R version 3.1.0 RC (2014-04-04 r65373) -- "Spring Dance"
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> #### d|ensity
> #### p|robability (cumulative)
> #### q|uantile
> #### r|andom number generation
> ####
> #### Functions for ``d/p/q/r''
>
> .ptime <- proc.time()
> F <- FALSE
> T <- TRUE
>
> ###-- these are identical in ./arith-true.R ["fixme": use source(..)]
> opt.conformance <- 0
> Meps <- .Machine $ double.eps
> xMax <- .Machine $ double.xmax
> options(rErr.eps = 1e-30)
> rErr <- function(approx, true, eps = .Options$rErr.eps)
+ {
+ if(is.null(eps)) { eps <- 1e-30; options(rErr.eps = eps) }
+ ifelse(Mod(true) >= eps,
+ 1 - approx / true, # relative error
+ true - approx) # absolute error (e.g. when true=0)
+ }
> ## Numerical equality: Here want "rel.error" almost always:
> All.eq <- function(x,y) {
+ all.equal.numeric(x,y, tolerance = 64*.Machine$double.eps,
+ scale = max(0, mean(abs(x), na.rm=TRUE)))
+ }
> if(!interactive()) set.seed(123)
>
> ## The prefixes of ALL the PDQ & R functions
> PDQRinteg <- c("binom", "geom", "hyper", "nbinom", "pois","signrank","wilcox")
> PDQR <- c(PDQRinteg, "beta", "cauchy", "chisq", "exp", "f", "gamma",
+ "lnorm", "logis", "norm", "t","unif","weibull")
> PQonly <- c("tukey")
>
> ###--- Discrete Distributions --- Consistency Checks pZZ = cumsum(dZZ)
>
> ##for(pre in PDQRinteg) { n <- paste("d",pre,sep=""); cat(n,": "); str(get(n))}
>
> ##__ 1. Binomial __
>
> ## Cumulative Binomial '==' Cumulative F :
> ## Abramowitz & Stegun, p.945-6; 26.5.24 AND 26.5.28 :
> n0 <- 50; n1 <- 16; n2 <- 20; n3 <- 8
> for(n in rbinom(n1, size = 2*n0, p = .4)) {
+ cat("n=",n,": ")
+ for(p in c(0,1,rbeta(n2, 2,4))) {
+ cat(".")
+ for(k in rbinom(n3, size = n, prob = runif(1))) {
+ ## For X ~ Bin(n,p), compute 1 - P[X > k] = P[X <= k] in three ways:
+ tst1 <- all.equal( pbinom(0:k, size = n, prob = p),
+ cumsum(dbinom(0:k, size = n, prob = p)))
+ tst <- all.equal(if(k==n || p==0) 1 else
+ pf((k+1)/(n-k)*(1-p)/p, df1=2*(n-k), df2=2*(k+1)),
+ sum(dbinom(0:k, size = n, prob = p)))
+ if(!isTRUE(tst1) || !isTRUE(tst)) {
+ cat("n=", n,"; p =",format(p),". k =",k)
+ if(!isTRUE(tst1)) cat("; tst1=",tst1)
+ if(!isTRUE(tst )) cat("; tst=", tst)
+ cat("\n")
+ }
+ }
+ }
+ cat("\n")
+ }
n= 39 : ......................
n= 50 : ......................
n= 35 : ......................
n= 44 : ......................
n= 39 : ......................
n= 44 : ......................
n= 37 : ......................
n= 45 : ......................
n= 44 : ......................
n= 40 : ......................
n= 33 : ......................
n= 42 : ......................
n= 41 : ......................
n= 38 : ......................
n= 38 : ......................
n= 47 : ......................
>
> ##__ 2. Geometric __
> for(pr in seq(1e-10,1,len=15)) { # p=0 is not a distribution
+ print(All.eq((dg <- dgeom(0:10, pr)),
+ pr * (1-pr)^(0:10)))
+ print(All.eq(cumsum(dg), pgeom(0:10, pr)))
+ }
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
>
> ##__ 3. Hypergeometric __
>
> m <- 10; n <- 7
> for(k in 2:m) {
+ x <- 0:(k+1)
+ print(All.eq(phyper(x, m, n, k), cumsum(dhyper(x, m, n, k))))
+ }
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
>
> ##__ 4. Negative Binomial __
>
> ## PR #842
> for(size in seq(0.8,2, by=.1))
+ print(all.equal(cumsum(dnbinom(0:7, size, .5)),
+ pnbinom(0:7, size, .5)))
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
> All.eq(pnbinom(c(1,3), .9, .5), c(0.777035760338812, 0.946945347071519))
[1] TRUE
>
> ##__ 5. Poisson __
>
> all(dpois(0:5,0) == c(1, rep(0,5)))
[1] TRUE
> all(dpois(0:5,0, log=TRUE) == c(0, rep(-Inf, 5)))
[1] TRUE
>
> ## Cumulative Poisson '==' Cumulative Chi^2 :
> ## Abramowitz & Stegun, p.941 : 26.4.21 (26.4.2)
> n1 <- 20; n2 <- 16
> for(lambda in rexp(n1))
+ for(k in rpois(n2, lambda)) {
+ tst <- all.equal(1 - pchisq(2*lambda, 2*(1+ 0:k)),
+ pp <- cumsum(dpois(0:k, lambda=lambda)),
+ tolerance = 100*Meps)
+ if(!isTRUE(tst))
+ cat("lambda=", format(lambda),". k =",k, " --> tst=", tst,"\n")
+ tst2 <- all.equal(pp, ppois(0:k, lambda=lambda), tolerance = 100*Meps)
+ if(!isTRUE(tst2))
+ cat("lambda=", format(lambda),". k =",k, " --> tst2=", tst2,"\n")
+ tst3 <- all.equal(1 - pp, ppois(0:k, lambda=lambda, lower.tail = FALSE))
+ if(!isTRUE(tst3))
+ cat("lambda=", format(lambda),". k =",k, " --> tst3=", tst3,"\n")
+ }
>
>
> ##__ 6. SignRank __
> for(n in rpois(32, lam=8)) {
+ x <- -1:(n + 4)
+ if(!isTRUE(eq <- All.eq(psignrank(x, n), cumsum(dsignrank(x, n)))))
+ print(eq)
+ }
>
> ##__ 7. Wilcoxon (symmetry & cumulative) __
> is.sym <- TRUE
> for(n in rpois(5, lam=6))
+ for(m in rpois(15, lam=8)) {
+ x <- -1:(n*m + 1)
+ fx <- dwilcox(x, n, m)
+ Fx <- pwilcox(x, n, m)
+ is.sym <- is.sym & all(fx == dwilcox(x, m, n))
+ if(!isTRUE(eq <- All.eq(Fx, cumsum(fx))))
+ print(eq)
+ }
> is.sym
[1] TRUE
>
>
> ###-------- Continuous Distributions ----------
>
> ##--- Gamma (incl. central chi^2) Density :
> x <- round(rgamma(100, shape = 2),2)
> for(sh in round(rlnorm(30),2)) {
+ Ga <- gamma(sh)
+ for(sig in round(rlnorm(30),2)) {
+ tst <- all.equal((d1 <- dgamma( x, shape = sh, scale = sig)),
+ (d2 <- dgamma(x/sig, shape = sh, scale = 1) / sig),
+ tolerance = 1e-14)## __ad interim__ was 1e-15
+ if(!isTRUE(tst))
+ cat("ERROR: dgamma() doesn't scale:",tst,"\n",
+ " x =", formatC(x),"\n shape,scale=",formatC(c(sh, sig)),"\n")
+ tst <- All.eq(d1, (d3 <- 1/(Ga * sig^sh) * x^(sh-1) * exp(-x/sig)))
+ if(!isTRUE(tst))
+ cat("NOT Equal:",tst,"\n x =", formatC(x),
+ "\n shape,scale=",formatC(c(sh, sig)),"\n")
+ }
+ }
> pgamma(1,Inf,scale=Inf) == 0
[1] TRUE
> ## Also pgamma(Inf,Inf) == 1 for which NaN was slightly more appropriate
> all(is.nan(c(pgamma(Inf, 1,scale=Inf),
+ pgamma(Inf,Inf,scale=Inf))))
[1] TRUE
Warning messages:
1: In pgamma(Inf, 1, scale = Inf) : NaNs produced
2: In pgamma(Inf, Inf, scale = Inf) : NaNs produced
> scLrg <- c(2,100, 1e300*c(.1, 1,10,100), 1e307, xMax, Inf)
> stopifnot(pgamma(Inf, 1, scale=xMax) == 1,
+ pgamma(xMax,1, scale=Inf) == 0,
+ all.equal(pgamma(1e300, 2, scale= scLrg, log=TRUE),
+ c(0, 0, -0.000499523968713701, -1.33089326820406,
+ -5.36470502873211, -9.91015144019122,
+ -32.9293385491433, -38.707517174609, -Inf),
+ tolerance = 2e-15)
+ )
>
> p <- 7e-4; df <- 0.9
> abs(1-c(pchisq(qchisq(p, df),df)/p, # was 2.31e-8 for R <= 1.8.1
+ pchisq(qchisq(1-p, df,lower=FALSE),df,lower=FALSE)/(1-p),# was 1.618e-11
+ pchisq(qchisq(log(p), df,log=TRUE),df, log=TRUE)/log(p), # was 3.181e-9
+ pchisq(qchisq(log1p(-p),df,log=T,lower=F),df, log=T,lower=F)/log1p(-p)
+ )# 32b-i386: (2.2e-16, 0,0, 3.3e-16); Opteron: (2.2e-16, 0,0, 2.2e-15)
+ ) < 1e-14
[1] TRUE TRUE TRUE TRUE
>
> ##-- non central Chi^2 :
> xB <- c(2000,1e6,1e50,Inf)
> for(df in c(0.1, 1, 10))
+ for(ncp in c(0, 1, 10, 100)) stopifnot(pchisq(xB, df=df, ncp=ncp) == 1)
> all.equal(qchisq(0.025,31,ncp=1,lower.tail=FALSE),# inf.loop PR#875
+ 49.7766246561514, tolerance = 1e-11)
[1] TRUE
> for(df in c(0.1, 0.5, 1.5, 4.7, 10, 20,50,100)) {
+ cat("df =", formatC(df, wid=3))
+ xx <- c(10^-(5:1), .9, 1.2, df + c(3,7,20,30,35,38))
+ pp <- pchisq(xx, df=df, ncp = 1) #print(pp)
+ dtol <- 1e-12 *(if(2 < df && df <= 50) 64 else if(df > 50) 20000 else 501)
+ print(all.equal(xx, qchisq(pp, df=df, ncp=1), tolerance = dtol))# TRUE
+ ##or print(mapply(rErr, xx, qchisq(pp, df=df,ncp=1)), digits = 3)
+ }
df = 0.1[1] TRUE
df = 0.5[1] TRUE
df = 1.5[1] TRUE
df = 4.7[1] TRUE
df = 10[1] TRUE
df = 20[1] TRUE
df = 50[1] TRUE
df = 100[1] TRUE
>
> ## p ~= 1 (<==> 1-p ~= 0) -- gave infinite loop in R <= 1.8.1 -- PR#6421
> options(warn=-1) # ignore warnings from R's version of log1p
> psml <- 2^-(10:54)
> q0 <- qchisq(psml, df=1.2, ncp=10, lower.tail=FALSE)
> q1 <- qchisq(1-psml, df=1.2, ncp=10) # inaccurate in the tail
> p0 <- pchisq(q0, df=1.2, ncp=10, lower.tail=FALSE)
> p1 <- pchisq(q1, df=1.2, ncp=10, lower.tail=FALSE)
> iO <- 1:30
> all.equal(q0[iO], q1[iO], tolerance = 1e-5)
[1] TRUE
> all.equal(p0[iO], psml[iO])
[1] TRUE
> options(warn=0)
>
> ##--- Beta (need more):
>
> ## big a & b (PR #643)
> summary(a <- rlnorm(20, 5.5))
Min. 1st Qu. Median Mean 3rd Qu. Max.
