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R : Copyright 2000, The R Development Core TeamVersion 1.1.0 Under development (unstable) (June 6, 2000)R is free software and comes with ABSOLUTELY NO WARRANTY.You are welcome to redistribute it under certain conditions.Type "?license" or "?licence" for distribution details.R is a collaborative project with many contributors.Type "?contributors" for a list.Type "demo()" for some demos, "help()" for on-line help, or"help.start()" for a 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()> ###-- 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)+ }> ## Short cut:> All.eq <- function(x,y) all.equal.numeric(x,y, tolerance = 1e-14)>> if(!interactive())+ .Random.seed <- c(0,rep(7654, 3))>> ###--- Discrete Distributions: Simple Consistency Checks pZZ = cumsum(dZZ)>> 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>> ## Currently, just Wilcoxon [should do this for all !]> 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))+ eq <- All.eq(Fx, cumsum(fx))+ if(!is.logical(eq) || !eq) print(eq)+ }> is.sym[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*(k+1)),+ pp <- sum(dpois(0:k, lambda=lambda)), tol = 100*Meps)+ if(!(is.logical(tst) && tst))+ cat("lambda=", format(lambda),". k =",k, " --> tst=", tst,"\n")+ tst2 <- all.equal(pp, ppois(k, lambda=lambda), tol = 100*Meps)+ if(!(is.logical(tst2) && tst2))+ cat("lambda=", format(lambda),". k =",k, " --> tst2=", tst2,"\n")+ tst3 <- all.equal(1 - pp, ppois(k, lambda=lambda, lower.tail=FALSE))+ if(!(is.logical(tst3) && tst3))+ cat("lambda=", format(lambda),". k =",k, " --> tst3=", tst3,"\n")+ }>> ##--- 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 two ways:+ 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(!(is.logical(tst) && tst))+ cat("n=", n,"; p =",format(p),". k =",k, " --> tst=",tst,"\n")+ }+ }+ cat("\n")+ }n= 33 : ......................n= 32 : ......................n= 33 : ......................n= 44 : ......................n= 39 : ......................n= 44 : ......................n= 41 : ......................n= 46 : ......................n= 30 : ......................n= 43 : ......................n= 44 : ......................n= 33 : ......................n= 37 : ......................n= 39 : ......................n= 37 : ......................n= 50 : ......................>> ##--- Gamma (incl. 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-15)+ if(!(is.logical(tst) && 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(!(is.logical(tst) && tst))+ cat("NOT Equal:",tst,"\n x =", formatC(x),+ "\n shape,scale=",formatC(c(sh, sig)),"\n")+ }+ }> pgamma(1,Inf,Inf) == 0[1] TRUE> all(is.nan(c(pgamma(Inf,1,Inf), pgamma(Inf,Inf,1), pgamma(Inf,Inf,Inf))))[1] TRUEWarning messages:1: NaNs produced in: pgamma(q, shape, scale, lower.tail, log.p)2: NaNs produced in: pgamma(q, shape, scale, lower.tail, log.p)3: NaNs produced in: pgamma(q, shape, scale, lower.tail, log.p)> pgamma(Inf,1,xMax) == 1 && pgamma(xMax,1,Inf) == 0[1] TRUE>> ##--- Normal (& Lognormal) :>> ## 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>> 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))> zsml <- z > -37.5 || !is.finite(z)> for(df in 1:10) if(!is.logical(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[zsml]), pnorm(z[zsml], 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 -2455.1672298[2,] -60 -Inf -1805.0130056[3,] -50 -Inf -1254.8305621[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>> 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>> ## The prefixes of ALL the PDQ & R functions> PDQR <- c("beta", "binom", "cauchy", "chisq", "exp", "f",+ "gamma", "geom", "hyper", "lnorm", "logis", "nbinom","norm",+ "pois","signrank","t","unif","weibull","wilcox")> PQonly <- c("tukey")>> ###===== Random numbers -- first, just output:>> .Random.seed <- c(0, 17292, 29447, 24113)> n <- 20> ## for(pre in PDQR) { n <- paste("r",pre,sep=""); cat(n,": "); str(get(n))}> (Rbeta <- rbeta (n, shape1 = .8, shape2 = 2) )[1] 0.0009459396 0.0749569119 0.7969897349 0.1173196499 0.2400710007[6] 0.1318767829 0.8277537923 0.0359498142 0.2403329282 0.0372784791[11] 0.3759311413 0.3581043996 0.2895229973 0.6005864534 0.1833902168[16] 0.1198140580 0.0706071423 0.0308280998 0.3591918485 0.4281862391> (Rbinom <- rbinom (n, size = 55, prob = pi/16) )[1] 12 10 11 13 16 6 5 12 13 9 17 12 14 5 5 13 10 12 16 6> (Rcauchy <- rcauchy (n, location = 12, scale = 2) )[1] 34.263018 12.540399 16.802259 11.789490 14.154479 10.462895 12.540308[8] 12.876666 12.537513 11.977077 11.807015 12.481572 11.954381 10.884467[15] 9.516357 12.935321 10.946190 14.864709 7.898369 12.782296> (Rchisq <- rchisq (n, df = 3) )[1] 4.1802057 2.2410473 8.3668743 0.7165071 4.6012338 0.7650135 2.1202187[8] 0.2898359 0.7288776 2.9159575 3.6549180 0.5587417 4.9140095 7.3809424[15] 5.6769453 4.9920360 0.6435048 1.1676611 3.5323618 1.6709330> (Rexp <- rexp (n, rate = 2) )[1] 0.07333367 2.13910864 1.32756654 0.11396374 0.07688834 1.38460882[7] 0.55605730 0.34815444 0.72538091 0.76093410 0.51832937 0.94396435[13] 0.11381630 0.65756405 0.29282882 0.04916325 0.18753254 0.76833277[19] 1.01917017 2.40313307> (Rf <- rf (n, df1 = 12, df2 = 6) )[1] 1.0482995 7.1506346 0.6029385 2.0461847 1.1851525 4.3366766 0.9365564[8] 1.2584864 0.4850872 1.4162343 1.3338440 1.1810704 0.7064808 0.5176781[15] 2.1632007 2.9753123 1.2178294 1.7130207 0.5469131 0.1313706> (Rgamma <- rgamma (n, shape = 2, scale = 5) )[1] 3.9092623 21.6048906 11.5498513 11.1312817 4.6122029 14.6707798[7] 0.9007555 18.9112724 9.4168271 30.4501082 4.5755714 3.2193519[13] 24.5309039 5.2340905 14.5772699 1.0665363 11.3585753 8.7876520[19] 8.0414810 14.5654162> (Rgeom <- rgeom (n, prob = pi/16) )[1] 4 2 5 1 10 0 0 1 0 17 0 3 1 3 3 6 0 0 2 1> (Rhyper <- rhyper (n, m = 40, n = 30, k = 20) )[1] 16 11 11 15 11 13 13 12 13 10 10 7 11 14 13 9 14 13 13 11> (Rlnorm <- rlnorm (n, meanlog = -1, sdlog = 3) )[1] 2.210483e-01 1.713584e-01 2.116665e+00 8.205068e+01 3.111700e+00[6] 1.883586e+00 4.389370e+00 7.965253e-04 3.287647e-01 4.398008e-01[11] 3.554842e+00 5.186465e-01 2.313136e+00 2.371949e-01 4.293877e-01[16] 6.577897e-02 2.645768e-01 9.650592e-02 3.650006e-01 7.501445e-02> (Rlogis <- rlogis (n, location = 12, scale = 2) )[1] 17.000716 14.220747 22.693644 13.270882 4.748890 16.217577 12.780626[8] 