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R version 2.6.0 Under development (unstable) (2007-06-23 r42031)Copyright (C) 2007 The R Foundation for Statistical ComputingISBN 3-900051-07-0R 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>> ###-- 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)), tol= 100*Meps)+ if(!isTRUE(tst))+ cat("lambda=", format(lambda),". k =",k, " --> tst=", tst,"\n")+ tst2 <- all.equal(pp, ppois(0:k, lambda=lambda), tol = 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),+ tol = 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] TRUEWarning messages:1: In pgamma(q, shape, scale, lower.tail, log.p) : NaNs produced2: In pgamma(q, shape, scale, lower.tail, log.p) : 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), tol=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(log(1-p),df,log=T,lower=F),df, log=T,lower=F)/log(1-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, tol= 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), tol = dtol))# TRUE+ ##or print(mapply(rErr, xx, qchisq(pp, df=df,ncp=1)), digits = 3)+ }df = 0.1[1] TRUEdf = 0.5[1] TRUEdf = 1.5[1] TRUEdf = 4.7[1] TRUEdf = 10[1] TRUEdf = 20[1] TRUEdf = 50[1] TRUEdf = 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], 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)), tol = 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] TRUEWarning messages:1: In qnorm(p, mean, sd, lower.tail, log.p) : NaNs produced2: In qnorm(p, mean, sd, lower.tail, log.p) : 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 0p.s0 0 0.0 0.0 0.0 1.0 1.0 1.0 1.0 1.0 1d.sI 0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0p.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),+ tol = 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), tol= 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), tol= 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]), tol= 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] -0.49984609 -1.67069848 -0.72975770 0.00988246 1.51883080 -1.10548091[7] -0.07391342 -0.32360193 -0.30921048 1.07149067 -1.73064836 -0.28314148[13] 0.20530996 0.40967685 3.13749439 0.18707089 -1.15413093 0.97040836[19] -1.34067841 1.43776923> ## 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), tol = 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), tol = 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),tol=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), tol = 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 (log(1- Pbeta), shape1 = .8, shape2 = 2, lower=F, log=T))[1] TRUE> All.eq(Rbinom, qbinom (log(1- Pbinom), size = 55, prob = pi/16, lower=F, log=T))[1] TRUE> All.eq(Rcauchy, qcauchy (log(1- Pcauchy), location = 12, scale = 2, lower=F, log=T))[1] TRUE> All.eq(Rchisq, qchisq (log(1- Pchisq), df = 3, lower=F, log=T))[1] TRUE> All.eq(Rexp, qexp (log(1- Pexp), rate = 2, lower=F, log=T))[1] TRUE> All.eq(Rf, qf (log(1- Pf), df1 = 12, df2 = 6, lower=F, log=T))[1] TRUE> All.eq(Rgamma, qgamma (log(1- Pgamma), shape = 2, scale = 5, lower=F, log=T))[1] TRUE> All.eq(Rgeom, qgeom (log(1- Pgeom), prob = pi/16, lower=F, log=T))[1] TRUE> All.eq(Rhyper, qhyper (log(1- Phyper), m = 40, n = 30, k = 20, lower=F, log=T))[1] TRUE> All.eq(Rlnorm, qlnorm (log(1- Plnorm), meanlog = -1, sdlog = 3, lower=F, log=T))[1] TRUE> All.eq(Rlogis, qlogis (log(1- Plogis), location = 12, scale = 2, lower=F, log=T))[1] TRUE> All.eq(Rnbinom, qnbinom (log(1- Pnbinom), size = 7, prob = .01, lower=F, log=T))[1] TRUE> All.eq(Rnorm, qnorm (log(1- Pnorm), mean = -1, sd = 3, lower=F, log=T))[1] TRUE> All.eq(Rpois, qpois (log(1- Ppois), lambda = 12, lower=F, log=T))[1] TRUE> All.eq(Rsignrank, qsignrank(log(1- Psignrank), n = 