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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] TRUE Warning messages: 1: In pgamma(q, shape, scale, lower.tail, log.p) : NaNs produced 2: 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] TRUE df = 0.5[1] TRUE df = 1.5[1] TRUE df = 4.7[1] TRUE df = 10[1] TRUE df = 20[1] TRUE df = 50[1] TRUE df = 100[1] TRUE > > ## p ~= 1 (<==> 1-p ~= 0) -- gave infinite loop in R <= 1.8.1 -- PR#6421 > options(warn=-1) # ignore warnings from R's version of log1p > psml <- 2^-(10:54) > q0 <- qchisq(psml, df=1.2, ncp=10, lower.tail=FALSE) > q1 <- qchisq(1-psml, df=1.2, ncp=10) # inaccurate in the tail > p0 <- pchisq(q0, df=1.2, ncp=10, lower.tail=FALSE) > p1 <- pchisq(q1, df=1.2, ncp=10, lower.tail=FALSE) > iO <- 1:30 > all.equal(q0[iO], q1[iO], 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] TRUE Warning messages: 1: In qnorm(p, mean, sd, lower.tail, log.p) : NaNs produced 2: 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 0 p.s0 0 0.0 0.0 0.0 1.0 1.0 1.0 1.0 1.0 1 d.sI 0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0 p.sI 0 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1 > > ## 3 Test data from Wichura (1988) : > all.equal(qnorm(c( 0.25, .001, 1e-20)), + c(-0.6744897501960817, -3.090232306167814, -9.262340089798408), + 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*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 > 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) > > ## 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} > qg <- qgamma(10:123*1e-12, shape=19) > qg2<- qgamma(1:100 * 1e-9, shape=11) > stopifnot(diff(qg, diff=2) < -6e-6, + diff(qg2,diff=2) < -6e-6, + All.eq(qg [1], 2.35047385139143), + All.eq(qg2[30], 1.11512318734547)) > ## was non-continuous in R 2.6.2 and earlier > > f2 <- c(0.5, 1:4) > stopifnot(df(0, 1, f2) == Inf, + df(0, 2, f2) == 1, + df(0, 3, f2) == 0) > ## only the last one was ok in R 2.2.1 and earlier > > x0 <- -2 * 10^-c(22,10,7,5) > stopifnot(pbinom(x0, size = 3, prob = 0.1) == 0, + dbinom(x0, 3, 0.1) == 0) # d*() warns about non-integer Warning messages: 1: In dbinom(x0, 3, 0.1) : non-integer x = -0.000000 2: In dbinom(x0, 3, 0.1) : non-integer x = -0.000020 > ## very small negatives were rounded to 0 in R 2.2.1 and earlier > > ## dbeta(*, ncp): > db.x <- c(0, 5, 80, 405, 1280, 3125, 6480, 12005, 20480, 32805, + 50000, 73205, 103680, 142805, 192080, 253125, 327680) > a <- rlnorm(100) > stopifnot(All.eq(a, dbeta(0, 1, a, ncp=0)), + dbeta(0, 0.9, 2.2, ncp = c(0, a)) == Inf, + All.eq(65536 * dbeta(0:16/16, 5,1), db.x), + All.eq(exp(16 * log(2) + dbeta(0:16/16, 5,1, log=TRUE)), db.x) + ) > ## the first gave 0, the 2nd NaN in R <= 2.3.0; others use 'TRUE' values > > ## 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)) + ) > > ## qt(p ~ 0, df=1) - PR#9804 > p <- 10^(-10:-20) > qtp <- qt(p, df = 1) > ## relative error < 10^-14 : > stopifnot(abs(1 - p / pt(qtp, df=1)) < 1e-14) > > ## Similarly for df = 2 --- both for p ~ 0 *and* p ~ 1/2 > ## P ~ 0 > stopifnot(all.equal(qt(-740, df=2, log=TRUE), -exp(370)/sqrt(2))) > ## P ~ 1 (=> p ~ 0.5): > p.5 <- 0.5 + 2^(-5*(5:8)) > p.5 - 0.5 [1] 2.980232e-08 9.313226e-10 2.910383e-11 9.094947e-13 > stopifnot(all.equal(qt(p.5, df = 2), + c(8.429369702179e-08, 2.634178031931e-09, + 8.231806349784e-11, 2.572439484308e-12))) > ## qt(, log = TRUE) is now more finite and monotone (again!): > stopifnot(all.equal(qt(-1000, df = 4, log=TRUE), + -4.930611e108, tol = 1e-6)) > qtp <- qt(-(20:850), df=1.2, log=TRUE, lower=FALSE) > ##almost: stopifnot(all(abs(5/6 - diff(log(qtp))) < 1e-11)) > stopifnot(abs(5/6 - quantile(diff(log(qtp)), pr=c(0,0.995))) < 1e-11) > > ## close to df=1 (where Taylor steps are important!): > all.equal(-20, pt(qt(-20, df=1.02, log=TRUE), + df=1.02, log=TRUE), tol = 1e-12) [1] TRUE > stopifnot(diff(lq <- log(qt(-2^-(10:600), df=1.1, log=TRUE))) > 0.6) > lq1 <- log(qt(-2^-(20:600), df=1, log=TRUE)) > lq2 <- log(qt(-2^-(20:600), df=2, log=TRUE)) > stopifnot(mean(abs(diff(lq1) - log(2) )) < 1e-8, + mean(abs(diff(lq2) - log(sqrt(2)))) < 4e-8) > > ## pbeta(*, log=TRUE) {toms708} -- now improved tail behavior > x <- c(.01, .10, .25, .40, .55, .71, .98) > pbval <- c(-0.04605755624088, -0.3182809860569, -0.7503593555585, + -1.241555830932, -1.851527837938, -2.76044482378, -8.149862739881) > all.equal(pbeta(x, 0.8, 2, lower=FALSE, log=TRUE), pbval) [1] TRUE > all.equal(pbeta(1-x, 2, 0.8, log=TRUE), pbval) [1] TRUE > qq <- 2^(0:1022) > df.set <- c(0.1, 0.2, 0.5, 1, 1.2, 2.2, 5, 10, 20, 50, 100, 500) > for(nu in df.set) { + pqq <- pt(-qq, df = nu, log=TRUE) + stopifnot(is.finite(pqq)) + } > > All.eq(pt(2^-30, df=10), + 0.50000000036238542)# = .5+ integrate(dt, 0,2^-30, df=10, rel.tol=1e-20) [1] TRUE > > ## rbinom(*, size) gave NaN for large size up to R <= 2.6.1 > M <- .Machine$integer.max > set.seed(7) > tt <- table(rbinom(100, M, pr = 1e-9)) # had values in {0,2} only > t2 <- table(rbinom(100, 10*M, pr = 1e-10)) > stopifnot(names(tt) == 0:6, sum(tt) == 100, sum(t2) == 100) ## no NaN there > > > > cat("Time elapsed: ", proc.time() - .ptime,"\n") Time elapsed: 2.367 0.027 2.396 0 0 >