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R Under development (unstable) (2016-11-22 r71677) -- "Unsuffered Consequences"Copyright (C) 2016 The R Foundation for Statistical ComputingPlatform: x86_64-pc-linux-gnu (64-bit)R is free software and comes with ABSOLUTELY NO WARRANTY.You are welcome to redistribute it under certain conditions.Type 'license()' or 'licence()' for distribution details.R is a collaborative project with many contributors.Type 'contributors()' for more information and'citation()' on how to cite R or R packages in publications.Type 'demo()' for some demos, 'help()' for on-line help, or'help.start()' for an HTML browser interface to help.Type 'q()' to quit R.> #### d|ensity> #### p|robability (cumulative)> #### q|uantile> #### r|andom number generation> ####> #### Functions for ``d/p/q/r''>> .ptime <- proc.time()> F <- FALSE> T <- TRUE>> options(warn = 2)> ## ======== No warnings, unless explicitly asserted via> assertWarning <- tools::assertWarning>> as.nan <- function(x) { x[is.na(x) & !is.nan(x)] <- NaN ; x }> ###-- 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 = getOption("rErr.eps", 1e-30))+ {+ 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)) {+ for(p in c(0,1,rbeta(n2, 2,4))) {+ 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:+ stopifnot(all.equal( pbinom(0:k, size = n, prob = p),+ cumsum(dbinom(0:k, size = n, prob = p))),+ 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))))+ }+ }>> ##__ 2. Geometric __> for(pr in seq(1e-10,1,len=15)) # p=0 is not a distribution+ stopifnot(All.eq((dg <- dgeom(0:10, pr)),+ pr * (1-pr)^(0:10)),+ All.eq(cumsum(dg), pgeom(0:10, pr)))>>> ##__ 3. Hypergeometric __>> m <- 10; n <- 7> for(k in 2:m) {+ x <- 0:(k+1)+ stopifnot(All.eq(phyper(x, m, n, k), cumsum(dhyper(x, m, n, k))))+ }>> ##__ 4. Negative Binomial __>> ## PR #842> for(size in seq(0.8,2, by=.1))+ stopifnot(all.equal(cumsum(dnbinom(0:7, size, .5)),+ pnbinom(0:7, size, .5)))> stopifnot(All.eq(pnbinom(c(1,3), .9, .5),+ c(0.777035760338812, 0.946945347071519)))>> ##__ 5. Poisson __>> stopifnot(dpois(0:5,0) == c(1, rep(0,5)),+ dpois(0:5,0, log=TRUE) == c(0, rep(-Inf, 5)))>> ## 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))+ stopifnot(all.equal(1 - pchisq(2*lambda, 2*(1+ 0:k)),+ pp <- cumsum(dpois(0:k, lambda=lambda)),+ tolerance = 100*Meps),+ all.equal(pp, ppois(0:k, lambda=lambda), tolerance = 100*Meps),+ all.equal(1 - pp, ppois(0:k, lambda=lambda, lower.tail = FALSE)))>>> ##__ 6. SignRank __> for(n in rpois(32, lam=8)) {+ x <- -1:(n + 4)+ stopifnot(All.eq(psignrank(x, n), cumsum(dsignrank(x, n))))+ }>> ##__ 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))+ stopifnot(All.eq(Fx, cumsum(fx)))+ }> stopifnot(is.sym)>>> ###-------- 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))+ stopifnot(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+ ,+ All.eq(d1, (d3 <- 1/(Ga * sig^sh) * x^(sh-1) * exp(-x/sig)))+ )+ }>> stopifnot(pgamma(1,Inf,scale=Inf) == 