47, lower=F, log=T))[1] TRUE> All.eq(Rt, qt (log(1- Pt ), df = 11, lower=F, log=T))[1] TRUE> all.equal(Rt2, qt (log(1- Pt2), df = 1.01, lower=F, log=T), tol = 1e-2)[1] TRUE> All.eq(Runif, qunif (log(1- Punif), min = .2, max = 2, lower=F, log=T))[1] TRUE> All.eq(Rweibull, qweibull (log(1- Pweibull), shape = 3, scale = 2, lower=F, log=T))[1] TRUE> All.eq(Rwilcox, qwilcox (log(1- Pwilcox), m = 13, n = 17, lower=F, log=T))[1] TRUE>>> ## Check log( upper.tail ):> All.eq(log(1 - Pbeta), pbeta (Rbeta, shape1 = .8, shape2 = 2, lower=F, log=T))[1] TRUE> All.eq(log(1 - Pbinom), pbinom (Rbinom, size = 55, prob = pi/16, lower=F, log=T))[1] TRUE> All.eq(log(1 - Pcauchy), pcauchy (Rcauchy, location = 12, scale = 2, lower=F, log=T))[1] TRUE> All.eq(log(1 - Pchisq), pchisq (Rchisq, df = 3, lower=F, log=T))[1] TRUE> All.eq(log(1 - Pexp), pexp (Rexp, rate = 2, lower=F, log=T))[1] TRUE> All.eq(log(1 - Pf), pf (Rf, df1 = 12, df2 = 6, lower=F, log=T))[1] TRUE> All.eq(log(1 - Pgamma), pgamma (Rgamma, shape = 2, scale = 5, lower=F, log=T))[1] TRUE> All.eq(log(1 - Pgeom), pgeom (Rgeom, prob = pi/16, lower=F, log=T))[1] TRUE> All.eq(log(1 - Phyper), phyper (Rhyper, m = 40, n = 30, k = 20, lower=F, log=T))[1] TRUE> All.eq(log(1 - Plnorm), plnorm (Rlnorm, meanlog = -1, sdlog = 3, lower=F, log=T))[1] TRUE> All.eq(log(1 - Plogis), plogis (Rlogis, location = 12, scale = 2, lower=F, log=T))[1] TRUE> All.eq(log(1 - Pnbinom), pnbinom (Rnbinom, size = 7, prob = .01, lower=F, log=T))[1] TRUE> All.eq(log(1 - Pnorm), pnorm (Rnorm, mean = -1, sd = 3, lower=F, log=T))[1] TRUE> All.eq(log(1 - Ppois), ppois (Rpois, lambda = 12, lower=F, log=T))[1] TRUE> All.eq(log(1 - Psignrank), psignrank(Rsignrank, n = 47, lower=F, log=T))[1] TRUE> All.eq(log(1 - Pt), pt (Rt, df = 11, lower=F, log=T))[1] TRUE> All.eq(log(1 - Pt2), pt (Rt2,df = 1.01, lower=F, log=T))[1] TRUE> All.eq(log(1 - Punif), punif (Runif, min = .2, max = 2, lower=F, log=T))[1] TRUE> All.eq(log(1 - Pweibull), pweibull (Rweibull, shape = 3, scale = 2, lower=F, log=T))[1] TRUE> All.eq(log(1 - 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), tol = 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 <- 1e-23 ## PR#6757> stopifnot(all.equal(pr^ 12, pbinom(11, 12, prob= pr,lower=FALSE),+ tol= 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)), tol = 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)), tol = 1e-13)# 7e-16 (i386)[1] TRUE>> All.eq(dpois( 10, 2e-308, log=TRUE), -7100.13502718914)[1] TRUE> All.eq(dpois( 20, 3e-308, log=TRUE), -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> pf(x, 5, 1e6, ncp=1)[1] 0.065933194 0.470879987 0.978875867> pf(x, 5, 1e7, ncp=1)[1] 0.06593309 0.47088028 0.97887641> all.equal(pf(x, 5, 1e8, ncp=1), tol = 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)>> ## PR#7099 : pf() with large df1 or df2:> nu <- 2^seq(25,34, 0.5)> y <- 1e9*(pf(1,1,nu) - 0.68268949)> stopifnot(All.eq(pf(1,1,Inf), 0.68268949213708596),+ diff(y) > 0, # i.e. pf(1,1, *) is monotone increasing+ All.eq(y [1], -5.07420372386491),+ All.eq(y[19], 2.12300110824515))> ## not at all in R 2.1.0 or earlier>> 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>> 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-integerWarning messages:1: In dbinom(x0, 3, 0.1) : non-integer x = -0.0000002: 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):> a <- rlnorm(100)> stopifnot(All.eq(a, dbeta(0, 1, a, ncp=0)),+ dbeta(0, 0.9, 2.2, ncp = c(0, a)) == Inf+ )> ## the first gave 0, the 2nd NaN in R <= 2.3.0>> ## 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), tol = 1e-6),# (1.50 | 1.65)e-8+ All.eq(df(0, 2, 4, ncp=x), df(1e-300, 2, 4, ncp=x))+ )>> cat("Time elapsed: ", proc.time() - .ptime,"\n")Time elapsed: 3.95 0.03 3.98 NA NA>