54.74 180.10 279.40 530.70 785.00 2842.00
> summary(b <- rlnorm(20, 6.5))
Min. 1st Qu. Median Mean 3rd Qu. Max.
139.7 459.5 749.9 1113.0 1479.0 5387.0
> pab <- expand.grid(seq(0,1,by=.1), a, b)
> p <- pab[,1]; a <- pab[,2]; b <- pab[,3]
> all.equal(dbeta(p,a,b), exp(pab <- dbeta(p,a,b, log = TRUE)), tolerance = 1e-11)
[1] TRUE
> sample(pab, 50)
[1] 0.5684734 -15.3812126 -Inf -647.0316608 -1776.0647576
[6] -123.1916325 -740.1238787 -808.4127354 -290.7239227 -407.6274849
[11] -Inf -869.0911667 -79.1746272 -627.5238085 -Inf
[16] -1510.0939127 -20.4159620 -131.6241954 -702.3419988 -Inf
[21] 2.1972704 -Inf -Inf -495.6416130 -6.9542025
[26] -27.0548912 -708.4810716 -164.8837570 -413.8854518 -Inf
[31] -305.9834725 -20.0267958 -17.7687346 -158.4013464 -32.5114184
[36] -2906.1239814 -65.6720075 -20.6466431 -Inf -66.1286480
[41] -336.1188692 -Inf -Inf -142.1023940 -19.7794842
[46] -Inf -Inf -12.0769413 -Inf -124.4908467
>
>
> ##--- Normal (& Lognormal) :
>
> qnorm(0) == -Inf && qnorm(-Inf, log = TRUE) == -Inf
[1] TRUE
> qnorm(1) == Inf && qnorm(0, log = TRUE) == Inf
[1] TRUE
>
> is.nan(qnorm(1.1)) &&
+ is.nan(qnorm(-.1)) # + warn
[1] TRUE
Warning messages:
1: In qnorm(1.1) : NaNs produced
2: In qnorm(-0.1) : NaNs produced
>
> x <- c(-Inf, -1e100, 1:6, 1e200, Inf)
> rbind(d.s0 =dnorm(x,3,s=0), p.s0 = pnorm(x,3,s=0),
+ d.sI =dnorm(x,3,s=Inf), p.sI = pnorm(x,3,s=Inf))
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
d.s0 0 0.0 0.0 0.0 Inf 0.0 0.0 0.0 0.0 0
p.s0 0 0.0 0.0 0.0 1.0 1.0 1.0 1.0 1.0 1
d.sI 0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0
p.sI 0 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1
>
> ## 3 Test data from Wichura (1988) :
> all.equal(qnorm(c( 0.25, .001, 1e-20)),
+ c(-0.6744897501960817, -3.090232306167814, -9.262340089798408),
+ tolerance = 1e-15)
[1] TRUE
> # extreme tail -- available on log scale only:
> all.equal(qnorm(-1e5, log = TRUE), -447.1974945)
[1] TRUE
>
> z <- rnorm(1000); all.equal(pnorm(z), 1 - pnorm(-z), tolerance = 1e-15)
[1] TRUE
> z <- c(-Inf,Inf,NA,NaN, rt(1000, df=2))
> z.ok <- z > -37.5 | !is.finite(z)
> for(df in 1:10) if(!isTRUE(all.equal(pt(z, df), 1 - pt(-z,df), tolerance = 1e-15)))
+ cat("ERROR -- df = ", df, "\n")
> All.eq(pz <- pnorm(z), 1 - pnorm(z, lower=FALSE))
[1] TRUE
> All.eq(pz, pnorm(-z, lower=FALSE))
[1] TRUE
> All.eq(log(pz[z.ok]), pnorm(z[z.ok], log=TRUE))
[1] TRUE
> y <- seq(-70,0, by = 10)
> cbind(y, "log(pnorm(y))"= log(pnorm(y)), "pnorm(y, log=T)"= pnorm(y, log=TRUE))
y log(pnorm(y)) pnorm(y, log=T)
[1,] -70 -Inf -2455.1676378
[2,] -60 -Inf -1805.0135607
[3,] -50 -Inf -1254.8313611
[4,] -40 -Inf -804.6084420
[5,] -30 -454.3212440 -454.3212440
[6,] -20 -203.9171554 -203.9171554
[7,] -10 -53.2312852 -53.2312852
[8,] 0 -0.6931472 -0.6931472
> y <- c(1:15, seq(20,40, by=5))
> cbind(y, "log(pnorm(y))"= log(pnorm(y)), "pnorm(y, log=T)"= pnorm(y, log=TRUE),
+ "log(pnorm(-y))"= log(pnorm(-y)), "pnorm(-y, log=T)"= pnorm(-y, log=TRUE))
y log(pnorm(y)) pnorm(y, log=T) log(pnorm(-y)) pnorm(-y, log=T)
[1,] 1 -1.727538e-01 -1.727538e-01 -1.841022 -1.841022
[2,] 2 -2.301291e-02 -2.301291e-02 -3.783184 -3.783184
[3,] 3 -1.350810e-03 -1.350810e-03 -6.607726 -6.607726
[4,] 4 -3.167174e-05 -3.167174e-05 -10.360101 -10.360101
[5,] 5 -2.866516e-07 -2.866516e-07 -15.064998 -15.064998
[6,] 6 -9.865877e-10 -9.865876e-10 -20.736769 -20.736769
[7,] 7 -1.279865e-12 -1.279813e-12 -27.384307 -27.384307
[8,] 8 -6.661338e-16 -6.220961e-16 -35.013437 -35.013437
[9,] 9 0.000000e+00 -1.128588e-19 -43.628149 -43.628149
[10,] 10 0.000000e+00 -7.619853e-24 -53.231285 -53.231285
[11,] 11 0.000000e+00 -1.910660e-28 -63.824934 -63.824934
[12,] 12 0.000000e+00 -1.776482e-33 -75.410673 -75.410673
[13,] 13 0.000000e+00 -6.117164e-39 -87.989720 -87.989720
[14,] 14 0.000000e+00 -7.793537e-45 -101.563034 -101.563034
[15,] 15 0.000000e+00 -3.670966e-51 -116.131385 -116.131385
[16,] 20 0.000000e+00 -2.753624e-89 -203.917155 -203.917155
[17,] 25 0.000000e+00 -3.056697e-138 -316.639408 -316.639408
[18,] 30 0.000000e+00 -4.906714e-198 -454.321244 -454.321244
[19,] 35 0.000000e+00 -1.124911e-268 -616.975101 -616.975101
[20,] 40 0.000000e+00 0.000000e+00 -Inf -804.608442
> ## Symmetry:
> y <- c(1:50,10^c(3:10,20,50,150,250))
> y <- c(-y,0,y)
> for(L in c(FALSE,TRUE))
+ stopifnot(identical(pnorm(-y, log= L),
+ pnorm(+y, log= L, lower=FALSE)))
>
> ## Log norm
> All.eq(pz, plnorm(exp(z)))
[1] TRUE
>
>
> ###========== p <-> q Inversion consistency =====================
> ok <- 1e-5 < pz & pz < 1 - 1e-5
> all.equal(z[ok], qnorm(pz[ok]), tolerance = 1e-12)
[1] TRUE
>
> ###===== Random numbers -- first, just output:
>
> set.seed(123)
> # .Random.seed <- c(0L, 17292L, 29447L, 24113L)
> n <- 20
> ## for(pre in PDQR) { n <- paste("r",pre,sep=""); cat(n,": "); str(get(n))}
> (Rbeta <- rbeta (n, shape1 = .8, shape2 = 2) )
[1] 0.554206761 0.387924405 0.012541339 0.257889595 0.236064413 0.008248085
[7] 0.136510774 0.618501837 0.028641308 0.151670292 0.242835922 0.551824427
[13] 0.127798688 0.087335901 0.509096247 0.382121566 0.773494885 0.639404676
[19] 0.066559813 0.227487378
> (Rbinom <- rbinom (n, size = 55, prob = pi/16) )
[1] 7 13 15 10 12 7 10 9 13 10 13 13 13 10 13 12 12 3 11 8
> (Rcauchy <- rcauchy (n, location = 12, scale = 2) )
[1] 17.042930 6.592944 15.980645 12.728113 13.921385 8.570544 19.557102
[8] 10.430404 12.669798 21.641273 11.905411 11.301537 11.254793 13.226057
[15] 12.870672 8.167569 15.735143 8.272588 15.159679 13.338095
> (Rchisq <- rchisq (n, df = 3) )
[1] 3.8641030 1.8367371 1.2291085 1.9151780 2.5901414 0.4522238 0.1018120
[8] 0.4323865 2.4551985 2.8313420 1.1215175 8.6109152 0.2215425 2.6531221
[15] 2.9968656 2.1074132 4.2698236 8.2015930 1.1384935 0.6041294
> (Rexp <- rexp (n, rate = 2) )
[1] 0.719726313 0.232135206 0.134439025 0.731650280 0.768696989 0.002299563
[7] 0.554382734 0.149985159 0.596001504 0.557464352 0.033687945 0.240334360
[13] 0.785227170 0.129973053 0.928461144 0.231609810 0.118017873 0.591049711
[19] 0.029835686 0.201619221
> (Rf <- rf (n, df1 = 12, df2 = 6) )
[1] 0.7630651 0.4158576 0.4052799 0.8862771 2.0435073 2.5383554 0.8259689
