 3086 identical warnings: ## In qt(1 - 1 * (10^(-4 + ncp)), 35, ncp) : ## full precision may not have been achieved in 'pnt{final}' nc2 <- seq(-60, -55, by=1/32)/8 # {1/8, 1/32: exact decimals} r2 <- vapply(nc2, function(ncp) qt(1 - 10^(-4+ncp), df=35, ncp=ncp), 1.23) ## There were 52444 warnings (use warnings() to see them) summary(warnings()) ## 52444 identical warnings: .... ## lower.tail=FALSE r2. <- vapply(nc2, function(ncp) qt(10^(-4+ncp), df=35, ncp=ncp, lower.tail=FALSE), 1.23) all.equal(r2., r2) ## "Mean relative difference: 6.740991e-06" matplot(nc2, cbind(r2,r2.), type="l") ## not why such a locally linear function should not work... ## In case it matters: ## > sessionInfo() ## R Under development (unstable) (2022-06-05 r82452) ## Platform: x86_64-pc-linux-gnu (64-bit) ## Running under: Linux From Scratch r11.0-165 ##--> MM: exploring q2 <- seq(-3/4, -1/4, by=1/128) pq2 <- pt(q2, 35, ncp=-7, lower.tail=FALSE) # no warnings here! ### ==> Providing qtU() a simple uniroot() - based inversion of pt() qpqU <- qtU(pq2, 35, ncp=-7, lower.tail=FALSE, tol=1e-10) stopifnot(all.equal(q2, qpqU, tol=1e-9)) # perfect! ## and qtAppr() is really poor in comparison plot(pq2 ~ q2, type="l", col=2, lwd=2) qpq2 <- qt (pq2, 35, ncp=-7, lower.tail=FALSE) # 6262 warnings qAp2a <- qtAppr(pq2, 35, ncp=-7, lower.tail=FALSE, method="a") qAp2b <- qtAppr(pq2, 35, ncp=-7, lower.tail=FALSE, method="b") qAp2c <- qtAppr(pq2, 35, ncp=-7, lower.tail=FALSE, method="c") summary(warnings()) ## 6262 identical warnings: ... ## plot error -- interestingly *all* positive: .. then JUMP to Inf plot (qpq2-q2 ~ q2, type="l", col=2, lwd=2); abline(h=0,lty=2) ## but the approximations are *very* bad, compared: matplot(q2, cbind(qpq2, qAp2a,qAp2b,qAp2c)-q2, type="l", col=2:5, lwd=2) ## leave away the "b" which is particularly bad (here): matplot(q2, cbind(qpq2, qAp2a,qAp2c)-q2, type="l", col=2:5, lwd=2) ## absolute values & log-scale: matplot(q2, abs( cbind(qpq2, qAp2a,qAp2c) - q2), log="y", type="l", col=2:5, lwd=2) ## Idea: what about *starting* with qtAppr() and then do 1 or a few *Newton* steps? ## unfinished ... mat2 <- cbind(q2, pq2, qpq2, Dq=qpq2-q2, relE=relErrV(q2, qpq2)) tail(mat2, 10) ###=== Large df ; R currently uses "primitive" (<==> qtNappr(*, k=0)) for df > 1e20 ## ======== ##--- *CENTRAL* case (ncp == 0) ---------- if(!dev.interactive(orNone=TRUE)) { dev.off(); pdf("t-nonc_P-3.pdf") } ## originally from ../man/qtAppr.Rd : qts <- function(p, df, lower.tail = TRUE, log.p = FALSE, tolUa = 1e-14) { cbind(qt = qt (p, df=df, lower.tail=lower.tail, log.p=log.p) , qtU = qtU (p, df=df, lower.tail=lower.tail, log.p=log.p) , qtUa = qtU (p, df=df, lower.tail=lower.tail, log.p=log.p, tol = tolUa) , qtN0 = qtNappr(p, df=df, lower.tail=lower.tail, log.p=log.p, k=0) , qtN1 = qtNappr(p, df=df, lower.tail=lower.tail, log.p=log.p, k=1) , qtN2 = qtNappr(p, df=df, lower.tail=lower.tail, log.p=log.p, k=2) , qtN3 = qtNappr(p, df=df, lower.tail=lower.tail, log.p=log.p, k=3) , qtN4 = qtNappr(p, df=df, lower.tail=lower.tail, log.p=log.p, k=4) ) } p.qtNappr <- function(qtsR, p, ip, errMax = 1e-17, do.diff=FALSE) { stopifnot(is.matrix(qtsR), (n <- length(p)) == nrow(qtsR), 1 <= ip, ip <= n) (iN <- startsWith(prefix="qtN", colnames(qtsR))) # also "drop" qtU()'s result (dqN <- (qtsR[,iN] - qtsR[,1])[ip,]) if(do.diff) ## difference matplot(p[ip], dqN, type="l", col=2:6, main = paste0("difference qtNappr(p,df) - qt(p,df), df=",df), xlab=quote(p)) ## |difference| matplot(p[ip], pmax(abs(dqN), errMax), 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") ## Rel.err. matplot(p[ip], pmax(abs(dqN/qtsR[ip,"qt"]), errMax), 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") invisible(dqN) } 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[ii], qsp1c[ii,], type="l") # "all on top" p.qtNappr(qsp1c, p, ip=ii, do.diff=TRUE) df <- 2000 # <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< qsp1c <- qts(p, df=df) p.qtNappr(qsp1c, p, ip=ii) ## qtAppr is really worse even than k=2 here (df=2000): qtA <- qtAppr(p, df=df, ncp=0) lines(p, pmax(1e-17, abs(qtA/qsp1c[,1] - 1)), col=2, lwd=2, lty=2) legend(0.6, 1e-9, "qtAppr()", bty="n",cex=.8, col=2, lwd=2, lty=2) ## qtA is between k=1 and k=2 ... print(digits=4, cbind(p, cbind(qsp1c[,-1], qtA) - qsp1c[,1])[ii,]) df <- 1e8 # <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< large <<<<<<<<<<<< ## more p-values close to {0, 1} p. <- 2^-(2*(15:5)); p. <- sort(c(p., p, 1 - rev(p.))); plot(diff(p.)) qspL <- qts(p., df=df) ii <- which(is.finite(qspL[,"qt"])) p.qtNappr(qspL, p=p., ip=ii) lines(p., pmax(1e-17, abs(qspL[,"qtU"] /qspL[,"qt"] - 1)), col=2, lwd=2, lty=2) legend(0.6, 1e-9, "qtU(tol=1e-5)", bty="n",cex=.8, col=2, lwd=2, lty=2) lines(p., pmax(1e-17, abs(qspL[,"qtUa"]/qspL[,"qt"] - 1)), col=4, lwd=2, lty=3) legend(0.6, 1e-14, "qtU(tol=1e-14)", bty="n",cex=.8, col=4, lwd=2, lty=3) print(qspL, digits=6)