#### 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") } ##__ 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))) } ##__ 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)))) } ##__ 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))) All.eq(pnbinom(c(1,3), .9, .5), c(0.777035760338812, 0.946945347071519)) ##__ 5. Poisson __ all(dpois(0:5,0) == c(1, rep(0,5))) all(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)) { tst <- all.equal(1 - pchisq(2*lambda, 2*(1+ 0:k)), pp <- cumsum(dpois(0:k, lambda=lambda)), tolerance = 100*Meps) if(!isTRUE(tst)) cat("lambda=", format(lambda),". k =",k, " --> tst=", tst,"\n") tst2 <- all.equal(pp, ppois(0:k, lambda=lambda), tolerance = 100*Meps) if(!isTRUE(tst2)) cat("lambda=", format(lambda),". k =",k, " --> tst2=", tst2,"\n") tst3 <- all.equal(1 - pp, ppois(0:k, lambda=lambda, lower.tail = FALSE)) if(!isTRUE(tst3)) cat("lambda=", format(lambda),". k =",k, " --> tst3=", tst3,"\n") } ##__ 6. SignRank __ for(n in rpois(32, lam=8)) { x <- -1:(n + 4) if(!isTRUE(eq <- All.eq(psignrank(x, n), cumsum(dsignrank(x, n))))) print(eq) } ##__ 7. Wilcoxon (symmetry & cumulative) __ is.sym <- TRUE for(n in rpois(5, lam=6)) for(m in rpois(15, lam=8)) { x <- -1:(n*m + 1) fx <- dwilcox(x, n, m) Fx <- pwilcox(x, n, m) is.sym <- is.sym & all(fx == dwilcox(x, m, n)) if(!isTRUE(eq <- All.eq(Fx, cumsum(fx)))) print(eq) } is.sym ###-------- Continuous Distributions ---------- ##--- Gamma (incl. central chi^2) Density : x <- round(rgamma(100, shape = 2),2) for(sh in round(rlnorm(30),2)) { Ga <- gamma(sh) for(sig in round(rlnorm(30),2)) { tst <- all.equal((d1 <- dgamma( x, shape = sh, scale = sig)), (d2 <- dgamma(x/sig, shape = sh, scale = 1) / sig), tolerance = 1e-14)## __ad interim__ was 1e-15 if(!isTRUE(tst)) cat("ERROR: dgamma() doesn't scale:",tst,"\n", " x =", formatC(x),"\n shape,scale=",formatC(c(sh, sig)),"\n") tst <- All.eq(d1, (d3 <- 1/(Ga * sig^sh) * x^(sh-1) * exp(-x/sig))) if(!isTRUE(tst)) cat("NOT Equal:",tst,"\n x =", formatC(x), "\n shape,scale=",formatC(c(sh, sig)),"\n") } } pgamma(1,Inf,scale=Inf) == 0 ## 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)))) scLrg <- c(2,100, 1e300*c(.1, 1,10,100), 1e307, xMax, Inf) stopifnot(pgamma(Inf, 1, scale=xMax) == 1, pgamma(xMax,1, scale=Inf) == 0, all.equal(pgamma(1e300, 2, scale= scLrg, log=TRUE), c(0, 0, -0.000499523968713701, -1.33089326820406, -5.36470502873211, -9.91015144019122, -32.9293385491433, -38.707517174609, -Inf), tolerance = 2e-15) ) p <- 7e-4; df <- 0.9 abs(1-c(pchisq(qchisq(p, df),df)/p, # was 2.31e-8 for R <= 1.8.1 pchisq(qchisq(1-p, df,lower=FALSE),df,lower=FALSE)/(1-p),# was 1.618e-11 pchisq(qchisq(log(p), df,log=TRUE),df, log=TRUE)/log(p), # was 