8.498878 8.300536 9.565866 13.896595 14.425404 17.849150 8.886460[15] 9.756902 13.182288 15.045885 15.877290 14.773741 9.637214> (Rnbinom <- rnbinom (n, size = 7, prob = .01) )[1] 739 668 724 609 1052 404 660 381 321 839 901 292 572 684 883[16] 1133 503 241 686 787> (Rnorm <- rnorm (n, mean = -1, sd = 3) )[1] -0.61029370 2.35074871 2.49862518 0.28586195 1.56824133 4.02175812[7] -3.40412993 -2.65797756 -0.49471250 -1.78043063 -0.54327159 4.38579193[13] 0.90049087 -0.06666040 -0.03921922 4.00360536 2.36688992 1.44720283[19] 1.12529029 0.14489413> (Rpois <- rpois (n, lambda = 12) )[1] 11 14 13 7 12 9 19 8 9 12 10 17 11 13 11 12 14 8 6 14> (Rsignrank<- rsignrank(n, n = 47) )[1] 521 680 465 368 636 497 560 439 541 462 459 595 542 699 725 563 419 508 774[20] 496> (Rt <- rt (n, df = 11) )[1] -1.73939322 -0.53416778 1.61643746 -0.09307946 0.19731802 0.13339707[7] -0.99463560 -1.49665151 1.44899521 -0.72605060 -0.52184248 1.62334863[13] -1.07413952 0.18925757 -0.08940649 -0.90018242 -0.61911928 1.17444416[19] -1.06421053 -0.77380129> (Runif <- runif (n, min = .2, max = 2) )[1] 1.8416169 0.4703079 0.6841623 0.5048212 1.5620335 0.2738142 1.1573172[8] 0.3854089 1.0601866 1.9827946 0.7149722 1.3607511 1.3590662 1.3110552[15] 1.5017535 1.0945284 1.7290782 0.6057555 1.7500334 1.7121475> (Rweibull <- rweibull (n, shape = 3, scale = 2) )[1] 1.512345 2.645024 1.458062 2.484227 1.263118 2.081812 1.058986 1.104576[9] 1.644986 1.985632 2.127461 2.058949 2.524137 1.716619 1.875699 1.433992[17] 2.362451 2.861329 2.150789 1.239032> (Rwilcox <- rwilcox (n, m = 13, n = 17) )[1] 133 103 103 62 117 108 96 102 118 115 110 92 139 126 77 65 127 72 139[20] 124>> (Pbeta <- pbeta (Rbeta, shape1 = .8, shape2 = 2) )[1] 0.00685142 0.21898139 0.96943981 0.30726448 0.51350239 0.33510112[7] 0.97810553 0.12383254 0.51388356 0.12740407 0.68542852 0.66557175[13] 0.58183279 0.87756637 0.42564683 0.31211434 0.20917242 0.10975854[19] 0.66680467 0.73943522> (Pbinom <- pbinom (Rbinom, size = 55, prob = pi/16) )[1] 0.72571464 0.47335812 0.60646826 0.82208269 0.96870000 0.06545671[7] 0.02841293 0.72571464 0.82208269 0.34018132 0.98484929 0.72571464[13] 0.89271720 0.02841293 0.02841293 0.82208269 0.47335812 0.72571464[19] 0.96870000 0.06545671> (Pcauchy <- pcauchy (Rcauchy, location = 12, scale = 2) )[1] 0.9714812 0.5840012 0.8743873 0.4666191 0.7618305 0.2914213 0.5839878[8] 0.6314968 0.5835731 0.4963519 0.4693803 0.5752128 0.4927407 0.3380482[15] 0.2157967 0.6392424 0.3456390 0.8059952 0.1444131 0.6186825> (Pchisq <- pchisq (Rchisq, df = 3) )[1] 0.75734885 0.47609135 0.96099212 0.13068538 0.79656374 0.14218428[7] 0.45216677 0.03807139 0.13360847 0.59523480 0.69878142 0.09419094[13] 0.82179683 0.93930223 0.87156985 0.82761876 0.11359536 0.23922949[19] 0.68341090 0.35658474> (Pexp <- pexp (Rexp, rate = 2) )[1] 0.13641880 0.98613264 0.92971052 0.20381800 0.14253652 0.93728895[7] 0.67113719 0.50157835 0.76560834 0.78169633 0.64536236 0.84861495[13] 0.20358320 0.73156007 0.44326040 0.09364707 0.31275544 0.78490286[19] 0.86975531 0.99182166> (Pf <- pf (Rf, df1 = 12, df2 = 6) )[1] 0.493968892 0.987819917 0.214465434 0.804851021 0.560142915 0.958476676[7] 