[8] 1.5831584 0.6362728 0.7221707 1.1665204 0.4821276 1.1985013 0.8955592
[15] 1.5186453 0.7333794 0.3795005 0.7464008 1.5543317 5.8525268
> (Rgamma <- rgamma (n, shape = 2, scale = 5) )
[1] 11.843056 6.993596 12.413160 5.980309 5.970495 15.674812 7.382163
[8] 11.343789 5.392676 3.766812 15.331767 2.451095 13.046780 10.999054
[15] 9.929606 6.239055 3.588397 26.196937 3.647236 23.051571
> (Rgeom <- rgeom (n, prob = pi/16) )
[1] 8 2 4 2 3 5 5 4 1 8 1 6 1 6 8 3 8 7 5 0
> (Rhyper <- rhyper (n, m = 40, n = 30, k = 20) )
[1] 12 11 12 10 13 11 13 13 8 12 12 15 11 11 11 12 9 11 12 12
> (Rlnorm <- rlnorm (n, meanlog = -1, sdlog = 3) )
[1] 1.873201e-02 5.609747e+01 9.793124e-02 4.203730e-02 9.015415e-03
[6] 7.795979e-03 6.574837e-02 2.348924e+00 1.027326e+01 3.073340e+00
[11] 1.235670e-01 4.401014e-01 4.443226e-02 4.278129e-02 5.227596e+00
[16] 1.747860e-02 1.297853e+02 2.805625e-01 7.002040e-01 4.013193e-02
> (Rlogis <- rlogis (n, location = 12, scale = 2) )
[1] 10.139287 14.905198 7.466552 15.060325 11.415297 14.260524 13.347886
[8] 11.554385 13.039986 -3.344515 9.436395 13.741216 9.411881 14.951552
[15] 10.378811 13.579174 17.257244 7.933965 8.157203 13.491243
> (Rnbinom <- rnbinom (n, size = 7, prob = .01) )
[1] 563 315 519 707 614 637 560 1086 1088 842 522 787 576 673 607
[16] 443 1023 590 663 947
> (Rnorm <- rnorm (n, mean = -1, sd = 3) )
[1] -0.3732778 -5.6447208 -4.5214067 2.5364456 -2.2579283 -6.5577259
[7] 0.4577794 -2.4157378 1.5402158 1.7773118 -1.9624917 -4.9029209
[13] 6.3787040 1.3350950 1.0696427 -6.1133966 2.3161164 -1.9415584
[19] -2.0400782 2.1153214
> (Rpois <- rpois (n, lambda = 12) )
[1] 10 15 14 9 15 12 16 7 7 14 11 14 16 12 15 7 7 13 15 12
> (Rsignrank<- rsignrank(n, n = 47) )
[1] 531 522 385 583 731 461 542 590 485 582 571 475 702 537 395 571 603 515 477
[20] 609
> (Rt <- rt (n, df = 11) )
[1] -0.499846093 -1.670698483 -0.729757704 0.009882459 1.518830804
[6] -1.105480914 -0.073913420 -0.323601935 -0.309210484 1.071490670
[11] -1.730648357 -0.283141485 0.205309956 0.409676847 3.137494386
[16] 0.187070891 -1.154130926 0.970408359 -1.340678412 1.437769232
> ## Rt2 below (to preserve the following random numbers!)
> (Runif <- runif (n, min = .2, max = 2) )
[1] 1.3077353 0.6915398 0.3146723 0.3301459 1.7723202 0.8530850 1.9021199
[8] 1.2049741 1.1227901 1.9059890 0.3976429 1.6740336 0.8792158 1.2019539
[15] 0.7649681 0.3482172 1.6038876 0.5478225 1.3800538 0.9832647
> (Rweibull <- rweibull (n, shape = 3, scale = 2) )
[1] 2.3507552 0.3498804 1.8610901 3.0250054 0.8043691 1.0338586 1.5966081
[8] 2.1295946 1.1916615 0.7320637 1.0860054 2.5452588 0.6323843 1.6726976
[15] 3.4711560 2.5185134 3.4519830 1.1740161 1.2526410 0.8298120
> (Rwilcox <- rwilcox (n, m = 13, n = 17) )
[1] 111 96 66 97 152 131 167 97 99 160 73 97 59 97 109 103 100 108 94
[20] 100
> (Rt2 <- rt (n, df = 1.01))
[1] -0.94684312 -16.69743680 -0.31323806 -1.74808343 -0.03944642
[6] -2.24078160 -1.21454266 0.51014471 -9.72731278 -0.24196217
[11] 6.64038105 -0.37700450 -0.37380391 3.17514899 -0.17936741
[16] -1.16754001 0.73536283 -1.76538915 2.27417497 1.49299645
>
> (Pbeta <- pbeta (Rbeta, shape1 = .8, shape2 = 2) )
[1] 0.84605840 0.69836904 0.05389148 0.53895245 0.50764485 0.03861536
[7] 0.34373485 0.88868811 0.10358645 0.37126511 0.51751532 0.84433212
[13] 0.32741433 0.24605563 0.81153016 0.69214938 0.96181357 0.90094249
[19] 0.19989455 0.49493375
> (Pbinom <- pbinom (Rbinom, size = 55, prob = pi/16) )
[1] 0.128811009 0.822082686 0.939887897 0.473358125 0.725714645 0.128811009
[7] 0.473358125 0.340181320 0.822082686 0.473358125 0.822082686 0.822082686
[13] 0.822082686 0.473358125 0.822082686 0.725714645 0.725714645 0.002917878
[19] 0.606468256 0.221684127
> (Pcauchy <- pcauchy (Rcauchy, location = 12, scale = 2) )
[1] 0.8798165 0.1127710 0.8517979 0.6111354 0.7436195 0.1680556 0.9176468
[8] 0.2881958 0.6028646 0.9348927 0.4849570 0.3930511 0.3864691 0.6750527
[15] 0.6306957 0.1531019 0.8435165 0.1567581 0.8203732 0.6876911
> (Pchisq <- pchisq (Rchisq, df = 3) )
[1] 0.723482597 0.393026646 0.253968309 0.409802926 0.540779498 0.070744975
[7] 0.008380908 0.066526148 0.516557851 0.581633585 0.228117628 0.965062738
[13] 0.025961119 0.551747428 0.607891302 0.449585516 0.766236467 0.957975966
[19] 0.232209338 0.104513603
> (Pexp <- pexp (Rexp, rate = 2) )
[1] 0.762942518 0.371406447 0.235763552 0.768528971 0.785059488 0.004588567
[7] 0.670033937 0.259159790 0.696387483 0.672061341 0.065156264 0.381630263
[13] 0.792049319 0.228906859 0.843847516 0.370745580 0.210247557 0.693365692
[19] 0.057925925 0.331847235
> (Pf <- pf (Rf, df1 = 12, df2 = 6) )
[1] 0.32410218 0.09232470 0.08630331 0.40259602 0.80440198 0.86907389
[7] 0.36505563 0.70322821 0.23758524 0.29664970 0.55170692 0.13291359
[13] 0.56607937 0.40821091 0.68428769 0.30422992 0.07227779 0.31298492
[19] 0.69494050 0.97980295
> (Pgamma <- pgamma (Rgamma, shape = 2, scale = 5) )
[1] 0.6846621 0.4077251 0.7091198 0.3359489 0.3352389 0.8201234 0.4342559
[8] 0.6618778 0.2931040 0.1745504 0.8105456 0.0872524 0.7344167 0.6453838
[15] 0.5901700 0.3545801 0.1619732 0.9669094 0.1661030 0.9441847
> (Pgeom <- pgeom (Rgeom, prob = pi/16) )
[1] 0.8601686 0.4809591 0.6647753 0.4809591 0.5828725 0.7305965 0.7305965
[8] 0.6647753 0.3541459 0.8601686 0.3541459 0.7834938 0.3541459 0.7834938
[15] 0.8601686 0.5828725 0.8601686 0.8260047 0.7305965 0.1963495
> (Phyper <- phyper (Rhyper, m = 40, n = 30, k = 20) )
[1] 0.71494883 0.51295659 0.71494883 0.30864260 0.86627413 0.51295659
[7] 0.86627413 0.86627413 0.05904998 0.71494883 0.71494883 0.98680472
[13] 0.51295659 0.51295659 0.51295659 0.71494883 0.15132082 0.51295659
[19] 0.71494883 0.71494883
> (Plnorm <- plnorm (Rlnorm, meanlog = -1, sdlog = 3) )
[1] 0.16047510 0.95310121 0.32954742 0.23481968 0.10817853 0.09944582
[7] 0.28299287 0.73170766 0.86646776 0.76039954 0.35805697 0.52382260
[13] 0.24053070 0.23661975 0.81182742 0.15491172 0.97472581 0.46401663
[19] 0.58493655 0.23009692
> (Plogis <- plogis (Rlogis, location = 12, scale = 2) )