3.181e-9 pchisq(qchisq(log1p(-p),df,log=T,lower=F),df, log=T,lower=F)/log1p(-p) )# 32b-i386: (2.2e-16, 0,0, 3.3e-16); Opteron: (2.2e-16, 0,0, 2.2e-15) ) < 1e-14 ##-- non central Chi^2 : xB <- c(2000,1e6,1e50,Inf) for(df in c(0.1, 1, 10)) for(ncp in c(0, 1, 10, 100)) stopifnot(pchisq(xB, df=df, ncp=ncp) == 1) all.equal(qchisq(0.025,31,ncp=1,lower.tail=FALSE),# inf.loop PR#875 49.7766246561514, tolerance = 1e-11) for(df in c(0.1, 0.5, 1.5, 4.7, 10, 20,50,100)) { cat("df =", formatC(df, wid=3)) xx <- c(10^-(5:1), .9, 1.2, df + c(3,7,20,30,35,38)) pp <- pchisq(xx, df=df, ncp = 1) #print(pp) dtol <- 1e-12 *(if(2 < df && df <= 50) 64 else if(df > 50) 20000 else 501) print(all.equal(xx, qchisq(pp, df=df, ncp=1), tolerance = dtol))# TRUE ##or print(mapply(rErr, xx, qchisq(pp, df=df,ncp=1)), digits = 3) } ## p ~= 1 (<==> 1-p ~= 0) -- gave infinite loop in R <= 1.8.1 -- PR#6421 options(warn=-1) # ignore warnings from R's version of log1p psml <- 2^-(10:54) q0 <- qchisq(psml, df=1.2, ncp=10, lower.tail=FALSE) q1 <- qchisq(1-psml, df=1.2, ncp=10) # inaccurate in the tail p0 <- pchisq(q0, df=1.2, ncp=10, lower.tail=FALSE) p1 <- pchisq(q1, df=1.2, ncp=10, lower.tail=FALSE) iO <- 1:30 all.equal(q0[iO], q1[iO], tolerance = 1e-5) all.equal(p0[iO], psml[iO]) options(warn=0) ##--- Beta (need more): ## big a & b (PR #643) summary(a <- rlnorm(20, 5.5)) summary(b <- rlnorm(20, 6.5)) pab <- expand.grid(seq(0,1,by=.1), a, b) p <- pab[,1]; a <- pab[,2]; b <- pab[,3] all.equal(dbeta(p,a,b), exp(pab <- dbeta(p,a,b, log = TRUE)), tolerance = 1e-11) sample(pab, 50) ##--- Normal (& Lognormal) : qnorm(0) == -Inf && qnorm(-Inf, log = TRUE) == -Inf qnorm(1) == Inf && qnorm(0, log = TRUE) == Inf is.nan(qnorm(1.1)) && is.nan(qnorm(-.1)) # + warn 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)) ## 3 Test data from Wichura (1988) : all.equal(qnorm(c( 0.25, .001, 1e-20)), c(-0.6744897501960817, -3.090232306167814, -9.262340089798408), tolerance = 1e-15) # 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) z <- c(-Inf,Inf,NA,NaN, rt(1000, df=2)) z.ok <- z > -37.5 | !is.finite(z) for(df in 1:10) if(!isTRUE(all.equal(pt(z, df), 1 - pt(-z,df), tolerance = 1e-15))) cat("ERROR -- df = ", df, "\n") All.eq(pz <- pnorm(z), 1 - pnorm(z, lower=FALSE)) 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 <- 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)) ## 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))) ###========== p <-> q Inversion consistency ===================== ok <- 1e-5 < pz & pz < 1 - 1e-5 all.equal(z[ok], qnorm(pz[ok]), tolerance = 1e-12) ###===== 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) ) (Rbinom <- rbinom (n, size = 55, prob = pi/16) ) (Rcauchy <- rcauchy (n, location = 12, scale = 2) ) (Rchisq <- rchisq (n, df = 3) ) (Rexp <- rexp (n, rate = 2) ) (Rf <- rf (n, df1 = 12, df2 = 6) ) (Rgamma <- rgamma (n, shape = 2, scale = 5) ) (Rgeom <- rgeom (n, prob = pi/16) ) (Rhyper <- rhyper (n, m = 40, n = 30, k = 20) ) (Rlnorm <- rlnorm (n, meanlog = -1, sdlog = 3) ) (Rlogis <- rlogis (n, location = 12, scale = 2) ) (Rnbinom <- rnbinom (n, size = 7, prob = .01) ) (Rnorm <- rnorm (n, mean = -1, sd = 3) ) (Rpois <- rpois (n, lambda = 12) ) (Rsignrank<- rsignrank(n, n = 47) ) (Rt <- rt (n, df = 11) ) ## Rt2 below (to preserve the following random numbers!) (Runif <- runif (n, min = .2, max = 2) ) (Rweibull <- rweibull (n, shape = 3, scale = 2) ) (Rwilcox <- rwilcox (n, m = 13, n = 17) ) (Rt2 <- rt (n, df = 1.01)) (Pbeta <- pbeta (Rbeta, shape1 = .8, shape2 = 2) ) (Pbinom <- pbinom (Rbinom, size = 55, prob = pi/16) ) (Pcauchy <- pcauchy (Rcauchy, location = 12, scale = 2) ) (Pchisq <- pchisq (Rchisq, df = 3) ) (Pexp <- pexp (Rexp, rate = 2) ) (Pf <- pf (Rf, df1 = 12, df2 = 6) ) (Pgamma <- pgamma (Rgamma, shape = 2, scale = 5) ) (Pgeom <- pgeom (Rgeom, prob = pi/16) ) (Phyper <- phyper (Rhyper, m = 40, n = 30, k = 20) ) (Plnorm <- plnorm (Rlnorm, meanlog = -1, sdlog = 3) ) (Plogis <- plogis (Rlogis, location = 12, scale = 2) ) (Pnbinom <- pnbinom (Rnbinom, size = 7, prob = .01) ) (Pnorm <- pnorm (Rnorm, mean = -1, sd = 3) ) (Ppois <- ppois (Rpois, lambda = 12) ) (Psignrank<- psignrank(Rsignrank, n = 47) ) (Pt <- pt (Rt, df = 11) ) (Pt2 <- pt (Rt2, df = 1.01) ) (Punif <- punif (Runif, min = .2, max = 2) ) (Pweibull <- pweibull (Rweibull, shape = 3, scale = 2) ) (Pwilcox <- pwilcox (Rwilcox, m = 13, n = 17) ) dbeta (Rbeta, shape1 = .8, shape2 = 2) dbinom (Rbinom, size = 55, prob = pi/16) dcauchy (Rcauchy, location = 12, scale = 2) dchisq (Rchisq, df = 3) dexp (Rexp, rate = 2) df (Rf, df1 = 12, df2 = 6) dgamma (Rgamma, shape = 2, scale = 5) dgeom (Rgeom, prob = pi/16) dhyper (Rhyper, m = 40, n = 30, k = 20) dlnorm (Rlnorm, meanlog = -1, sdlog = 3) dlogis (Rlogis, location = 12, scale = 2) dnbinom (Rnbinom, size = 7, prob = .01) dnorm (Rnorm, mean = -1, sd = 3) dpois (Rpois, lambda = 12) dsignrank(Rsignrank, n = 47) dt (Rt, df = 11) dunif (Runif, min = .2, max = 2) dweibull (Rweibull, shape = 3, scale = 2) dwilcox (Rwilcox, m = 13, n = 17) ## Check q*(p*(.)) = identity All.eq(Rbeta, qbeta (Pbeta, shape1 = .8, shape2 = 2)) All.eq(Rbinom, qbinom (Pbinom, size = 55, prob = pi/16)) All.eq(Rcauchy, qcauchy (Pcauchy, location = 12, scale = 2)) All.eq(Rchisq, qchisq (Pchisq, df = 3)) All.eq(Rexp, qexp (Pexp, rate = 2)) All.eq(Rf, qf (Pf, df1 = 12, df2 = 6)) All.eq(Rgamma, qgamma (Pgamma, shape = 2, scale = 5)) All.eq(Rgeom, qgeom (Pgeom, prob = pi/16)) All.eq(Rhyper, qhyper (Phyper, m = 40, n = 30, k = 20)) All.eq(Rlnorm, qlnorm (Plnorm, meanlog = -1, sdlog = 3)) All.eq(Rlogis, qlogis (Plogis, location = 12, scale = 2)) All.eq(Rnbinom, qnbinom (Pnbinom, size = 7, prob = .01)) All.eq(Rnorm, qnorm (Pnorm, mean = -1, sd = 3)) All.eq(Rpois, qpois (Ppois, lambda = 12)) All.eq(Rsignrank, qsignrank(Psignrank, n = 47)) All.eq(Rt, qt (Pt, df = 11)) all.equal(Rt2, qt (Pt2, df = 1.01), tolerance = 1e-2) All.eq(Runif, qunif (Punif, min = .2, max = 2)) All.eq(Rweibull, qweibull (Pweibull, shape = 3, scale = 2)) All.eq(Rwilcox, qwilcox (Pwilcox, m = 13, n = 17)) ## Same with "upper tail": All.eq(Rbeta, qbeta (1- Pbeta, shape1 = .8, shape2 = 2, lower=F)) All.eq(Rbinom, qbinom (1- Pbinom, size = 55, prob = pi/16, lower=F)) All.eq(Rcauchy, qcauchy (1- Pcauchy, location = 12, scale = 2, lower=F)) All.eq(Rchisq, qchisq (1- Pchisq, df = 3, lower=F)) All.eq(Rexp, qexp (1- Pexp, rate = 2, lower=F)) All.eq(Rf, qf (1- Pf, df1 = 12, df2 = 6, lower=F)) All.eq(Rgamma, qgamma (1- Pgamma, shape = 2, scale = 5, lower=F)) All.eq(Rgeom, qgeom (1- Pgeom, prob = pi/16, lower=F)) All.eq(Rhyper, qhyper (1- Phyper, m = 40, n = 30, k = 20, lower=F)) All.eq(Rlnorm, qlnorm (1- Plnorm, meanlog = -1, sdlog = 3, lower=F)) All.eq(Rlogis, qlogis (1- Plogis, location = 12, scale = 2, lower=F)) All.eq(Rnbinom, qnbinom (1- Pnbinom, size = 7, prob = .01, lower=F)) All.eq(Rnorm, qnorm (1- Pnorm, mean = -1, sd = 3,lower=F)) All.eq(Rpois, qpois (1- Ppois, lambda = 12, lower=F)) All.eq(Rsignrank, qsignrank(1- Psignrank, n = 47, lower=F)) All.eq(Rt, qt (1- Pt, df = 11, lower=F)) all.equal(Rt2, qt (1- Pt2, df = 1.01, lower=F), tolerance = 1e-2) All.eq(Runif, qunif (1- Punif, min = .2, max = 2, lower=F)) All.eq(Rweibull, qweibull (1- Pweibull, shape = 3, scale = 2, lower=F)) All.eq(Rwilcox, qwilcox (1- Pwilcox, m = 13, n = 17, lower=F)) ## Check q*(p* ( log ), log) = identity All.eq(Rbeta, qbeta (log(Pbeta), shape1 = .8, shape2 = 2, log=TRUE)) All.eq(Rbinom, qbinom (log(Pbinom), size = 55, prob = pi/16, log=TRUE)) All.eq(Rcauchy, qcauchy (log(Pcauchy), location = 12, scale = 2, log=TRUE)) all.equal(Rchisq, qchisq (log(Pchisq), df = 3, log=TRUE),tolerance =1e-14) All.eq(Rexp, qexp (log(Pexp), rate = 2, log=TRUE)) All.eq(Rf, qf (log(Pf), df1= 12, df2= 6, log=TRUE)) All.eq(Rgamma, qgamma (log(Pgamma), shape = 2, scale = 5, log=TRUE)) All.eq(Rgeom, qgeom (log(Pgeom), prob = pi/16, log=TRUE)) All.eq(Rhyper, qhyper (log(Phyper), m = 40, n = 30, k = 20, log=TRUE)) All.eq(Rlnorm, qlnorm (log(Plnorm), meanlog = -1, sdlog = 3, log=TRUE)) All.eq(Rlogis, qlogis (log(Plogis), location = 12, scale = 2, log=TRUE)) All.eq(Rnbinom, qnbinom (log(Pnbinom), size = 7, prob = .01, log=TRUE)) All.eq(Rnorm, qnorm (log(Pnorm), mean = -1, sd = 3, log=TRUE)) All.eq(Rpois, qpois (log(Ppois), lambda = 12, log=TRUE)) All.eq(Rsignrank, qsignrank(log(Psignrank), n = 47, log=TRUE)) All.eq(Rt, qt (log(Pt), df = 11, log=TRUE)) all.equal(Rt2, qt (log(Pt2), df = 1.01, log=TRUE), tolerance = 1e-2) All.eq(Runif, qunif (log(Punif), min = .2, max = 2, log=TRUE)) All.eq(Rweibull, qweibull (log(Pweibull), shape = 3, scale = 2, log=TRUE)) All.eq(Rwilcox, qwilcox (log(Pwilcox), m = 13, n = 17, log=TRUE)) ## same q*(p* (log) log) with upper tail: All.eq(Rbeta, qbeta (log1p(-Pbeta), shape1 = .8, shape2 = 2, lower=F, log=T)) All.eq(Rbinom, qbinom (log1p(-Pbinom), size = 55, prob = pi/16, lower=F, log=T)) All.eq(Rcauchy, qcauchy (log1p(-Pcauchy), location = 12, scale = 2, lower=F, log=T)) All.eq(Rchisq, qchisq (log1p(-Pchisq), df = 3, lower=F, log=T)) All.eq(Rexp, qexp (log1p(-Pexp), rate = 2, lower=F, log=T)) All.eq(Rf, qf (log1p(-Pf), df1 = 12, df2 = 6, lower=F, log=T)) All.eq(Rgamma, qgamma (log1p(-Pgamma), shape = 2, scale = 5, lower=F, log=T)) All.eq(Rgeom, qgeom (log1p(-Pgeom), prob = pi/16, lower=F, log=T)) All.eq(Rhyper, qhyper (log1p(-Phyper), m = 40, n = 30, k = 20, lower=F, log=T)) All.eq(Rlnorm, qlnorm (log1p(-Plnorm), meanlog = -1, sdlog = 3, lower=F, log=T)) All.eq(Rlogis, qlogis (log1p(-Plogis), location = 12, scale = 2, lower=F, log=T)) All.eq(Rnbinom, qnbinom (log1p(-Pnbinom), size = 7, prob = .01, lower=F, log=T)) All.eq(Rnorm, qnorm (log1p(-Pnorm), mean = -1, sd = 3, lower=F, log=T)) All.eq(Rpois, qpois (log1p(-Ppois), lambda = 12, lower=F, log=T)) All.eq(Rsignrank, qsignrank(log1p(-Psignrank), n = 47, lower=F, log=T)) All.eq(Rt, qt (log1p(-Pt ), df = 11, lower=F, log=T)) all.equal(Rt2, qt (log1p(-Pt2), df = 1.01, lower=F, log=T), tolerance = 1e-2) All.eq(Runif, qunif (log1p(-Punif), min = .2, max = 2, lower=F, log=T)) All.eq(Rweibull, qweibull (log1p(-Pweibull), shape = 3, scale = 2, lower=F, log=T)) All.eq(Rwilcox, qwilcox (log1p(-Pwilcox), m = 13, n = 17, lower=F, log=T)) ## Check log( upper.tail ): All.eq(log1p(-Pbeta), pbeta (Rbeta, shape1 = .8, shape2 = 2, lower=F, log=T)) All.eq(log1p(-Pbinom), pbinom (Rbinom, size = 55, prob = pi/16, lower=F, log=T)) All.eq(log1p(-Pcauchy), pcauchy (Rcauchy, location = 12, scale = 2, lower=F, log=T)) All.eq(log1p(-Pchisq), pchisq (Rchisq, df = 3, lower=F, log=T)) All.eq(log1p(-Pexp), pexp (Rexp, rate = 2, lower=F, log=T)) All.eq(log1p(-Pf), pf (Rf, df1 = 12, df2 = 6, lower=F, log=T)) All.eq(log1p(-Pgamma), pgamma (Rgamma, shape = 2, scale = 5, lower=F, log=T)) All.eq(log1p(-Pgeom), pgeom (Rgeom, prob = pi/16, lower=F, log=T)) All.eq(log1p(-Phyper), phyper (Rhyper, m = 40, n = 30, k = 20, lower=F, log=T)) All.eq(log1p(-Plnorm), plnorm (Rlnorm, meanlog = -1, sdlog = 3, lower=F, log=T)) All.eq(log1p(-Plogis), plogis (Rlogis, location = 12, scale = 2, lower=F, log=T)) All.eq(log1p(-Pnbinom), pnbinom (Rnbinom, size = 7, prob = .01, lower=F, log=T)) All.eq(log1p(-Pnorm), pnorm (Rnorm, mean = -1, sd = 3, lower=F, log=T)) All.eq(log1p(-Ppois), ppois (Rpois, lambda = 12, lower=F, log=T)) All.eq(log1p(-Psignrank), psignrank(Rsignrank, n = 47, lower=F, log=T)) All.eq(log1p(-Pt), pt (Rt, df = 11, lower=F, log=T)) All.eq(log1p(-Pt2), pt (Rt2,df = 1.01, lower=F, log=T)) All.eq(log1p(-Punif), punif (Runif, min = .2, max = 2, lower=F, log=T)) All.eq(log1p(-Pweibull), pweibull (Rweibull, shape = 3, scale = 2, lower=F, log=T)) All.eq(log1p(-Pwilcox), pwilcox (Rwilcox, m = 13, n = 17, lower=F, log=T)) ### (Extreme) tail tests added more recently: All.eq(1, -1e-17/ pexp(qexp(-1e-17, log=TRUE),log=TRUE)) abs(pgamma(30,100, lower=FALSE, log=TRUE) + 7.3384686328784e-24) < 1e-36 All.eq(1, pcauchy(-1e20) / 3.18309886183791e-21) All.eq(1, pcauchy(+1e15, log=TRUE) / -3.18309886183791e-16)## PR#6756 x <- 10^(ex <- c(1,2,5*(1:5),50,100,200,300,Inf)) for(a in x[ex > 10]) ## improve pt() : cbind(x,t= pt(-x, df=1), C=pcauchy(-x)) print(all.equal(pt(-a, df=1), pcauchy(-a), tolerance = 1e-15)) ## for PR#7902: ex <- -c(rev(1/x), ex) All.eq(-x, qcauchy(pcauchy(-x))) All.eq(+x, qcauchy(pcauchy(+x, log=TRUE), log=TRUE)) All.eq(1/x, pcauchy(qcauchy(1/x))) All.eq(ex, pcauchy(qcauchy(ex, log=TRUE), log=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))) 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) all.equal(pgamma(x, 0.1, log = TRUE) - 0.1*log(x), rep(0.0498724412598364, length(x)), tolerance = 1e-13)# 7e-16 (i386) All.eq(dpois( 10*1:2, 3e-308, log=TRUE), c(-7096.08037610806, -14204.2875435307)) All.eq(dpois(1e20, 1e-290, log=TRUE), -7.12801378828154e+22) ## 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) df(x, 3, Inf) pf(x, 3, 1e6) pf(x, 3, Inf) df(x, 1e6, 5) df(x, Inf, 5) pf(x, 1e6, 5) pf(x, Inf, 5) df(x, Inf, Inf)# (0, Inf, 0) - since 2.1.1 pf(x, Inf, Inf)# (0, 1/2, 1) pf(x, 5, Inf, ncp=0) all.equal(pf(x, 5, 1e6, ncp=1), tolerance = 1e-6, c(0.065933194, 0.470879987, 0.978875867)) all.equal(pf(x, 5, 1e7, ncp=1), tolerance = 1e-6, c(0.06593309, 0.47088028, 0.97887641)) all.equal(pf(x, 5, 1e8, ncp=1), tolerance = 1e-6, c(0.0659330751, 0.4708802996, 0.9788764591)) pf(x, 5, Inf, ncp=1) dt(1, Inf) dt(1, Inf, ncp=0) dt(1, Inf, ncp=1) dt(1, 1e6, ncp=1) dt(1, 1e7, ncp=1) dt(1, 1e8, ncp=1) dt(1, 1e10, ncp=1) # = Inf ## 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)) ## small 'x' used to suffer from cancellation options(oo) x <- c(outer(1:12, 10^c(-3:2, 