0)> ## Also pgamma(Inf,Inf) == 1 for which NaN was slightly more appropriate> assertWarning(stopifnot(+ is.nan(c(pgamma(Inf, 1,scale=Inf),+ pgamma(Inf,Inf,scale=Inf)))))> 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> stopifnot(+ 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+ )>> ##-- 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)> stopifnot(all.equal(qchisq(0.025,31,ncp=1,lower.tail=FALSE),# inf.loop PR#875+ 49.7766246561514, tolerance = 1e-11))> for(df in c(0.1, 0.5, 1.5, 4.7, 10, 20,50,100)) {+ 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)+ stopifnot(all.equal(xx, qchisq(pp, df=df, ncp=1), tolerance = dtol))+ }>> ## p ~= 1 (<==> 1-p ~= 0) -- gave infinite loop in R <= 1.8.1 -- PR#6421> 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> stopifnot(all.equal(q0[iO], q1[iO], tolerance = 1e-5),# 9.86e-8+ all.equal(p0[iO], psml[iO])) # 1.07e-13>> ##--- Beta (need more):>> ## big a & b (PR #643)> stopifnot(is.finite(a <- rlnorm(20, 5.5)), a > 0,+ is.finite(b <- rlnorm(20, 6.5)), b > 0)> pab <- expand.grid(seq(0,1,by=.1), a, b)> p <- pab[,1]; a <- pab[,2]; b <- pab[,3]> stopifnot(all.equal(dbeta(p,a,b),+ exp(pab <- dbeta(p,a,b, log = TRUE)), tolerance = 1e-11))> sp <- sample(pab, 50)> if(!interactive())+ stopifnot(which(isI <- sp == -Inf) ==+ c(3, 11, 15, 20, 22, 23, 30, 39, 42, 43, 46, 47, 49),+ all.equal(range(sp[!isI]), c(-2906.123981, 2.197270387))+ )>>> ##--- Normal (& Lognormal) :>> stopifnot(+ qnorm(0) == -Inf, qnorm(-Inf, log = TRUE) == -Inf,+ qnorm(1) == Inf, qnorm( 0, log = TRUE) == Inf)>> assertWarning(stopifnot(+ is.nan(qnorm(1.1)),+ is.nan(qnorm(-.1))))>> x <- c(-Inf, -1e100, 1:6, 1e200, Inf)> stopifnot(+ dnorm(x,3,s=0) == c(0,0,0,0, Inf, 0,0,0,0,0),+ pnorm(x,3,s=0) == c(0,0,0,0, 1 , 1,1,1,1,1),+ dnorm(x,3,s=Inf) == 0,+ pnorm(x,3,s=Inf) == c(0, rep(0.5, 8), 1))>> ## 3 Test data from Wichura (1988) :> stopifnot(+ all.equal(qnorm(c( 0.25, .001, 1e-20)),+ c(-0.6744897501960817, -3.090232306167814, -9.262340089798408),+ tolerance = 1e-15)+ , ## extreme tail -- available on log scale only:+ all.equal(qnorm(-1e5, log = TRUE), -447.1974945)+ )>> 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) stopifnot(all.equal(pt(z, df), 1 - pt(-z,df), tolerance = 1e-15))>> stopifnot(All.eq(pz <- pnorm(z), 1 - pnorm(z, lower=FALSE)),+ All.eq(pz, pnorm(-z, lower=FALSE)),+ All.eq(log(pz[z.ok]), pnorm(z[z.ok], log=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> stopifnot(All.eq(pz, plnorm(exp(z))))>>> ###========== 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.eq(Rt2, qt (Pt2, df = 1.01))[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.eq(Rt2, qt (1- Pt2, df = 1.01, lower=F))[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.eq(Rchisq, qchisq (log(Pchisq), df = 3, log=TRUE))[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.eq(Rt2, qt (log(Pt2), df = 1.01, log=TRUE))[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.eq(Rt2, qt (log1p(-Pt2), df = 1.01, lower=F, log=T))[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))+ stopifnot(all.equal(pt(-a, df=1), pcauchy(-a), tolerance = 1e-15))> ## 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) # ==> d*() warns about non-integer:> assertWarning(fx0 <- dbinom(x0, size = 3, prob = 0.1))> stopifnot(fx0 == 0, pbinom(x0, size = 3, prob = 0.1) == 0)>> ## 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))> 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!):> stopifnot(all.equal(-20, pt(qt(-20, df=1.02, log=TRUE),+ df=1.02, log=TRUE), tolerance = 1e-12),+ 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)> stopifnot(all.equal(pbeta(x, 0.8, 2, lower=FALSE, log=TRUE), pbval),+ all.equal(pbeta(1-x, 2, 0.8, log=TRUE), pbval))> 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))+ }> ## PR#14230 -- more extreme beta cases {should no longer rely on denormalized}> x <- (256:512)/1024> P <- pbeta(x, 3, 2200, lower.tail=FALSE, log.p=TRUE)> stopifnot(is.finite(P), P < -600,+ -.001 < (D3P <- diff(P, diff = 3)), D3P < 0, diff(D3P) < 0)> ## all but the first 43 where -Inf in R <= 2.9.1> stopifnot(All.eq(pt(2^-30, df=10),+ 0.50000000036238542))> ## = .5+ integrate(dt, 0,2^-30, df=10, rel.tol=1e-20)>> ## 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(q0 <- qbeta(-1e10, 50,40, log.p=TRUE)),+ 1 == qbeta(-1e10, 2, 3, log.p=TRUE, lower=FALSE))> ## 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)> mu <- sort(outer(1:7, 10^c(0:10,50*(1:6))))> NB <- dnbinom(5, size=1e305, mu=mu, log=TRUE)> P <- dpois (5, mu, log=TRUE)> stopifnot(abs(rErr(NB,P)) < 9*Meps)# seen 2.5*> ## wrong in 3.1.0 and earlier>>> ## 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)> ## left "extreme tail"> lp <- pchisq(2^-(0:200), 100, 1, log=TRUE)> stopifnot(is.finite(lp), lp < -184,+ all.equal(lp[201], -7115.10693158))> dlp <- diff(lp)> dd <- abs(dlp[-(1:30)] - -34.65735902799)> stopifnot(-34.66 < dlp, dlp < -34.41, dd < 1e-8)# 2.2e-10 64bit Lnx> ## underflowed to -Inf much too early in R <= 3.1.0> for(e in c(0, 2e-16))# continuity at 80 (= branch point)+ stopifnot(all.equal(pchisq(1:2, 1.01, ncp = 80*(1-e), log=TRUE),+ c(-34.57369629, -31.31514671)))>> ## 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)> ##> ## "stair function" effect (from denormalized numbers)> a <- 43779; b <- 0.06728> x. <- .9833 + (0:100)*1e-6> px <- pbeta(x., a,b, log=TRUE) # plot(x., px) # -> "stair"> d2. <- diff(dpx <- diff(px))> stopifnot(all.equal(px[1], -746.0986886924, tol=1e-12),+ 0.0445741 < dpx, dpx < 0.0445783,+ -4.2e-8 < d2., d2. < -4.18e-8)> ## were way off in R <= 3.1.0>> c0 <- system.time(p0 <- pbeta( .9999, 1e30, 1.001, log=TRUE))> cB <- max(.001, c0[[1]])# base time> c1 <- system.time(p1 <- pbeta(1- 1e-9, 1e30, 1.001, log=TRUE))> c2 <- system.time(p2 <- pbeta(1-1e-12, 1e30, 1.001, log=TRUE))> stopifnot(all.equal(p0, -1.000050003333e26, tol=1e-10),+ all.equal(p1, -1e21, tol = 1e-6),+ all.equal(p2, -9.9997788e17),+ c(c1[[1]], c2[[1]]) < 1000*cB)> ## (almost?) infinite loop in R <= 3.1.0>>> ## pbinom(), dbinom(), dhyper(),.. : R allows "almost integer" n> for (FUN in c(function(n) dbinom(1,n,0.5), function(n) pbinom(1,n,0.5),+ function(n) dpois(n, n), function(n) dhyper(n+1, n+5,n+5, n)))+ try( lapply(sample(10000, size=1000), function(M) {+ ## invisible(lapply(sample(10000, size=1000), function(M) {+ n <- (M/100)*10^(2:20); if(anyNA(P <- FUN(n)))+ stop("NA for M=",M, "; 10ex=",paste((2:20)[is.na(P)], collapse=", "))}))> ## check was too tight for large n in R <= 3.1.0 (PR#15734)>> ## [dpqr]beta(*, a,b) where a and/or b are Inf> stopifnot(pbeta(.1, Inf, 40) == 0,+ pbeta(.5, 40, Inf) == 1,+ pbeta(.4, Inf,Inf) == 0,+ pbeta(.5, Inf,Inf) == 1,+ ## gave infinite loop (or NaN) in R <= 3.1.0+ qbeta(.9, Inf, 100) == 1, # Inf.loop+ qbeta(.1, Inf, Inf) == 1/2)# NaN + Warning> ## range check (in "close" cases):> assertWarning(qN <- qbeta(2^-(10^(1:3)), 2,3, log.p=TRUE))> assertWarning(qn <- qbeta(c(-.1, -1e-300, 1.25), 2,3))> stopifnot(is.nan(qN), is.nan(qn))>> ## lognormal boundary case sdlog = 0:> p <- (0:8)/8; x <- 2^(-10:10)> stopifnot(all.equal(qlnorm(p, meanlog=1:2, sdlog=0),+ qlnorm(p, meanlog=1:2, sdlog=1e-200)),+ dlnorm(x, sdlog=0) == ifelse(x == 1, Inf, 0))>> ## qbeta(*, a,b) when a,b << 1 : can easily fail> qbeta(2^-28, 0.125, 2^-26) # 1000 Newton it + warning[1] 9.094955e-13> a <- 1/8; b <- 2^-(4:200); alpha <- b/4> qq <- qbeta(alpha, a,b)# gave warnings intermediately> pp <- pbeta(qq, a,b)> stopifnot(pp > 0, diff(pp) < 0, ## pbeta(qbeta(alpha,*),*) == alpha:+ abs(1 - pp/alpha) < 4e-15)# seeing 2.2e-16>> ## orig. qbeta() using *many* Newton steps; case where we "know the truth"> a <- 25; b <- 6; x <- 2^-c(3:15, 100, 200, 250, 300+100*(0:7))> pb <- c(## via Rmpfr's roundMpfr(pbetaI(x, a,b, log.p=TRUE, precBits = 2048), 64) :+ -40.7588797271766572448, -57.7574063441183625303, -74.9287878018119846216,+ -92.1806244636893542185, -109.471318248524419364, -126.781111923947395655,+ -144.100375042814531426, -161.424352961544612370, -178.750683324909148353,+ -196.078188674895169383, -213.406281209657976525, -230.734667259724367416,+ -248.063200048177428608, -1721.00081201679567511, -3453.86876341665894863,+ -4320.30273911659058550, -5186.73671481652222237, -6919.60466621638549567,+ -8652.47261761624876897, -10385.3405690161120427, -12118.2085204159753165,+ -13851.0764718158385902, -15583.9444232157018631, -17316.8123746155651368)> stopifnot(all.equal(pb, pbeta(x,a,b, log.p=TRUE), tol=8e-16))# seeing {1.5|1.6|2.0}e-16> qp <- qbeta(pb, a,b, log.p=TRUE)> ## x == qbeta(pbeta(x, *), *) :> stopifnot(qp > 0, all.equal(x, qp, tol= 1e-15))# seeing {2.4|3.3}e-16>> ## qbeta(), PR#15755> a1 <- 0.0672788; b1 <- 226390> p <- 0.6948886> qp <- qbeta(p, a1,b1)> stopifnot(qp < 2e-8, # was '1' (with a warning) in R <= 3.1.0+ All.eq(p, pbeta(qp, a1,b1)))> ## less extreme example, same phenomenon:> a <- 43779; b <- 