[1] 0.2828523996 0.8103980662 0.0939166071 0.8220300712 0.4274282807
[6] 0.7558872597 0.6623855066 0.4445273969 0.6271461844 0.0004653491
[11] 0.2172435140 0.7048722457 0.2151664882 0.8139337213 0.3077638743
[16] 0.6877426857 0.9326810790 0.1157796648 0.1277056879 0.6782238574
> (Pnbinom <- pnbinom (Rnbinom, size = 7, prob = .01) )
[1] 0.34539127 0.04498289 0.27616523 0.57177025 0.42755552 0.46435569
[7] 0.34059007 0.91950991 0.92034921 0.74409223 0.28078034 0.68045166
[13] 0.36627489 0.52069882 0.41628414 0.16798077 0.88887110 0.38885301
[19] 0.50524696 0.83942274
> (Pnorm <- pnorm (Rnorm, mean = -1, sd = 3) )
[1] 0.58273974 0.06078223 0.12023713 0.88076412 0.33749500 0.03197163
[7] 0.68649035 0.31849459 0.80142958 0.82271739 0.37416939 0.09663374
[13] 0.99304478 0.78182307 0.75486546 0.04414707 0.86550081 0.37681641
[19] 0.36441108 0.85046748
> (Ppois <- ppois (Rpois, lambda = 12) )
[1] 0.3472294 0.8444157 0.7720245 0.2423922 0.8444157 0.5759652 0.8987090
[8] 0.0895045 0.0895045 0.7720245 0.4615973 0.7720245 0.8987090 0.5759652
[15] 0.8444157 0.0895045 0.0895045 0.6815356 0.8444157 0.5759652
> (Psignrank<- psignrank(Rsignrank, n = 47) )
[1] 0.36663746 0.33169387 0.02921581 0.58098562 0.96191228 0.14044836
[7] 0.41083885 0.60942723 0.20480997 0.57688473 0.53132960 0.17614726
[13] 0.92806867 0.39057277 0.03720129 0.53132960 0.66065154 0.30545809
[19] 0.18167093 0.68340768
> (Pt <- pt (Rt, df = 11) )
[1] 0.31351556 0.06147847 0.24039508 0.50385399 0.92149468 0.14627181
[7] 0.47120313 0.37615421 0.38146974 0.84655256 0.05571530 0.39116296
[13] 0.57946003 0.65504612 0.99527474 0.57249461 0.13645270 0.82365134
[19] 0.10352754 0.91083331
> (Pt2 <- pt (Rt2, df = 1.01) )
[1] 0.25806154 0.01854802 0.40317545 0.16447686 0.48742623 0.13262717
[7] 0.21849621 0.65049354 0.03193557 0.42428168 0.95323005 0.38499762
[13] 0.38589278 0.90387188 0.44339511 0.22470366 0.70232456 0.16312545
[19] 0.86911865 0.81300659
> (Punif <- punif (Runif, min = .2, max = 2) )
[1] 0.61540849 0.27307767 0.06370684 0.07230328 0.87351121 0.36282500
[7] 0.94562215 0.55831894 0.51266115 0.94777166 0.10980160 0.81890755
[13] 0.37734211 0.55664105 0.31387119 0.08234288 0.77993754 0.19323473
[19] 0.65558547 0.43514705
> (Pweibull <- pweibull (Rweibull, shape = 3, scale = 2) )
[1] 0.802851673 0.005339578 0.553257134 0.968573336 0.062983442 0.129016046
[7] 0.398753717 0.700984544 0.190653725 0.047857615 0.147946029 0.872690264
[13] 0.031117650 0.442898991 0.994635553 0.864236080 0.994152898 0.183125938
[19] 0.217836412 0.068933714
> (Pwilcox <- pwilcox (Rwilcox, m = 13, n = 17) )
[1] 0.51643195 0.28170171 0.03248009 0.29584273 0.96068495 0.80746123
[7] 0.99230879 0.29584273 0.32502911 0.98253011 0.06135823 0.29584273
[13] 0.01564120 0.29584273 0.48356805 0.38647276 0.34003726 0.46716310
[19] 0.25440692 0.34003726
>
> dbeta (Rbeta, shape1 = .8, shape2 = 2)
[1] 0.7223732 1.0651681 3.4136000 1.4013424 1.4682944 3.7281631 1.8517657
[8] 0.6047651 2.8467121 1.7813410 1.4470712 0.7268595 1.8952825 2.1401041
[15] 0.8090938 1.0785126 0.3433595 0.5678419 2.3110352 1.4958106
> dbinom (Rbinom, size = 55, prob = pi/16)
[1] 0.063354298 0.096368041 0.047170694 0.133176805 0.119246389 0.063354298
[7] 0.133176805 0.118497194 0.096368041 0.133176805 0.096368041 0.096368041
[13] 0.096368041 0.133176805 0.096368041 0.119246389 0.119246389 0.002298608
[19] 0.133110131 0.092873118
> dcauchy (Rcauchy, location = 12, scale = 2)
[1] 0.021630824 0.019154376 0.032078738 0.140529537 0.082766881 0.040391664
[7] 0.010417640 0.098492589 0.143104718 0.006566194 0.158799748 0.141854068
[13] 0.139752618 0.115681430 0.133797911 0.034066567 0.035463730 0.035578069
[19] 0.045526265 0.109942144
> dchisq (Rchisq, df = 3)
[1] 0.11359392 0.21581974 0.23922560 0.21190393 0.17584488 0.21398711
[7] 0.12097660 0.21132681 0.18315290 0.16296267 0.24114522 0.01579786
[13] 0.16808590 0.17245292 0.15434146 0.20191378 0.09748429 0.01891934
[19] 0.24090989 0.22923980
> dexp (Rexp, rate = 2)
[1] 0.4741150 1.2571871 1.5284729 0.4629421 0.4298810 1.9908229 0.6599321
[8] 1.4816804 0.6072250 0.6558773 1.8696875 1.2367395 0.4159014 1.5421863
[15] 0.3123050 1.2585088 1.5795049 0.6132686 1.8841481 1.3363055
> df (Rf, df1 = 12, df2 = 6)
[1] 0.664042111 0.576090730 0.562274916 0.607318497 0.167959027 0.100284610
[7] 0.637144174 0.282709564 0.693573655 0.678003283 0.457493848 0.643454728
[13] 0.441386316 0.602499118 0.304762080 0.674516557 0.524996364 0.670131912
[19] 0.292349736 0.008558897
> dgamma (Rgamma, shape = 2, scale = 5)
[1] 0.044345445 0.069072391 0.041471424 0.072333682 0.072356861 0.027275042
[7] 0.067458566 0.046936330 0.073360525 0.070933729 0.028572738 0.060051063
[13] 0.038400284 0.048758423 0.054515175 0.071657454 0.070028735 0.005557423
[19] 0.070344309 0.009173320
> dgeom (Rgeom, prob = pi/16)
[1] 0.03416390 0.12681315 0.08190279 0.12681315 0.10191344 0.06582121
[7] 0.06582121 0.08190279 0.15779640 0.03416390 0.15779640 0.05289725
[13] 0.15779640 0.05289725 0.03416390 0.10191344 0.03416390 0.04251090
[19] 0.06582121 0.19634954
> dhyper (Rhyper, m = 40, n = 30, k = 20)
[1] 0.20199224 0.20431399 0.20199224 0.15732178 0.15132529 0.20431399
[7] 0.15132529 0.15132529 0.04108936 0.20199224 0.20199224 0.03541012
[13] 0.20431399 0.20431399 0.20431399 0.20199224 0.09227084 0.20431399
[19] 0.20199224 0.20199224
> dlnorm (Rlnorm, meanlog = -1, sdlog = 3)
[1] 4.3380954158 0.0005822436 1.2319845653 2.4357017049 6.8694422907
[6] 7.4733929149 1.7154009417 0.0467724227 0.0069920836 0.0336865537
[11] 1.0073244481 0.3016205039 2.3349967070 2.4034440204 0.0172006936
[16] 4.5426601250 0.0001514856 0.4720497698 0.1855964185 2.5226771709
> dlogis (Rlogis, location = 12, scale = 2)
[1] 0.1014234598 0.0768265203 0.0425481390 0.0731483166 0.1223666728
[6] 0.0922608552 0.1118154736 0.1234613952 0.1169169239 0.0002325663
[11] 0.0850243848 0.1040136815 0.0844349353 0.0757228093 0.1065226360
[16] 0.1073763420 0.0313935419 0.0511873670 0.0556984726 0.1091181283
> dnbinom (Rnbinom, size = 7, prob = .01)
[1] 0.0016012889 0.0006114669 0.0015340423 0.0014659766 0.0016087509