6:9, 10*(2:30)))) for(nu in c(.75, 1.2, 4.5, 999, 1e50)) { lfx <- dt(x, df=nu, log=TRUE) stopifnot(is.finite(lfx), All.eq(exp(lfx), dt(x, df=nu))) }## dt(1e160, 1.2, log=TRUE) was -Inf up to R 2.15.2 ## pf() with large df1 or df2 ## (was said to be PR#7099, but that is about non-central pchisq) nu <- 2^seq(25, 34, 0.5) target <- pchisq(1, 1) # 0.682... y <- pf(1, 1, nu) stopifnot(All.eq(pf(1, 1, Inf), target), diff(c(y, target)) > 0, # i.e. pf(1, 1, *) is monotone increasing abs(y[1] - (target - 7.21129e-9)) < 1e-11) # computed value ## non-monotone in R <= 2.1.0 stopifnot(pgamma(Inf, 1.1) == 1) ## didn't not terminate in R 2.1.x (only) ## qgamma(q, *) should give {0,Inf} for q={0,1} sh <- c(1.1, 0.5, 0.2, 0.15, 1e-2, 1e-10) stopifnot(Inf == qgamma(1, sh)) stopifnot(0 == qgamma(0, sh)) ## the first gave Inf, NaN, and 99.425 in R 2.1.1 and earlier ## In extreme left tail {PR#11030} p <- 10:123*1e-12 qg <- qgamma(p, shape=19) qg2<- qgamma(1:100 * 1e-9, shape=11) stopifnot(diff(qg, diff=2) < -6e-6, diff(qg2,diff=2) < -6e-6, abs(1 - pgamma(qg, 19)/ p) < 1e-13, All.eq(qg [1], 2.35047385139143), All.eq(qg2[30], 1.11512318734547)) ## was non-continuous in R 2.6.2 and earlier f2 <- c(0.5, 1:4) stopifnot(df(0, 1, f2) == Inf, df(0, 2, f2) == 1, df(0, 3, f2) == 0) ## only the last one was ok in R 2.2.1 and earlier x0 <- -2 * 10^-c(22,10,7,5) stopifnot(pbinom(x0, size = 3, prob = 0.1) == 0, dbinom(x0, 3, 0.1) == 0) # d*() warns about non-integer ## very small negatives were rounded to 0 in R 2.2.1 and earlier ## dbeta(*, ncp): db.x <- c(0, 5, 80, 405, 1280, 3125, 6480, 12005, 20480, 32805, 50000, 73205, 103680, 142805, 192080, 253125, 327680) a <- rlnorm(100) stopifnot(All.eq(a, dbeta(0, 1, a, ncp=0)), dbeta(0, 0.9, 2.2, ncp = c(0, a)) == Inf, All.eq(65536 * dbeta(0:16/16, 5,1), db.x), All.eq(exp(16 * log(2) + dbeta(0:16/16, 5,1, log=TRUE)), db.x) ) ## the first gave 0, the 2nd NaN in R <= 2.3.0; others use 'TRUE' values stopifnot(all.equal(dbeta(0.8, 0.5, 5, ncp=1000),# was way too small in R <= 2.6.2 3.001852308909e-35), all.equal(1, integrate(dbeta, 0,1, 0.8, 0.5, ncp=1000)$value, tolerance = 1e-4), all.equal(1, integrate(dbeta, 0,1, 0.5, 200, ncp=720)$value), all.equal(1, integrate(dbeta, 0,1, 125, 200, ncp=2000)$value) ) ## df(*, ncp): x <- seq(0, 10, length=101) h <- 1e-7 dx.h <- (pf(x+h, 7, 5, ncp= 2.5) - pf(x-h, 7, 5, ncp= 2.5)) / (2*h) stopifnot(all.equal(dx.h, df(x, 7, 5, ncp= 2.5), tolerance = 1e-6),# (1.50 | 1.65)e-8 All.eq(df(0, 2, 4, ncp=x), df(1e-300, 2, 4, ncp=x)) ) ## qt(p ~ 0, df=1) - PR#9804 p <- 10^(-10:-20) qtp <- qt(p, df = 1) ## relative error < 10^-14 : stopifnot(abs(1 - p / pt(qtp, df=1)) < 1e-14) ## Similarly for df = 2 --- both for p ~ 0 *and* p ~ 