0.06728> stopifnot(All.eq(0.695, pbeta(qbeta(0.695, b,a), b,a)))> x <- -exp(seq(0, 14, by=2^-9))> ct <- system.time(qx <- qbeta(x, a,b, log.p=TRUE))[[1]]> pqx <- pbeta(qx, a,b, log=TRUE)> stopifnot(all.equal(x, pqx, tol= 2e-15)) # seeing {3.51|3.54}e-16> ## note that qx[x > -exp(2)] is too close to 1 to get full accuracy:> ## i2 <- x > -exp(2); all.equal(x[i2], pqx[i2], tol= 0)#-> 5.849e-12> if(ct > 0.5) { cat("system.time:\n"); print(ct) }# lynne(2013): 0.048> ## was Inf, and much slower, for R <= 3.1.0> x3 <- -(15450:15700)/2^11> pq3 <- pbeta(qbeta(x3, a,b, log.p=TRUE), a,b, log=TRUE)> stopifnot(mean(abs(pq3-x3)) < 4e-12,# 1.46e-12+ max (abs(pq3-x3)) < 8e-12)# 2.95e-12> ##> .a <- .2; .b <- .03; lp <- -(10^-(1:323))> qq <- qbeta(lp, .a,.b, log=TRUE) # warnings in R <= 3.1.0> assertWarning(qN <- qbeta(.5, 2,3, log.p=TRUE))> assertWarning(qn <- qbeta(c(-.1, 1.25), 2,3))> stopifnot(1-qq < 1e-15, is.nan(qN), is.nan(qn))# typically qq == 1 exactly> ## failed in intermediate versions> ##> a <- 2^-8; b <- 2^(200:500)> pq <- pbeta(qbeta(1/8, a, b), a, b)> stopifnot(abs(pq - 1/8) < 1/8)> ## whereas qbeta() would underflow to 0 "too early", for R <= 3.1.0> #> ## very extreme tails on both sides> x <- c(1e-300, 1e-12, 1e-5, 0.1, 0.21, 0.3)> stopifnot(0 == qbeta(x, 2^-12, 2^-10))## gave warnings> a <- 10^-(8:323)> qb <- qbeta(0.95, a, 20)> ## had warnings and wrong value +1; also NaN> ct2 <- system.time(q2 <- qbeta(0.95, a,a))[1]> stopifnot(is.finite(qb), qb < 1e-300, q2 == 1)> if(ct2 > 0.020) { cat("system.time:\n"); print(ct2) }> ## had warnings and was much slower for R <= 3.1.0>> ## qt(p, df= Inf, ncp) <==> qnorm(p, m=ncp)> p <- (0:32)/32> stopifnot(all.equal(qt(p, df=Inf, ncp=5), qnorm(p, m=5)))> ## qt(*, df=Inf, .) gave NaN in R <= 3.2.1>> ## rhyper(*, <large>); PR#16489> ct3 <- system.time(N <- rhyper(100, 8000, 1e9-8000, 1e6))[1]> table(N)N1 2 3 4 5 6 7 8 9 10 11 12 13 141 1 2 5 8 14 16 17 8 13 5 5 3 2> summary(N)Min. 1st Qu. Median Mean 3rd Qu. Max.1.00 6.00 8.00 7.85 10.00 14.00> stopifnot(abs(mean(N) - 8) < 1.5)> if(ct3 > 0.02) { cat("system.time:\n"); print(ct3) }> ## N were all 0 and took very long for R <= 3.2.1> set.seed(17)> stopifnot(rhyper(1, 3024, 27466, 251) == 25,+ rhyper(1, 329, 3059, 225) == 22)> ## failed for a day after a "thinko" in the above bug fix.>> ## *chisq(*, df=0, ncp=0) == Point mass at 0> stopifnot(rchisq(32, df=0, ncp=0) == 0,+ dchisq((0:16)/16, df=0, ncp=0) == c(Inf, rep(0, 16)))> ## gave all NaN's for R <= 3.2.2>> ## pchisq(*, df=0, ncp > 0, log.p=TRUE) :> th <- 10*c(1:9,10^c(1:3,7))> pp <- pchisq(0, df = 0, ncp=th, log.p=TRUE)> stopifnot(all.equal(pp, -th/2, tol=1e-15))> ## underflowed at about th ~= 60 in R <= 3.2.2>> ## pnbinom (-> C's bratio())> op <- options(warn = 1)# -- NaN's giving warnings> L <- 1e308; p <- suppressWarnings(pnbinom(L, L, mu = 5)) # NaN or 1 (for 64 / 32 bit)> is.nan(p) || p == 1[1] TRUE> ## gave infinite loop on some 64b platforms in R <= 