0.432468011 0.591675896 0.134821391 0.651135448 0.621437022 0.558309651[13] 0.285976592 0.156229064 0.823256086 0.904828024 0.574518093 0.737314198[19] 0.175931308 0.001535476> (Pgamma <- pgamma (Rgamma, shape = 2, scale = 5) )[1] 0.18469983 0.92930080 0.67143844 0.65178456 0.23573834 0.79079896[7] 0.01440388 0.89110187 0.56150517 0.98393855 0.23305255 0.13654797[13] 0.95629013 0.28145838 0.78786294 0.01975953 0.66257119 0.52440798[19] 0.47774483 0.78748817> (Pgeom <- pgeom (Rgeom, prob = pi/16) )[1] 0.6647753 0.4809591 0.7305965 0.3541459 0.9096893 0.1963495 0.1963495[8] 0.3541459 0.1963495 0.9804472 0.1963495 0.5828725 0.3541459 0.5828725[15] 0.5828725 0.7834938 0.1963495 0.1963495 0.4809591 0.3541459> (Phyper <- phyper (Rhyper, m = 40, n = 30, k = 20) )[1] 0.99744478 0.51295659 0.51295659 0.98680472 0.51295659 0.86627413[7] 0.86627413 0.71494883 0.86627413 0.30864260 0.30864260 0.01796063[13] 0.51295659 0.95139461 0.86627413 0.15132082 0.95139461 0.86627413[19] 0.86627413 0.51295659> (Plnorm <- plnorm (Rlnorm, meanlog = -1, sdlog = 3) )[1] 0.43258714 0.39949055 0.72014780 0.96426235 0.76168186 0.70691429[7] 0.79571003 0.02042329 0.48505472 0.52373190 0.77520644 0.54557496[13] 0.73001823 0.44184582 0.52055037 0.28304536 0.45625441 0.32778045[19] 0.49895528 0.29804735> (Plogis <- plogis (Rlogis, location = 12, scale = 2) )[1] 0.92416690 0.75219874 0.99525932 0.65372216 0.02594333 0.89175446[7] 0.59635809 0.14797648 0.13590439 0.22845301 0.72077271 0.77077670[13] 0.94904799 0.17411062 0.24572404 0.64362757 0.82097136 0.87420320[19] 0.80009214 0.23480185> (Pnbinom <- pnbinom (Rnbinom, size = 7, prob = .01) )[1] 0.61736655 0.51299455 0.59632006 0.41950678 0.90402573 0.12140233[7] 0.50057859 0.09750529 0.04876502 0.74086686 0.80178090 0.03225449[13] 0.35983768 0.53748456 0.78532898 0.93731909 0.25190327 0.01316344[19] 0.54051099 0.68045166> (Pnorm <- pnorm (Rnorm, mean = -1, sd = 3) )[1] 0.5516781 0.8679849 0.8782349 0.6659003 0.8040234 0.9529268 0.2114568[8] 0.2902484 0.5668772 0.3973765 0.5605023 0.9636942 0.7367954 0.6221427[15] 0.6256145 0.9523291 0.8691318 0.7926739 0.7606609 0.6486326> (Ppois <- ppois (Rpois, lambda = 12) )[1] 0.46159733 0.77202453 0.68153563 0.08950450 0.57596525 0.24239216[7] 0.97872023 0.15502778 0.24239216 0.57596525 0.34722942 0.93703370[13] 0.46159733 0.68153563 0.46159733 0.57596525 0.77202453 0.15502778[19] 0.04582231 0.77202453> (Psignrank<- psignrank(Rsignrank, n = 47) )[1] 0.32789203 0.89007439 0.15011688 0.01887176 0.77676020 0.24256054[7] 0.48536796 0.09474662 0.40676506 0.14282568 0.13577317 0.62940572[13] 0.41083885 0.92356606 0.95623357 0.49790926 0.06352849 0.28016760[19] 0.98752692 0.23928001> (Pt <- pt (Rt, df = 11) )[1] 0.05491642 0.30192660 0.93285384 0.46375705 0.57641129 0.55185574[7] 0.17064488 0.08130751 0.91237971 0.24148527 0.30606307 0.93360050[13] 0.15287884 0.57333111 0.46518298 0.19365307 0.27422273 0.86749203[19] 0.15501842 0.22767857> (Punif <- punif (Runif, min = .2, max = 2) )[1] 0.91200940 0.15017105 0.26897908 0.16934513 0.75668526 0.04100786[7] 0.53184287 0.10300496 0.47788145 0.99044142 0.28609568 0.64486175[13] 0.64392568 0.61725289 0.72319637 