[6] 0.0015899813 0.0015985567 0.0004225217 0.0004186957 0.0010719579
[11] 0.0015404998 0.0012444310 0.0016100854 0.0015372764 0.0016118864
[16] 0.0012822374 0.0005567006 0.0016141573 0.0015545113 0.0007531913
> dnorm (Rnorm, mean = -1, sd = 3)
[1] 0.130110398 0.040112199 0.066772988 0.066380399 0.121789513 0.023907371
[7] 0.118172118 0.118967900 0.092918367 0.086632832 0.126309889 0.057050581
[13] 0.006458952 0.098226734 0.104819164 0.031112294 0.072188495 0.126589892
[19] 0.125224300 0.077557930
> dpois (Rpois, lambda = 12)
[1] 0.10483726 0.07239112 0.09048890 0.08736438 0.07239112 0.11436792
[7] 0.05429334 0.04368219 0.04368219 0.09048890 0.11436792 0.09048890
[13] 0.05429334 0.11436792 0.07239112 0.04368219 0.04368219 0.10557038
[19] 0.07239112 0.11436792
> dsignrank(Rsignrank, n = 47)
[1] 0.0039429676 0.0038018425 0.0007151971 0.0041008982 0.0009032331
[6] 0.0023508267 0.0040737886 0.0040318201 0.0029828933 0.0041090875
[11] 0.0041704554 0.0027222983 0.0014781510 0.0040203094 0.0008698303
[16] 0.0041704554 0.0038520708 0.0036731856 0.0027749926 0.0037486048
> dt (Rt, df = 11)
[1] 0.340823726 0.100413165 0.293668976 0.389968983 0.124439520 0.207270198
[7] 0.388829635 0.368437694 0.370255866 0.214961569 0.091948602 0.373362745
[13] 0.381142118 0.356119169 0.008424401 0.382627645 0.196428441 0.238239645
[19] 0.157280563 0.138777161
> dunif (Runif, min = .2, max = 2)
[1] 0.5555556 0.5555556 0.5555556 0.5555556 0.5555556 0.5555556 0.5555556
[8] 0.5555556 0.5555556 0.5555556 0.5555556 0.5555556 0.5555556 0.5555556
[15] 0.5555556 0.5555556 0.5555556 0.5555556 0.5555556 0.5555556
> dweibull (Rweibull, shape = 3, scale = 2)
[1] 0.40854433 0.04566100 0.58026142 0.10784049 0.22734704 0.34911114
[7] 0.57475176 0.50853257 0.43099423 0.19135106 0.37684466 0.30928353
[13] 0.14529960 0.58452097 0.02423843 0.32292685 0.02612818 0.42221582
[19] 0.46023762 0.24042036
> dwilcox (Rwilcox, m = 13, n = 17)
[1] 0.016431951 0.013817085 0.003042940 0.014141025 0.003817298 0.011603922
[7] 0.001015666 0.014141025 0.014737869 0.002001138 0.005024180 0.014141025
[13] 0.001667195 0.014141025 0.016404947 0.015691594 0.015008152 0.016351073
[19] 0.013124724 0.015008152
>
> ## Check q*(p*(.)) = identity
> All.eq(Rbeta, qbeta (Pbeta, shape1 = .8, shape2 = 2))
[1] TRUE
> All.eq(Rbinom, qbinom (Pbinom, size = 55, prob = pi/16))
[1] TRUE
> All.eq(Rcauchy, qcauchy (Pcauchy, location = 12, scale = 2))
[1] TRUE
> All.eq(Rchisq, qchisq (Pchisq, df = 3))
[1] TRUE
> All.eq(Rexp, qexp (Pexp, rate = 2))
[1] TRUE
> All.eq(Rf, qf (Pf, df1 = 12, df2 = 6))
[1] TRUE
> All.eq(Rgamma, qgamma (Pgamma, shape = 2, scale = 5))
[1] TRUE
> All.eq(Rgeom, qgeom (Pgeom, prob = pi/16))
[1] TRUE
> All.eq(Rhyper, qhyper (Phyper, m = 40, n = 30, k = 20))
[1] TRUE
> All.eq(Rlnorm, qlnorm (Plnorm, meanlog = -1, sdlog = 3))
[1] TRUE
> All.eq(Rlogis, qlogis (Plogis, location = 12, scale = 2))
[1] TRUE
> All.eq(Rnbinom, qnbinom (Pnbinom, size = 7, prob = .01))
[1] TRUE
> All.eq(Rnorm, qnorm (Pnorm, mean = -1, sd = 3))
[1] TRUE
> All.eq(Rpois, qpois (Ppois, lambda = 12))
[1] TRUE
> All.eq(Rsignrank, qsignrank(Psignrank, n = 47))
[1] TRUE
> All.eq(Rt, qt (Pt, df = 11))
[1] TRUE
> all.equal(Rt2, qt (Pt2, df = 1.01), tolerance = 1e-2)
[1] TRUE
> All.eq(Runif, qunif (Punif, min = .2, max = 2))
[1] TRUE
> All.eq(Rweibull, qweibull (Pweibull, shape = 3, scale = 2))
[1] TRUE
> All.eq(Rwilcox, qwilcox (Pwilcox, m = 13, n = 17))
[1] TRUE
>
> ## Same with "upper tail":
> All.eq(Rbeta, qbeta (1- Pbeta, shape1 = .8, shape2 = 2, lower=F))
[1] TRUE
> All.eq(Rbinom, qbinom (1- Pbinom, size = 55, prob = pi/16, lower=F))
[1] TRUE
> All.eq(Rcauchy, qcauchy (1- Pcauchy, location = 12, scale = 2, lower=F))
[1] TRUE
> All.eq(Rchisq, qchisq (1- Pchisq, df = 3, lower=F))
[1] TRUE
> All.eq(Rexp, qexp (1- Pexp, rate = 2, lower=F))
[1] TRUE
> All.eq(Rf, qf (1- Pf, df1 = 12, df2 = 6, lower=F))
[1] TRUE
> All.eq(Rgamma, qgamma (1- Pgamma, shape = 2, scale = 5, lower=F))
[1] TRUE
> All.eq(Rgeom, qgeom (1- Pgeom, prob = pi/16, lower=F))
[1] TRUE
> All.eq(Rhyper, qhyper (1- Phyper, m = 40, n = 30, k = 20, lower=F))
[1] TRUE
> All.eq(Rlnorm, qlnorm (1- Plnorm, meanlog = -1, sdlog = 3, lower=F))
[1] TRUE
> All.eq(Rlogis, qlogis (1- Plogis, location = 12, scale = 2, lower=F))
[1] TRUE
> All.eq(Rnbinom, qnbinom (1- Pnbinom, size = 7, prob = .01, lower=F))
[1] TRUE
> All.eq(Rnorm, qnorm (1- Pnorm, mean = -1, sd = 3,lower=F))
[1] TRUE
> All.eq(Rpois, qpois (1- Ppois, lambda = 12, lower=F))
[1] TRUE
> All.eq(Rsignrank, qsignrank(1- Psignrank, n = 47, lower=F))
[1] TRUE
> All.eq(Rt, qt (1- Pt, df = 11, lower=F))
[1] TRUE
> all.equal(Rt2, qt (1- Pt2, df = 1.01, lower=F), tolerance = 1e-2)
[1] TRUE
> All.eq(Runif, qunif (1- Punif, min = .2, max = 2, lower=F))
[1] TRUE
> All.eq(Rweibull, qweibull (1- Pweibull, shape = 3, scale = 2, lower=F))
[1] TRUE
> All.eq(Rwilcox, qwilcox (1- Pwilcox, m = 13, n = 17, lower=F))
[1] TRUE
>
> ## Check q*(p* ( log ), log) = identity
> All.eq(Rbeta, qbeta (log(Pbeta), shape1 = .8, shape2 = 2, log=TRUE))
[1] TRUE
> All.eq(Rbinom, qbinom (log(Pbinom), size = 55, prob = pi/16, log=TRUE))
[1] TRUE
> All.eq(Rcauchy, qcauchy (log(Pcauchy), location = 12, scale = 2, log=TRUE))
[1] TRUE
> all.equal(Rchisq, qchisq (log(Pchisq), df = 3, log=TRUE),tolerance =1e-14)
[1] TRUE
> All.eq(Rexp, qexp (log(Pexp), rate = 2, log=TRUE))
[1] TRUE
> All.eq(Rf, qf (log(Pf), df1= 12, df2= 6, log=TRUE))
[1] TRUE
> All.eq(Rgamma, qgamma (log(Pgamma), shape = 2, scale = 5, log=TRUE))
[1] TRUE
> All.eq(Rgeom, qgeom (log(Pgeom), prob = pi/16, log=TRUE))
[1] TRUE
> All.eq(Rhyper, qhyper (log(Phyper), m = 40, n = 30, k = 20, log=TRUE))
[1] TRUE
> All.eq(Rlnorm, qlnorm (log(Plnorm), meanlog = -1, sdlog = 3, log=TRUE))
[1] TRUE
> All.eq(Rlogis, qlogis (log(Plogis), location = 12, scale = 2, log=TRUE))
[1] TRUE
> All.eq(Rnbinom, qnbinom (log(Pnbinom), size = 7, prob = .01, log=TRUE))
[1] TRUE
> All.eq(Rnorm, qnorm (log(Pnorm), mean = -1, sd = 3, log=TRUE))
[1] TRUE
> All.eq(Rpois, qpois (log(Ppois), lambda = 12, log=TRUE))
[1] TRUE
> All.eq(Rsignrank, qsignrank(log(Psignrank), n = 47, log=TRUE))