3.2.3>> ## [dpqr]nbinom(*, mu, size=Inf) -- PR#16727> L <- 1e308; mu <- 5; pp <- (0:16)/16> x <- c(0:3, 1e10, 1e100, L, Inf)> (d <- dnbinom(x, mu = mu, size = Inf)) # gave NaN (for 0 and L)[1] 0.006737947 0.033689735 0.084224337 0.140373896 0.000000000 0.000000000[7] 0.000000000 0.000000000> (p <- pnbinom(x, mu = mu, size = Inf)) # gave all NaN[1] 0.006737947 0.040427682 0.124652019 0.265025915 1.000000000 1.000000000[7] 1.000000000 1.000000000> (q <- qnbinom(pp, mu = mu, size = Inf)) # gave all NaN[1] 0 2 3 3 3 4 4 4 5 5 6 6 6 7 8 9 Inf> set.seed(1); NI <- rnbinom(32, mu = mu, size = Inf)# gave all NaN> set.seed(1); N2 <- rnbinom(32, mu = mu, size = L )> stopifnot(all.equal(d, dpois(x, mu)),+ all.equal(p, ppois(x, mu)),+ q == qpois(pp, mu),+ identical(NI, N2))> options(op)> ## size = Inf -- mostly gave NaN in R <= 3.2.3>> ## qpois(p, *) for invalid 'p' should give NaN -- PR#16972> stopifnot(is.nan(suppressWarnings(c(qpois(c(-2,3, NaN), 3), qpois(1, 3, log.p=TRUE),+ qpois(.5, 0, log.p=TRUE), qpois(c(-1,pi), 0)))))> ## those in the 2nd line gave 0 in R <= 3.3.1> ## Similar but different for qgeom():> stopifnot(qgeom((0:8)/8, prob=1) == 0, ## p=1 gave Inf in R <= 3.3.1+ is.nan(suppressWarnings(qgeom(c(-1/4, 1.1), prob=1))))>> ## all our RNG r<dist>() functions:> ##' catch all: value and warnings or error <-- demo(error.catching) :> tryCatch.W.E <- function(expr) {+ W <- NULL+ w.handler <- function(w){ # warning handler+ W <<- w+ invokeRestart("muffleWarning")+ }+ list(value = withCallingHandlers(tryCatch(expr, error = function(e) e),+ warning = w.handler),+ warning = W)+ }> .stat.ns <- asNamespace("stats")> Ns <- 4> for(dist in PDQR) {+ fn <- paste0("r",dist)+ cat(sprintf("%-9s(%d, ..): ", fn, Ns))+ F <- get(fn, envir = .stat.ns)+ nArg <- length(fms <- formals(F))+ if(dist %in% c("nbinom", "gamma")) ## cannot specify *both* 'prob' & 'mu' / 'rate' & 'scale'+ nArg <- nArg - 1+ nA1 <- nArg - 1 # those beside the first (= 'n' mostly)+ expected <- rep(if(dist %in% PDQRinteg) NA_integer_ else NaN, Ns)+ for(ia in seq_len(nA1)) {+ aa <- rep(list(1), nA1)+ aa[[ia]] <- NA+ cat(ia,"")+ R <- tryCatch.W.E( do.call(F, c(Ns, aa)) )+ if(!inherits(R$warning, "simpleWarning")) cat(" .. did *NOT* give a warning! ")+ if(!(identical(R$value, expected))) { ## allow NA/NaN mismatch in these cases for now:+ if(!(dist %in% c("beta","f","t") && all(is.na(R$value))))+ cat(" .. not giving expected NA/NaN's ")+ }+ }+ cat(" [Ok]\n")+ }rbinom (4, ..): 1 2 [Ok]rgeom (4, ..): 1 [Ok]rhyper (4, ..): 1 2 3 [Ok]rnbinom (4, ..): 1 2 [Ok]rpois (4, ..): 1 [Ok]rsignrank(4, ..): 1 [Ok]rwilcox (4, ..): 1 2 [Ok]rbeta (4, ..): 1 2 3 [Ok]rcauchy (4, ..): 1 2 [Ok]rchisq (4, ..): 1 2 [Ok]rexp (4, ..): 1 [Ok]rf (4, ..): 1 2 3 [Ok]rgamma (4, ..): 1 2 [Ok]rlnorm (4, ..): 1 2 [Ok]rlogis (4, ..): 1 2 [Ok]rnorm (4, ..): 1 2 [Ok]rt (4, ..): 1 2 [Ok]runif (4, ..): 1 2 [Ok]rweibull (4, ..): 1 2 [Ok]>>> cat("Time elapsed: ", proc.time() - .ptime,"\n")Time elapsed: 1.346 0.012 1.363 0 0>