1/2 ## P ~ 0 stopifnot(all.equal(qt(-740, df=2, log=TRUE), -exp(370)/sqrt(2))) ## P ~ 1 (=> p ~ 0.5): p.5 <- 0.5 + 2^(-5*(5:8)) p.5 - 0.5 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, tolerance = 1e-6)) qtp <- qt(-(20:850), df=1.2, log=TRUE, lower=FALSE) ##almost: stopifnot(all(abs(5/6 - diff(log(qtp))) < 1e-11)) stopifnot(abs(5/6 - quantile(diff(log(qtp)), pr=c(0,0.995))) < 1e-11) ## close to df=1 (where Taylor steps are important!): all.equal(-20, pt(qt(-20, df=1.02, log=TRUE), df=1.02, log=TRUE), tolerance = 1e-12) stopifnot(diff(lq <- log(qt(-2^-(10:600), df=1.1, log=TRUE))) > 0.6) lq1 <- log(qt(-2^-(20:600), df=1, log=TRUE)) lq2 <- log(qt(-2^-(20:600), df=2, log=TRUE)) stopifnot(mean(abs(diff(lq1) - log(2) )) < 1e-8, mean(abs(diff(lq2) - log(sqrt(2)))) < 4e-8) ## Case, where log.p=TRUE was fine, but log.p=FALSE (default) gave NaN: lp <- 40:406 stopifnot(all.equal(lp, -pt(qt(exp(-lp), 1.2), 1.2, log=TRUE), tolerance = 4e-16)) ## pbeta(*, log=TRUE) {toms708} -- now improved tail behavior x <- c(.01, .10, .25, .40, .55, .71, .98) pbval <- c(-0.04605755624088, -0.3182809860569, -0.7503593555585, -1.241555830932, -1.851527837938, -2.76044482378, -8.149862739881) all.equal(pbeta(x, 0.8, 2, lower=FALSE, log=TRUE), pbval) 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)) } ## 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(qbeta(-1e10, 50,40, log.p=TRUE)), is.finite(qbeta(-1e10, 2, 3, lower=FALSE, log.p=TRUE))) ## infinite loop or NaN in R <= 2.7.0 ## phyper(x, 0,0,0), notably for huge x stopifnot(all(phyper(c(0:3, 1e67), 0,0,0) == 1)) ## practically infinite loop and NaN in R <= 2.7.1 (PR#11813) ## plnorm(<= 0, . , log.p=TRUE) stopifnot(plnorm(-1:0, lower.tail=FALSE, log.p=TRUE) == 0, plnorm(-1:0, lower.tail=TRUE, log.p=TRUE) == -Inf) ## was wrongly == 'log.p=FALSE' up to R <= 2.7.1 (PR#11867) ## pchisq(df=0) was wrong in 2.7.1; then, upto 2.10.1, P*(0,0) gave 1 stopifnot(pchisq(c(-1,0,1), df=0) == c(0,0,1), pchisq(c(-1,0,1), df=0, lower.tail=FALSE) == c(1,1,0), ## for ncp >= 80, gave values >= 1 in 2.10.0 pchisq(500:700, 1.01, ncp = 80) <= 1) ## dnbinom for extreme size and/or mu : mu <- 20 d <- dnbinom(17, mu=mu, size = 1e11*2^(1:10)) - dpois(17, lambda=mu) stopifnot(d < 0, diff(d) > 0, d[1] < 1e-10) ## was wrong up to 2.7.1 ## The fix to the above, for x = 0, had a new cancellation problem mu <- 1e12 * 2^(0:20) stopifnot(all.equal(1/(1+mu), dnbinom(0, size = 1, mu = mu), tolerance = 1e-13)) ## was wrong in 2.7.2 (only) 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, ,, .., 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 ## 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) cat("Time elapsed: ", proc.time() - .ptime,"\n")