[1] TRUE
> All.eq(Rt, qt (log(Pt), df = 11, log=TRUE))
[1] TRUE
> all.equal(Rt2, qt (log(Pt2), df = 1.01, log=TRUE), tolerance = 1e-2)
[1] TRUE
> All.eq(Runif, qunif (log(Punif), min = .2, max = 2, log=TRUE))
[1] TRUE
> All.eq(Rweibull, qweibull (log(Pweibull), shape = 3, scale = 2, log=TRUE))
[1] TRUE
> All.eq(Rwilcox, qwilcox (log(Pwilcox), m = 13, n = 17, log=TRUE))
[1] TRUE
>
> ## same q*(p* (log) log) with upper tail:
>
> All.eq(Rbeta, qbeta (log1p(-Pbeta), shape1 = .8, shape2 = 2, lower=F, log=T))
[1] TRUE
> All.eq(Rbinom, qbinom (log1p(-Pbinom), size = 55, prob = pi/16, lower=F, log=T))
[1] TRUE
> All.eq(Rcauchy, qcauchy (log1p(-Pcauchy), location = 12, scale = 2, lower=F, log=T))
[1] TRUE
> All.eq(Rchisq, qchisq (log1p(-Pchisq), df = 3, lower=F, log=T))
[1] TRUE
> All.eq(Rexp, qexp (log1p(-Pexp), rate = 2, lower=F, log=T))
[1] TRUE
> All.eq(Rf, qf (log1p(-Pf), df1 = 12, df2 = 6, lower=F, log=T))
[1] TRUE
> All.eq(Rgamma, qgamma (log1p(-Pgamma), shape = 2, scale = 5, lower=F, log=T))
[1] TRUE
> All.eq(Rgeom, qgeom (log1p(-Pgeom), prob = pi/16, lower=F, log=T))
[1] TRUE
> All.eq(Rhyper, qhyper (log1p(-Phyper), m = 40, n = 30, k = 20, lower=F, log=T))
[1] TRUE
> All.eq(Rlnorm, qlnorm (log1p(-Plnorm), meanlog = -1, sdlog = 3, lower=F, log=T))
[1] TRUE
> All.eq(Rlogis, qlogis (log1p(-Plogis), location = 12, scale = 2, lower=F, log=T))
[1] TRUE
> All.eq(Rnbinom, qnbinom (log1p(-Pnbinom), size = 7, prob = .01, lower=F, log=T))
[1] TRUE
> All.eq(Rnorm, qnorm (log1p(-Pnorm), mean = -1, sd = 3, lower=F, log=T))
[1] TRUE
> All.eq(Rpois, qpois (log1p(-Ppois), lambda = 12, lower=F, log=T))
[1] TRUE
> All.eq(Rsignrank, qsignrank(log1p(-Psignrank), n = 47, lower=F, log=T))
[1] TRUE
> All.eq(Rt, qt (log1p(-Pt ), df = 11, lower=F, log=T))
[1] TRUE
> all.equal(Rt2, qt (log1p(-Pt2), df = 1.01, lower=F, log=T), tolerance = 1e-2)
[1] TRUE
> All.eq(Runif, qunif (log1p(-Punif), min = .2, max = 2, lower=F, log=T))
[1] TRUE
> All.eq(Rweibull, qweibull (log1p(-Pweibull), shape = 3, scale = 2, lower=F, log=T))
[1] TRUE
> All.eq(Rwilcox, qwilcox (log1p(-Pwilcox), m = 13, n = 17, lower=F, log=T))
[1] TRUE
>
>
> ## Check log( upper.tail ):
> All.eq(log1p(-Pbeta), pbeta (Rbeta, shape1 = .8, shape2 = 2, lower=F, log=T))
[1] TRUE
> All.eq(log1p(-Pbinom), pbinom (Rbinom, size = 55, prob = pi/16, lower=F, log=T))
[1] TRUE
> All.eq(log1p(-Pcauchy), pcauchy (Rcauchy, location = 12, scale = 2, lower=F, log=T))
[1] TRUE
> All.eq(log1p(-Pchisq), pchisq (Rchisq, df = 3, lower=F, log=T))
[1] TRUE
> All.eq(log1p(-Pexp), pexp (Rexp, rate = 2, lower=F, log=T))
[1] TRUE
> All.eq(log1p(-Pf), pf (Rf, df1 = 12, df2 = 6, lower=F, log=T))
[1] TRUE
> All.eq(log1p(-Pgamma), pgamma (Rgamma, shape = 2, scale = 5, lower=F, log=T))
[1] TRUE
> All.eq(log1p(-Pgeom), pgeom (Rgeom, prob = pi/16, lower=F, log=T))
[1] TRUE
> All.eq(log1p(-Phyper), phyper (Rhyper, m = 40, n = 30, k = 20, lower=F, log=T))
[1] TRUE
> All.eq(log1p(-Plnorm), plnorm (Rlnorm, meanlog = -1, sdlog = 3, lower=F, log=T))
[1] TRUE
> All.eq(log1p(-Plogis), plogis (Rlogis, location = 12, scale = 2, lower=F, log=T))
[1] TRUE
> All.eq(log1p(-Pnbinom), pnbinom (Rnbinom, size = 7, prob = .01, lower=F, log=T))
[1] TRUE
> All.eq(log1p(-Pnorm), pnorm (Rnorm, mean = -1, sd = 3, lower=F, log=T))
[1] TRUE
> All.eq(log1p(-Ppois), ppois (Rpois, lambda = 12, lower=F, log=T))
[1] TRUE
> All.eq(log1p(-Psignrank), psignrank(Rsignrank, n = 47, lower=F, log=T))
[1] TRUE
> All.eq(log1p(-Pt), pt (Rt, df = 11, lower=F, log=T))
[1] TRUE
> All.eq(log1p(-Pt2), pt (Rt2,df = 1.01, lower=F, log=T))
[1] TRUE
> All.eq(log1p(-Punif), punif (Runif, min = .2, max = 2, lower=F, log=T))
[1] TRUE
> All.eq(log1p(-Pweibull), pweibull (Rweibull, shape = 3, scale = 2, lower=F, log=T))
[1] TRUE
> All.eq(log1p(-Pwilcox), pwilcox (Rwilcox, m = 13, n = 17, lower=F, log=T))
[1] TRUE
>
>
> ### (Extreme) tail tests added more recently:
> All.eq(1, -1e-17/ pexp(qexp(-1e-17, log=TRUE),log=TRUE))
[1] TRUE
> abs(pgamma(30,100, lower=FALSE, log=TRUE) + 7.3384686328784e-24) < 1e-36
[1] TRUE
> All.eq(1, pcauchy(-1e20) / 3.18309886183791e-21)
[1] TRUE
> All.eq(1, pcauchy(+1e15, log=TRUE) / -3.18309886183791e-16)## PR#6756
[1] TRUE
> x <- 10^(ex <- c(1,2,5*(1:5),50,100,200,300,Inf))
> for(a in x[ex > 10]) ## improve pt() : cbind(x,t= pt(-x, df=1), C=pcauchy(-x))
+ print(all.equal(pt(-a, df=1), pcauchy(-a), tolerance = 1e-15))
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
> ## for PR#7902:
> ex <- -c(rev(1/x), ex)
> All.eq(-x, qcauchy(pcauchy(-x)))
[1] TRUE
> All.eq(+x, qcauchy(pcauchy(+x, log=TRUE), log=TRUE))
[1] TRUE
> All.eq(1/x, pcauchy(qcauchy(1/x)))
[1] TRUE
> All.eq(ex, pcauchy(qcauchy(ex, log=TRUE), log=TRUE))
[1] TRUE
> II <- c(-Inf,Inf)
> stopifnot(pcauchy(II) == 0:1, qcauchy(0:1) == II,
+ pcauchy(II, log=TRUE) == c(-Inf,0),
+ qcauchy(c(-Inf,0), log=TRUE) == II)
> ## PR#15521 :
> p <- 1 - 1/4096
> stopifnot(all.equal(qcauchy(p), 1303.7970381453319163, tolerance = 1e-14))
>
> pr <- 1e-23 ## PR#6757
> stopifnot(all.equal(pr^ 12, pbinom(11, 12, prob= pr,lower=FALSE),
+ tolerance = 1e-12, scale= 1e-270))
> ## pbinom(.) gave 0 in R 1.9.0
> pp <- 1e-17 ## PR#6792
> stopifnot(all.equal(2*pp, pgeom(1, pp), scale= 1e-20))
> ## pgeom(.) gave 0 in R 1.9.0
>
> x <- 10^(100:295)
> sapply(c(1e-250, 1e-25, 0.9, 1.1, 101, 1e10, 1e100),
+ function(shape)
+ All.eq(-x, pgamma(x, shape=shape, lower=FALSE, log=TRUE)))
[1] TRUE TRUE TRUE TRUE TRUE TRUE TRUE
> x <- 2^(-1022:-900)
> ## where all completely off in R 2.0.1
> all.equal(pgamma(x, 10, log = TRUE) - 10*log(x),
+ rep(-15.104412573076, length(x)), tolerance = 1e-12)# 3.984e-14 (i386)
[1] TRUE
> all.equal(pgamma(x, 0.1, log = TRUE) - 0.1*log(x),
+ rep(0.0498724412598364, length(x)), tolerance = 1e-13)# 7e-16 (i386)
[1] TRUE
>
> All.eq(dpois( 10*1:2, 3e-308, log=TRUE),
+ c(-7096.08037610806, -14204.2875435307))
[1] TRUE
> All.eq(dpois(1e20, 1e-290, log=TRUE), -7.12801378828154e+22)
[1] TRUE
> ## all gave -Inf in R 2.0.1
>
>
> ## Inf df in pf etc.
> # apparently pf(df2=Inf) worked in 2.0.1 (undocumented) but df did not.