0.49696022 0.84948787 0.22541974[19] 0.86112968 0.84008195> (Pweibull <- pweibull (Rweibull, shape = 3, scale = 2) )[1] 0.3510356 0.9010483 0.3212279 0.8528629 0.2226838 0.6762575 0.1379570[8] 0.1550349 0.4267381 0.6241639 0.6998987 0.6641379 0.8660439 0.4686384[15] 0.5617197 0.3082951 0.8075955 0.9465114 0.7116742 0.2116163> (Pwilcox <- pwilcox (Rwilcox, m = 13, n = 17) )[1] 0.82946462 0.38647276 0.38647276 0.02165645 0.61352724 0.46716310[7] 0.28170171 0.37078117 0.62921883 0.58162230 0.50000000 0.22852300[13] 0.88566977 0.74559308 0.08487632 0.02943715 0.75871780 0.05633406[19] 0.88566977 0.71829829>> dbeta (Rbeta, shape1 = .8, shape2 = 2)[1] 5.7913365 2.2364651 0.3059074 1.9511551 1.4556855 1.8746068 0.2575917[8] 2.6997454 1.4548665 2.6765264 1.0928822 1.1350760 1.3109191 0.6368984[15] 1.6508280 1.9374717 2.2740086 2.7988135 1.1324661 0.9756421> dbinom (Rbinom, size = 55, prob = pi/16)[1] 0.11924639 0.13317680 0.13311013 0.09636804 0.02881210 0.03704378[7] 0.01819424 0.11924639 0.09636804 0.11849719 0.01614929 0.11924639[13] 0.07063452 0.01819424 0.01819424 0.09636804 0.13317680 0.11924639[19] 0.02881210 0.03704378> dcauchy (Rcauchy, location = 12, scale = 2)[1] 0.001274152 0.148326009 0.023524757 0.157411041 0.073667654 0.100055114[7] 0.148329389 0.133504054 0.148433568 0.159134039 0.157686757 0.150433150[13] 0.159072180 0.121390049 0.062607155 0.130593337 0.124570555 0.052153921[19] 0.030572357 0.138035910> dchisq (Rchisq, df = 3)[1] 0.10087621 0.19475936 0.01759333 0.23601129 0.08574365 0.23802581[7] 0.20123372 0.18580144 0.23657215 0.15852893 0.12265734 0.22552069[13] 0.07578172 0.02705295 0.05562044 0.07345850 0.23198007 0.24044408[19] 0.12820355 0.22364083> dexp (Rexp, rate = 2)[1] 1.72716241 0.02773472 0.14057897 1.59236400 1.71492696 0.12542210[7] 0.65772563 0.99684330 0.46878331 0.43660735 0.70927528 0.30277010[13] 1.59283360 0.53687987 1.11347921 1.81270585 1.37448911 0.43019428[19] 0.26048939 0.01635668> df (Rf, df1 = 12, df2 = 6)[1] 0.519937867 0.004371848 0.692823649 0.167468952 0.448061444 0.022237248[7] 0.580744146 0.412297607 0.645723076 0.343445106 0.377992320 0.450116704[13] 0.682397435 0.666844216 0.147593859 0.066212131 0.431848359 0.243311742[19] 0.680166477 0.051540741> dgamma (Rgamma, shape = 2, scale = 5)[1] 0.071548510 0.011482464 0.045859471 0.048056741 0.073342809 0.031205091[7] 0.030090422 0.017225367 0.057283478 0.002759218 0.073295320 0.067639298[13] 0.007261834 0.073497725 0.031591526 0.034466522 0.046858733 0.060624349[19] 0.064405338 0.031640760> dgeom (Rgeom, prob = pi/16)[1] 0.081902787 0.126813148 0.065821212 0.157796399 0.022064895 0.196349541[7] 0.196349541 0.157796399 0.196349541 0.004777187 0.196349541 0.101913445[13] 0.157796399 0.101913445 0.101913445 0.052897247 0.196349541 0.196349541[19] 0.126813148 0.157796399> dhyper (Rhyper, m = 40, n = 30, k = 20)[1] 0.01064006 0.20431399 0.20431399 0.03541012 0.20431399 0.15132529[7] 0.15132529 0.20199224 0.15132529 0.15732178 0.15732178 0.01379223[13] 0.20431399 0.08512048 0.15132529 0.09227084 0.08512048 0.15132529[19] 0.15132529 0.20431399> dlnorm (Rlnorm, meanlog = -1, sdlog = 3)[1] 5.929818e-01 7.512774e-01 