> x <- c(1/pi, 1, pi)
> oo <- options(digits = 8)
> df(x, 3, 1e6)
[1] 0.72553184 0.46254030 0.03300701
> df(x, 3, Inf)
[1] 0.725532165 0.462540989 0.033006719
> pf(x, 3, 1e6)
[1] 0.18784423 0.60837436 0.97585435
> pf(x, 3, Inf)
[1] 0.18784423 0.60837482 0.97585479
>
> df(x, 1e6, 5)
[1] 0.158602071 0.610206081 0.061036395
> df(x, Inf, 5)
[1] 0.15859792 0.61020761 0.06103637
> pf(x, 1e6, 5)
[1] 0.0077295711 0.4158807972 0.9022692409
> pf(x, Inf, 5)
[1] 0.0077292503 0.4158801870 0.9022693759
>
> df(x, Inf, Inf)# (0, Inf, 0) - since 2.1.1
[1] 0 Inf 0
> pf(x, Inf, Inf)# (0, 1/2, 1)
[1] 0.0 0.5 1.0
>
> pf(x, 5, Inf, ncp=0)
[1] 0.097730624 0.584119813 0.992270750
> all.equal(pf(x, 5, 1e6, ncp=1), tolerance = 1e-6,
+ c(0.065933194, 0.470879987, 0.978875867))
[1] TRUE
> all.equal(pf(x, 5, 1e7, ncp=1), tolerance = 1e-6,
+ c(0.06593309, 0.47088028, 0.97887641))
[1] TRUE
> all.equal(pf(x, 5, 1e8, ncp=1), tolerance = 1e-6,
+ c(0.0659330751, 0.4708802996, 0.9788764591))
[1] TRUE
> pf(x, 5, Inf, ncp=1)
[1] 0.065933078 0.470880318 0.978876467
>
> dt(1, Inf)
[1] 0.24197072
> dt(1, Inf, ncp=0)
[1] 0.24197072
> dt(1, Inf, ncp=1)
[1] 0.39894228
> dt(1, 1e6, ncp=1)
[1] 0.39894208
> dt(1, 1e7, ncp=1)
[1] 0.39894226
> dt(1, 1e8, ncp=1)
[1] 0.39894227
> dt(1, 1e10, ncp=1) # = Inf
[1] 0.39894228
> ## Inf valid as from 2.1.1: df(x, 1e16, 5) was way off in 2.0.1.
>
> sml.x <- c(10^-c(2:8,100), 0)
> cbind(x = sml.x, `dt(x,*)` = dt(sml.x, df = 2, ncp=1))
x dt(x,*)
[1,] 1e-02 0.21686052
[2,] 1e-03 0.21468294
[3,] 1e-04 0.21446517
[4,] 1e-05 0.21444339
[5,] 1e-06 0.21444121
[6,] 1e-07 0.21444100
[7,] 1e-08 0.21444097
[8,] 1e-100 0.21444097
[9,] 0e+00 0.21444097
> ## small 'x' used to suffer from cancellation
> options(oo)
> x <- c(outer(1:12, 10^c(-3:2, 6:9, 10*(2:30))))
> for(nu in c(.75, 1.2, 4.5, 999, 1e50)) {
+ lfx <- dt(x, df=nu, log=TRUE)
+ stopifnot(is.finite(lfx), All.eq(exp(lfx), dt(x, df=nu)))
+ }## dt(1e160, 1.2, log=TRUE) was -Inf up to R 2.15.2
>
> ## pf() with large df1 or df2
> ## (was said to be PR#7099, but that is about non-central pchisq)
> nu <- 2^seq(25, 34, 0.5)
> target <- pchisq(1, 1) # 0.682...
> y <- pf(1, 1, nu)
> stopifnot(All.eq(pf(1, 1, Inf), target),
+ diff(c(y, target)) > 0, # i.e. pf(1, 1, *) is monotone increasing
+ abs(y[1] - (target - 7.21129e-9)) < 1e-11) # computed value
> ## non-monotone in R <= 2.1.0
>
> stopifnot(pgamma(Inf, 1.1) == 1)
> ## didn't not terminate in R 2.1.x (only)
>
> ## qgamma(q, *) should give {0,Inf} for q={0,1}
> sh <- c(1.1, 0.5, 0.2, 0.15, 1e-2, 1e-10)
> stopifnot(Inf == qgamma(1, sh))
> stopifnot(0 == qgamma(0, sh))
> ## the first gave Inf, NaN, and 99.425 in R 2.1.1 and earlier
>
> ## In extreme left tail {PR#11030}
> p <- 10:123*1e-12
> qg <- qgamma(p, shape=19)
> qg2<- qgamma(1:100 * 1e-9, shape=11)
> stopifnot(diff(qg, diff=2) < -6e-6,
+ diff(qg2,diff=2) < -6e-6,
+ abs(1 - pgamma(qg, 19)/ p) < 1e-13,
+ All.eq(qg [1], 2.35047385139143),
+ All.eq(qg2[30], 1.11512318734547))
> ## was non-continuous in R 2.6.2 and earlier
>
> f2 <- c(0.5, 1:4)
> stopifnot(df(0, 1, f2) == Inf,
+ df(0, 2, f2) == 1,
+ df(0, 3, f2) == 0)
> ## only the last one was ok in R 2.2.1 and earlier
>
> x0 <- -2 * 10^-c(22,10,7,5)
> stopifnot(pbinom(x0, size = 3, prob = 0.1) == 0,
+ dbinom(x0, 3, 0.1) == 0) # d*() warns about non-integer
Warning messages:
1: In dbinom(x0, 3, 0.1) : non-integer x = -0.000000
2: In dbinom(x0, 3, 0.1) : non-integer x = -0.000020
> ## very small negatives were rounded to 0 in R 2.2.1 and earlier
>
> ## dbeta(*, ncp):
> db.x <- c(0, 5, 80, 405, 1280, 3125, 6480, 12005, 20480, 32805,
+ 50000, 73205, 103680, 142805, 192080, 253125, 327680)
> a <- rlnorm(100)
> stopifnot(All.eq(a, dbeta(0, 1, a, ncp=0)),
+ dbeta(0, 0.9, 2.2, ncp = c(0, a)) == Inf,
+ All.eq(65536 * dbeta(0:16/16, 5,1), db.x),
+ All.eq(exp(16 * log(2) + dbeta(0:16/16, 5,1, log=TRUE)), db.x)
+ )
> ## the first gave 0, the 2nd NaN in R <= 2.3.0; others use 'TRUE' values
> stopifnot(all.equal(dbeta(0.8, 0.5, 5, ncp=1000),# was way too small in R <= 2.6.2
+ 3.001852308909e-35),
+ all.equal(1, integrate(dbeta, 0,1, 0.8, 0.5, ncp=1000)$value,
+ tolerance = 1e-4),
+ all.equal(1, integrate(dbeta, 0,1, 0.5, 200, ncp=720)$value),
+ all.equal(1, integrate(dbeta, 0,1, 125, 200, ncp=2000)$value)
+ )
>
> ## df(*, ncp):
> x <- seq(0, 10, length=101)
> h <- 1e-7
> dx.h <- (pf(x+h, 7, 5, ncp= 2.5) - pf(x-h, 7, 5, ncp= 2.5)) / (2*h)
> stopifnot(all.equal(dx.h, df(x, 7, 5, ncp= 2.5), tolerance = 1e-6),# (1.50 | 1.65)e-8
+ All.eq(df(0, 2, 4, ncp=x), df(1e-300, 2, 4, ncp=x))
+ )
>
> ## qt(p ~ 0, df=1) - PR#9804
> p <- 10^(-10:-20)
> qtp <- qt(p, df = 1)
> ## relative error < 10^-14 :
> stopifnot(abs(1 - p / pt(qtp, df=1)) < 1e-14)
>
> ## Similarly for df = 2 --- both for p ~ 0 *and* p ~ 1/2
> ## P ~ 0
> stopifnot(all.equal(qt(-740, df=2, log=TRUE), -exp(370)/sqrt(2)))
> ## P ~ 1 (=> p ~ 0.5):
> p.5 <- 0.5 + 2^(-5*(5:8))
> p.5 - 0.5
[1] 2.980232e-08 9.313226e-10 2.910383e-11 9.094947e-13
> stopifnot(all.equal(qt(p.5, df = 2),
+ c(8.429369702179e-08, 2.634178031931e-09,
+ 8.231806349784e-11, 2.572439484308e-12)))
> ## qt(<large>, log = TRUE) is now more finite and monotone (again!):
> stopifnot(all.equal(qt(-1000, df = 4, log=TRUE),
+ -4.930611e108, tolerance = 1e-6))
> qtp <- qt(-(20:850), df=1.2, log=TRUE, lower=FALSE)
> ##almost: stopifnot(all(abs(5/6 - diff(log(qtp))) < 1e-11))
> stopifnot(abs(5/6 - quantile(diff(log(qtp)), pr=c(0,0.995))) < 1e-11)
>
> ## close to df=1 (where Taylor steps are important!):
> all.equal(-20, pt(qt(-20, df=1.02, log=TRUE),
+ df=1.02, log=TRUE), tolerance = 1e-12)
[1] TRUE
> stopifnot(diff(lq <- log(qt(-2^-(10:600), df=1.1, log=TRUE))) > 0.6)
> lq1 <- log(qt(-2^-(20:600), df=1, log=TRUE))
> lq2 <- log(qt(-2^-(20:600), df=2, log=TRUE))
> stopifnot(mean(abs(diff(lq1) - log(2) )) < 1e-8,
+ mean(abs(diff(lq2) - log(sqrt(2)))) < 4e-8)
> ## Case, where log.p=TRUE was fine, but log.p=FALSE (default) gave NaN:
> lp <- 40:406
> stopifnot(all.equal(lp, -pt(qt(exp(-lp), 1.2), 1.2, log=TRUE), tolerance = 4e-16))
>
>
> ## pbeta(*, log=TRUE) {toms708} -- now improved tail behavior
> x <- c(.01, .10, .25, .40, .55, .71, .98)
> pbval <- c(-0.04605755624088, -0.3182809860569, -0.7503593555585,
+ -1.241555830932, -1.851527837938, -2.76044482378, -8.149862739881)
> all.equal(pbeta(x, 0.8, 2, lower=FALSE, log=TRUE), pbval)
[1] TRUE
> all.equal(pbeta(1-x, 2, 0.8, log=TRUE), pbval)
[1] TRUE
> qq <- 2^(0:1022)
> df.set <- c(0.1, 0.2, 0.5, 1, 1.2, 2.2, 5, 10, 20, 50, 100, 500)
> for(nu in df.set) {
+ pqq <- pt(-qq, df = nu, log=TRUE)
+ stopifnot(is.finite(pqq))
+ }
> ##
> All.eq(pt(2^-30, df=10),
+ 0.50000000036238542)# = .5+ integrate(dt, 0,2^-30, df=10, rel.tol=1e-20)
[1] TRUE
>
> ## rbinom(*, size) gave NaN for large size up to R <= 2.6.1
> M <- .Machine$integer.max
> set.seed(7)
> tt <- table(rbinom(100, M, pr = 1e-9)) # had values in {0,2} only
> t2 <- table(rbinom(100, 10*M, pr = 1e-10))
> stopifnot(names(tt) == 0:6, sum(tt) == 100, sum(t2) == 100) ## no NaN there
>
> ## qf() with large df1, df2 and/or small p:
> x <- 0.01; f1 <- 1e60; f2 <- 1e90
> stopifnot(qf(1/4, Inf, Inf) == 1,
+ all.equal(1, 1e-18/ pf(qf(1e-18, 12,50), 12,50), tolerance = 1e-10),
+ abs(x - qf(pf(x, f1,f2, log.p=TRUE), f1,f2, log.p=TRUE)) < 1e-4)
>
> ## qbeta(*, log.p) for "border" case:
> stopifnot(is.finite(qbeta(-1e10, 50,40, log.p=TRUE)),
+ is.finite(qbeta(-1e10, 2, 3, lower=FALSE, log.p=TRUE)))
> ## infinite loop or NaN in R <= 2.7.0
>
> ## phyper(x, 0,0,0), notably for huge x
> stopifnot(all(phyper(c(0:3, 1e67), 0,0,0) == 1))
> ## practically infinite loop and NaN in R <= 2.7.1 (PR#11813)
>
> ## plnorm(<= 0, . , log.p=TRUE)
> stopifnot(plnorm(-1:0, lower.tail=FALSE, log.p=TRUE) == 0,
+ plnorm(-1:0, lower.tail=TRUE, log.p=TRUE) == -Inf)
> ## was wrongly == 'log.p=FALSE' up to R <= 2.7.1 (PR#11867)
>
>
> ## pchisq(df=0) was wrong in 2.7.1; then, upto 2.10.1, P*(0,0) gave 1
> stopifnot(pchisq(c(-1,0,1), df=0) == c(0,0,1),
+ pchisq(c(-1,0,1), df=0, lower.tail=FALSE) == c(1,1,0),
+ ## for ncp >= 80, gave values >= 1 in 2.10.0
+ pchisq(500:700, 1.01, ncp = 80) <= 1)
>
> ## dnbinom for extreme size and/or mu :
> mu <- 20
> d <- dnbinom(17, mu=mu, size = 1e11*2^(1:10)) - dpois(17, lambda=mu)
> stopifnot(d < 0, diff(d) > 0, d[1] < 1e-10)
> ## was wrong up to 2.7.1
> ## The fix to the above, for x = 0, had a new cancellation problem
> mu <- 1e12 * 2^(0:20)
> stopifnot(all.equal(1/(1+mu), dnbinom(0, size = 1, mu = mu), tolerance = 1e-13))
> ## was wrong in 2.7.2 (only)
>
>
> ## Non-central F for large x
> x <- 1e16 * 1.1 ^ (0:20)
> dP <- diff(pf(x, df1=1, df2=1, ncp=20, lower.tail=FALSE, log=TRUE))
> stopifnot(-0.047 < dP, dP < -0.0455)
> ## pf(*, log) jumped to -Inf prematurely in 2.8.0 and earlier
>
>
> ## Non-central Chi^2 density for large x
> stopifnot(0 == dchisq(c(Inf, 1e80, 1e50, 1e40), df=10, ncp=1))
> ## did hang in 2.8.0 and earlier (PR#13309).
>
>
> ## qbinom() .. particularly for large sizes, small prob:
> p.s <- c(.01, .001, .1, .25)
> pr <- (2:20)*1e-7
> sizes <- 1000*(5000 + c(0,6,16)) + 279
> k.s <- 0:15; q.xct <- rep(k.s, each=length(pr))
> for(sz in sizes) {
+ for(p in p.s) {
+ qb <- qbinom(p=p, size = sz, prob=pr)
+ pb <- qpois (p=p, lambda = sz * pr)
+ stopifnot(All.eq(qb, pb))
+ }
+ pp.x <- outer(pr, k.s, function(pr, q) pbinom(q, size = sz, prob=pr))
+ qq.x <- apply(pp.x, 2, function(p) qbinom(p, size = sz, prob=pr))
+ stopifnot(qq.x == q.xct)
+ }
> ## do_search() in qbinom() contained a thinko up to 2.9.0 (PR#13711)
>
>
> ## pbeta(x, a,b, log=TRUE) for small x and a is ~ log-linear
> x <- 2^-(200:10)
> for(a in c(1e-8, 1e-12, 16e-16, 4e-16))
+ for(b in c(0.6, 1, 2, 10)) {
+ dp <- diff(pbeta(x, a, b, log=TRUE)) # constant approximately
+ stopifnot(sd(dp) / mean(dp) < 0.0007)
+ }
> ## had accidental cancellation '1 - w'
>
> ## qgamma(p, a) for small a and (hence) small p
> ## pgamma(x, a) for very very small a
> a <- 2^-seq(10,1000, .25)
> q.1c <- qgamma(1e-100,a,lower.tail=FALSE)
> q.3c <- qgamma(1e-300,a,lower.tail=FALSE)
> p.1c <- pgamma(q.1c[q.1c > 0], a[q.1c > 0], lower.tail=FALSE)
> p.3c <- pgamma(q.3c[q.3c > 0], a[q.3c > 0], lower.tail=FALSE)
> x <- 1+1e-7*c(-1,1); pg <- pgamma(x, shape = 2^-64, lower.tail=FALSE)
> stopifnot(qgamma(.99, .00001) == 0,
+ abs(pg[2] - 1.18928249197237758088243e-20) < 1e-33,
+ abs(diff(pg) + diff(x)*dgamma(1, 2^-64)) < 1e-13 * mean(pg),
+ abs(1 - p.1c/1e-100) < 10e-13,# max = 2.243e-13 / 2.442 e-13
+ abs(1 - p.3c/1e-300) < 28e-13)# max = 7.057e-13
> ## qgamma() was wrong here, orders of magnitude up to R 2.10.0
> ## pgamma() had inaccuracies, e.g.,
> ## pgamma(x, shape = 2^-64, lower.tail=FALSE) was discontinuous at x=1
>
> stopifnot(all(qpois((0:8)/8, lambda=0) == 0))
> ## gave Inf as p==1 was checked *before* lambda==0
>
> ## extreme tail of non-central chisquare
> stopifnot(all.equal(pchisq(200, 4, ncp=.001, log.p=TRUE), -3.851e-42))
> ## jumped to zero too early up to R 2.10.1 (PR#14216)
>
> ## logit() == qlogit() on the right extreme:
> x <- c(10:80, 80 + 5*(1:24), 200 + 20*(1:25))
> stopifnot(All.eq(x, qlogis(plogis(x, log.p=TRUE),
+ log.p=TRUE)))
> ## qlogis() gave Inf much too early for R <= 2.12.1
> ## Part 2:
> x <- c(x, seq(700, 800, by=10))
> stopifnot(All.eq(x, qlogis(plogis(x, lower=FALSE, log.p=TRUE),
+ lower=FALSE, log.p=TRUE)))
> # plogis() underflowed to -Inf too early for R <= 2.15.0
>
> ## log upper tail pbeta():
> x <- (25:50)/128
> pbx <- pbeta(x, 1/2, 2200, lower.tail=FALSE, log.p=TRUE)
> d2p <- diff(dp <- diff(pbx))
> b <- 2200*2^(0:50)
> y <- log(-pbeta(.28, 1/2, b, lower.tail=FALSE, log.p=TRUE))
> stopifnot(-1094 < pbx, pbx < -481.66,
+ -29 < dp, dp < -20,
+ -.36 < d2p, d2p < -.2,
+ all.equal(log(b), y+1.113, tolerance = .00002)
+ )
> ## pbx had two -Inf; y was all Inf for R <= 2.15.3; PR#15162
>
> ## dnorm(x) for "large" |x|
> stopifnot(abs(1 - dnorm(35+3^-9)/ 3.933395747534971e-267) < 1e-15)
> ## has been losing up to 8 bit precision for R <= 3.0.x
>
> ## pbeta(x, <small a>,<small b>, .., log):
> ldp <- diff(log(diff(pbeta(0.5, 2^-(90+ 1:25), 2^-60, log.p=TRUE))))
> stopifnot(abs(ldp - log(1/2)) < 1e-9)
> ## pbeta(*, log) lost all precision here, for R <= 3.0.x (PR#15641)
>
>
> cat("Time elapsed: ", proc.time() - .ptime,"\n")
Time elapsed: 0.657 0.009 0.668 0 0
>