5.299802e-02 3.193275e-04 3.317380e-02[6] 6.087639e-02 2.153230e-02 2.062527e+01 4.042023e-01 3.018308e-01[11] 2.810781e-02 2.547247e-01 4.764589e-02 5.546720e-01 3.092876e-01[16] 1.714756e+00 4.995922e-01 1.247473e+00 3.643290e-01 1.540428e+00> dlogis (Rlogis, location = 12, scale = 2)[1] 0.035041221 0.093197897 0.002359104 0.113184749 0.012635138 0.048264223[7] 0.120357559 0.063039721 0.058717192 0.088131115 0.100629705 0.088339990[13] 0.024177951 0.071898055 0.092671867 0.114685561 0.073488693 0.054985983[19] 0.079972353 0.089834971> dnbinom (Rnbinom, size = 7, prob = .01)[1] 0.0013843572 0.0015461270 0.0014240958 0.0016111183 0.0004916298[6] 0.0010965512 0.0015593115 0.0009750548 0.0006438980 0.0010814585[11] 0.0008880238 0.0004914120 0.0016079010 0.0015162429 0.0009432177[16] 0.0003394350 0.0014949693 0.0002632443 0.0015121979 0.0012444310> dnorm (Rnorm, mean = -1, sd = 3)[1] 0.13186348 0.07126843 0.06736890 0.12130968 0.09218225 0.03275978[7] 0.09645747 0.11414715 0.13110785 0.12855632 0.13144855 0.02654173[13] 0.10880369 0.12669831 0.12633298 0.03309267 0.07084041 0.09534418[19] 0.10346857 0.12364112> dpois (Rpois, lambda = 12)[1] 0.11436792 0.09048890 0.10557038 0.04368219 0.11436792 0.08736438[7] 0.01613672 0.06552328 0.08736438 0.11436792 0.10483726 0.03832471[13] 0.11436792 0.10557038 0.11436792 0.11436792 0.09048890 0.06552328[19] 0.02548128 0.09048890> dsignrank(Rsignrank, n = 47)[1] 0.0037844241 0.0020112978 0.0024569596 0.0004983669 0.0031591482[6] 0.0032805288 0.0041778794 0.0017851857 0.0040639142 0.0023773231[11] 0.0022979400 0.0039702993 0.0040737886 0.0015466398 0.0010084031[16] 0.0041812581 0.0013244780 0.0035300054 0.0003600738 0.0032566009> dt (Rt, df = 11)[1] 0.09076142 0.33444332 0.10857032 0.38815185 0.38180921 0.38622576[7] 0.23261963 0.12825909 0.13672995 0.29449489 0.33676702 0.10750499[13] 0.21435914 0.38245649 0.38829368 0.25458773 0.31753985 0.19196797[19] 0.21661975 0.28375142> 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] 0.5566140 0.2596054 0.5411369 0.3405149 0.4650686 0.5261555 0.3625272[8] 0.3865994 0.5817133 0.5556831 0.5093571 0.5339287 0.3200512 0.5871773[15] 0.5782419 0.5333911 0.4026909 0.1642208 0.5001614 0.4538727> dwilcox (Rwilcox, m = 13, n = 17)[1] 0.010799421 0.015691594 0.015691594 0.002185415 0.015873676 0.016351073[7] 0.013817085 0.015486075 0.015691594 0.016163798 0.016431951 0.012383032[13] 0.008348499 0.013477697 0.006413577 0.002809857 0.013124724 0.004704515[19] 0.008348499 0.014141025>> ## 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.equal( Rt, qt (Pt, df = 11), tol = 1e-7 )## !! (386-lx:2.48e-9)[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.equal(Rt, qt (1- Pt, df = 11, lower=F), tol = 1e-7)[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.equal(Rt, qt (log(Pt), df = 11, log=TRUE), tol = 1e-7)[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.equal(Rt, qt (log(1- Pt), df = 11, lower=F, log=T), tol = 1e-7)#ok{prec.}[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 - 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>> cat("Time elapsed: ", proc.time() - .ptime,"\n")Time elapsed: 10.17 0.15 10.37 0 0>