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Type 'q()' to quit R. > invisible(options(echo = TRUE)) > library("mvtnorm") > set.seed(290875) > > chk <- function(...) isTRUE(all.equal(...)) > > # correlation matrices for unequal variances were wrong > # from Pamela Ohman-Strickland > > a <- 4.048 > shi <- -9 > slo <- -10 > mu <- -5 > sig <- matrix(c(1,1,1,2),ncol=2) > pmvnorm(lower=c(-a,slo),upper=c(a,shi),mean=c(mu,2*mu),sigma=sig) [1] 0.04210555 attr(,"error") [1] 1e-15 attr(,"msg") [1] "Normal Completion" > > # check if set.seed works (starting from 0.5-7) > n <- 5 > lower <- -1 > upper <- 3 > df <- 4 > corr <- diag(5) > corr[lower.tri(corr)] <- 0.5 > delta <- rep(0, 5) > set.seed(290875) > prob1 <- pmvt(lower=lower, upper=upper, delta=delta, df=df, corr=corr) > set.seed(290875) > prob2 <- pmvt(lower=lower, upper=upper, delta=delta, df=df, corr=corr) > stopifnot(chk(prob1, prob2)) > > # confusion for univariate probabilities when sigma is a matrix > # by Jerome Asselin > a <- pmvnorm(lower=-Inf,upper=2,mean=0,sigma=matrix(1.5)) > attributes(a) <- NULL > stopifnot(chk(a, pnorm(2, sd=sqrt(1.5)))) > a <- pmvnorm(lower=-Inf,upper=2,mean=0,sigma=matrix(.5)) > attributes(a) <- NULL > stopifnot(chk(a, pnorm(2, sd=sqrt(.5)))) > a <- pmvnorm(lower=-Inf,upper=2,mean=0,sigma=.5) > attributes(a) <- NULL > stopifnot(chk(a, pnorm(2, sd=sqrt(.5)))) > > # log argument added by Jerome Asselin > dmvnorm(x=c(0,0), mean=c(1,1),log=TRUE) [1] -2.837877 > dmvnorm(x=c(0,0), mean=c(25,25),log=TRUE) [1] -626.8379 > dmvnorm(x=c(0,0), mean=c(30,30),log=TRUE) [1] -901.8379 > stopifnot(chk(dmvnorm(x=0, mean=30, log=TRUE), + dnorm (0, 30, log=TRUE))) > > stopifnot( + chk(dmvnorm(x=c(0,0), mean =c(30,30),log=TRUE) -> f., + dmvt (x=c(0,0), delta=c(30,30),log=TRUE, df=Inf)) + , + chk(f., dmvt(x=c(0,0), delta=c(30,30),log=TRUE, df=10000), + tolerance = 0.09) + ) > > # large df > pnorm(2)^2 [1] 0.9550173 > pmvt(lower=c(-Inf,-Inf), upper=c(2,2), delta=c(0, 0), df=25, corr=diag(2)) [1] 0.9446454 attr(,"error") [1] 1e-15 attr(,"msg") [1] "Normal Completion" > pmvt(lower=c(-Inf,-Inf), upper=c(2,2), delta=c(0, 0), df=250, corr=diag(2)) [1] 0.9539846 attr(,"error") [1] 1e-15 attr(,"msg") [1] "Normal Completion" > pmvt(lower=c(-Inf,-Inf), upper=c(2,2), delta=c(0, 0), df=1340, corr=diag(2)) [1] 0.9548248 attr(,"error") [1] 1e-15 attr(,"msg") [1] "Normal Completion" > pmvt(lower=c(-Inf,-Inf), upper=c(2,2), delta=c(0, 0), df=2500, corr=diag(2)) [1] 0.9549141 attr(,"error") [1] 1e-15 attr(,"msg") [1] "Normal Completion" > pmvt(lower=c(-100,-100), upper=c(2,2), delta=c(0, 0), df=2500, corr=diag(2)) [1] 0.9549141 attr(,"error") [1] 1e-15 attr(,"msg") [1] "Normal Completion" > > # df = 0 > pmvt(lower=c(-Inf,-Inf), upper=c(2,2), delta=c(0, 0), df=0, corr=diag(2)) [1] 0.9550173 attr(,"error") [1] 1e-15 attr(,"msg") [1] "Normal Completion" > pmvt(lower=-Inf, upper = 2, delta=0, df=0, corr=1) upper 0.9772499 attr(,"error") [1] 0 attr(,"msg") [1] "univariate: using pnorm" > pnorm(2) [1] 0.9772499 > > # larger dimensions > pnorm(2)^2 [1] 0.9550173 > pmvnorm(lower=rep(-Inf, 2), upper=rep(2,2), sigma = diag(2)) [1] 0.9550173 attr(,"error") [1] 1e-15 attr(,"msg") [1] "Normal Completion" > pnorm(2)^90 [1] 0.1260393 > pmvnorm(lower=rep(-Inf, 90), upper=rep(2,90), sigma = diag(90)) [1] 0.1260393 attr(,"error") [1] 0 attr(,"msg") [1] "Normal Completion" > pnorm(2)^199 [1] 0.01025932 > pmvnorm(lower=rep(-Inf, 199), upper=rep(2,199), sigma = diag(199)) [1] 0.01025932 attr(,"error") [1] 0 attr(,"msg") [1] "Normal Completion" > > # larger dimensions, again. Spotted by Chihiro Kuroki > # Alan's fix to MVCHNC solves this problem > cr = matrix(0.5, nr = 4, nc = 4) > diag(cr) = 1 > cr [,1] [,2] [,3] [,4] [1,] 1.0 0.5 0.5 0.5 [2,] 0.5 1.0 0.5 0.5 [3,] 0.5 0.5 1.0 0.5 [4,] 0.5 0.5 0.5 1.0 > a <- pmvt(low = -rep(1, 4), upp = rep(1, 4), df = 999, corr = cr) > b <- pmvt(low = -rep(1, 4), upp = rep(1, 4), df = 4999, corr = cr) > b [1] 0.2893192 attr(,"error") [1] 8.971803e-05 attr(,"msg") [1] "Normal Completion" > attributes(a) <- NULL > attributes(b) <- NULL > stopifnot(chk(round(a, 3), round(b, 3))) > > # cases where the support is the empty set tried to compute something. > # spotted by Peter Thomson > stopifnot(chk(c(pmvnorm(upper=c(-Inf,1))), 0)) > stopifnot(chk(c(pmvnorm(lower=c(Inf,1))), 0)) > stopifnot(chk(c(pmvnorm(lower=c(-2,0),upper=c(-1,1),corr=matrix(rep(1,4),2,2))), 0)) > > # bugged Fritz (long time ago) > stopifnot(chk(c(pmvnorm(-Inf, c(Inf, 0), 0, diag(2))), + c(pmvnorm(-Inf, c(Inf, 0), 0)))) > > # this is a bug in `mvtdst' nobody was able to fix yet :-( > stopifnot(chk(c(pmvnorm(lo=c(-Inf,-Inf), up=c(Inf,Inf), mean=c(0,0))), 1)) > > ### check for correct random seed initialization > ### problem reported by Karen Conneely > dm <- 250000 > iters <- 2 > corr <- .7 > dim <- 100 > abserr <- .0000035 > cutoff <- -5.199338 > mn <- rep(0,dim) > mat <- diag(dim) > for (i in 1:dim) { + for (j in 1:(i-1)) { + mat[i,j]=mat[j,i]=corr^(i-j) + } + } > ll <- rep(cutoff, dim) > mn <- rep(0, dim) > p <- matrix(0, iters,1) > > set.seed(290875) > for (i in 1:iters) { + pp <- pmvnorm(lower=ll, sigma=mat, maxpts=dm, abseps=abserr) + p[i] <- 1-pp + } > stopifnot(abs(p[1] - p[2]) < 2 * abserr) > ptmp <- p > set.seed(290875) > for (i in 1:iters) { + pp <- pmvnorm(lower=ll, sigma=mat, maxpts=dm, abseps=abserr) + p[i] <- 1-pp + } > stopifnot(chk(p, ptmp)) > > ### same for algoritm = Miwa > > pmvnormM <- function(...) pmvnorm(..., algorithm = Miwa()) > > a <- 4.048 > shi <- -9 > slo <- -10 > mu <- -5 > sig <- matrix(c(1,1,1,2),ncol=2) > pmvnormM(lower=c(-a,slo),upper=c(a,shi),mean=c(mu,2*mu),sigma=sig) [1] 0.04210555 attr(,"error") [1] NA attr(,"msg") [1] "Normal Completion" > > # check if set.seed works (starting from 0.5-7) > n <- 5 > lower <- -1 > upper <- 3 > df <- 4 > corr <- diag(5) > corr[lower.tri(corr)] <- 0.5 > delta <- rep(0, 5) > set.seed(290875) > prob1 <- pmvnormM(lower=lower, upper=upper, mean = delta, corr=corr) > set.seed(290875) > prob2 <- pmvnormM(lower=lower, upper=upper, mean = delta, corr=corr) > stopifnot(chk(prob1, prob2)) > > # confusion for univariate probabilities when sigma is a matrix > # by Jerome Asselin > a <- pmvnormM(lower=-Inf,upper=2,mean=0,sigma=matrix(1.5)) > attributes(a) <- NULL > stopifnot(chk(a, pnorm(2, sd=sqrt(1.5)))) > a <- pmvnormM(lower=-Inf,upper=2,mean=0,sigma=matrix(.5)) > attributes(a) <- NULL > stopifnot(chk(a, pnorm(2, sd=sqrt(.5)))) > a <- pmvnormM(lower=-Inf,upper=2,mean=0,sigma=.5) > attributes(a) <- NULL > stopifnot(chk(a, pnorm(2, sd=sqrt(.5)))) > > > # cases where the support is the empty set tried to compute something. > # spotted by Peter Thomson > stopifnot(chk(c(pmvnormM(upper=c(-Inf,1))), 0)) > stopifnot(chk(c(pmvnormM(lower=c(Inf,1))), 0)) > > # bugged Fritz (long time ago) > stopifnot(chk(pmvnormM(-Inf, c(Inf, 0), 0, diag(2)), + pmvnormM(-Inf, c(Inf, 0), 0))) > > # this is a bug in `mvtdst' nobody was able to fix yet :-( > stopifnot(chk(c(pmvnormM(lo=c(-Inf,-Inf), up=c(Inf,Inf), mean=c(0,0))), 1)) > > ### check for correct random seed initialization > ### problem reported by Karen Conneely > dm <- 250000 > iters <- 2 > corr <- .7 > dim <- 10 > abserr <- .0000035 > cutoff <- -5.199338 > mn <- rep(0,dim) > mat <- diag(dim) > for (i in 1:dim) { + for (j in 1:(i-1)) { + mat[i,j]=mat[j,i]=corr^(i-j) + } + } > ll <- rep(cutoff, dim) > mn <- rep(0, dim) > p <- matrix(0, iters,1) > > set.seed(290875) > for (i in 1:iters) { + pp <- pmvnormM(lower=ll, sigma=mat, maxpts=dm, abseps=abserr) + p[i] <- 1-pp + } > stopifnot(abs(p[1] - p[2]) < 2 * abserr) > ptmp <- p > set.seed(290875) > for (i in 1:iters) { + pp <- pmvnormM(lower=ll, sigma=mat, maxpts=dm, abseps=abserr) + p[i] <- 1-pp + } > stopifnot(chk(p, ptmp)) > > ### was == 1; spotted by Alex Lenkoski > stopifnot(chk(c(pmvnorm(c(-Inf, -Inf, 0, 0))), 0.25)) > > ############################# > ## testing rmvt und pmvt > ############################# > set.seed(290875) > n <- 100000 > df <- rpois(1,1/rexp(1,1))+1 > dim <- rpois(1,runif(1,0,10))+2 > mn <- rnorm(dim,0,4) ##rep(0,dim) > sigma <- matrix(runif(dim^2,-1,1), nrow = dim, ncol = dim) > sigma <- crossprod(sigma)+diag(dim) > d <- runif(dim, 0.3, 20) > sigma <- diag(d)%*%sigma%*%diag(d) > corrMat <- cov2cor(sigma) > > ## sigma handed over > sims1 <- rmvt(n, sigma = sigma, delta = mn, df=df, type = "shifted", pre0.9_9994 = TRUE) > sims2 <- rmvt(n, sigma = sigma, delta = mn, df=df, type = "Kshirsagar", pre0.9_9994 = TRUE) > lower <- mn-d*2 > upper <- mn+d*3 > comp <- function(x, lower, upper){ + all(x>lower) & all(x ind1 <- apply(sims1, 1, comp, lower=lower, upper=upper) > mean(ind1) #Monte Carlo Integration [1] 0.247 > pmvt(lower, upper, sigma = sigma, delta=mn, df=df, type = "shifted") [1] 0.2459638 attr(,"error") [1] 8.791565e-05 attr(,"msg") [1] "Normal Completion" > ind2 <- apply(sims2, 1, comp, lower=lower, upper=upper) > mean(ind2) [1] 0.24729 > pmvt(lower, upper, sigma = sigma, delta=mn, df=df, type = "Kshirsagar") [1] 0.2462984 attr(,"error") [1] 6.430839e-05 attr(,"msg") [1] "Normal Completion" > > ## corrMat handed over > sims1 <- rmvt(n, sigma = corrMat, delta = mn, df=df, type = "shifted", pre0.9_9994 = TRUE) > sims2 <- rmvt(n, sigma = corrMat, delta = mn, df=df, type = "Kshirsagar", pre0.9_9994 = TRUE) > lower <- mn-d*0.5 > upper <- mn+d > comp <- function(x, lower, upper){ + all(x>lower) & all(x ind1 <- apply(sims1, 1, comp, lower=lower, upper=upper) > mean(ind1) #Monte Carlo Integration [1] 0.99669 > pmvt(lower, upper, corr = corrMat, delta=mn, df=df, type = "shifted") [1] 0.996872 attr(,"error") [1] 2.461945e-05 attr(,"msg") [1] "Normal Completion" > ind2 <- apply(sims2, 1, comp, lower=lower, upper=upper) > mean(ind2) [1] 0.98827 > pmvt(lower, upper, corr = corrMat, delta=mn, df=df, type = "Kshirsagar") [1] 0.9882905 attr(,"error") [1] 6.344011e-05 attr(,"msg") [1] "Normal Completion" > > ### approx_interval for tail = "upper" went wild > ### spotted by Ravi Varadhan > m <- 10 > rho <- 0.1 > k <- 2 > alpha <- 0.05 > cc_z <- numeric(m) > var <- matrix(c(1,rho,rho,1), nrow=2, ncol=2, byrow=T) > i <- 1 > q1 <- qmvnorm((k*(k-1))/(m*(m-1))*alpha, tail="upper", sigma=var, + ptol=0.00001)$quantile > q2 <- qmvnorm((k*(k-1))/(m*(m-1))*alpha, tail="upper", sigma=var, + interval = c(0, 5), ptol=0.00001)$quantile > stopifnot(chk(round(q1, 4), round(q2, 4))) > > ### grrr, still problems in approx_interval > qmvnorm(.95, sigma = tcrossprod(c(0.009, 0.75, 0.25)))$quantile [1] 1.23364 > > ### qmvt(..., df = 0, ...) didn't work > ### spotted by Ulrich Halekoh > stopifnot(is.finite(qmvt(.95, df = 0, corr = matrix(1))$quantile)) > > ### spotted by and fixed > ### in mvtdst.f by Alan 2013-05-29 > corr <- matrix(1, ncol = 2, nrow = 2) > p <- c(pmvnorm(lower=c(-Inf,-Inf),upper=c(1.96,1.96),mean=c(1.72,1.72),corr=corr), + pmvt(lower=c(-Inf,-Inf),upper=c(1.96,1.96),delta=c(1.72,1.72),df=0,corr=corr), + pmvt(lower=c(-Inf,-Inf),upper=c(1.96,1.96) - c(1.72,1.72),df=0,corr=corr), + pmvt(lower=c(-Inf,-Inf),upper=c(1.96,1.96) - c(1.72,1.72),df=100,corr=corr), + pmvt(lower=c(-Inf,-Inf),upper=c(1.96,1.96), delta=c(1.72,1.72),df=100,corr=corr)) > stopifnot(all(abs(p - mean(p)) < 1 / 100)) > > ### spotted and fixed by Xuefei Mi > m <- 3 > S <- diag(m) > S[2, 1] <- S[1, 2] <- 1/4 > S[3, 1] <- S[1, 3] <- 1/5 > S[3, 2] <- S[2, 3] <- 1/3 > # NaN was given. > p <- pmvnorm(lower=c(-Inf, 0, 0), upper=c(0, Inf, Inf), mean=c(0, 0, 0), + sigma=S, algorithm = Miwa()) > stopifnot(!is.na(p)) > > ### introduced with dmvnorm in 0.9-9999 > set.seed(29) > ### dmvnorm up to 0.9-9997 > d1 <- function(x, mean, sigma) { + distval <- mahalanobis(x, center = mean, cov = sigma) + logdet <- sum(log(eigen(sigma, symmetric=TRUE, + only.values=TRUE)$values)) + -(ncol(x)*log(2*pi) + logdet + distval)/2 + } > ### current version > d2 <- function(...) dmvnorm(..., log = TRUE) > > for (i in 1:100) { + p <- sample(2:10, 1) + Sigma <- tcrossprod(matrix(runif(p^2) * 2, ncol = p)) + x <- matrix(rnorm(p), nr = 1) + m <- runif(p) + ld1 <- d1(x=x, mean=m, sigma=Sigma) + ld2 <- d2(x=x, mean=m, sigma=Sigma) + + stopifnot(chk(ld1, ld2, tol = .Machine$double.eps^(1/3))) + } > > ### --- Singular Sigma --- Now treated the same as dnorm(*, sd=0): "Inf or 0" > L <- diag(10*(1:4)) > L[lower.tri(L)] <- 1:6 > L[3,3] <- 0 # to make it singular > L [,1] [,2] [,3] [,4] [1,] 10 0 0 0 [2,] 1 20 0 0 [3,] 2 4 0 0 [4,] 3 5 6 40 > (Sig <- tcrossprod(L)) [,1] [,2] [,3] [,4] [1,] 100 10 20 30 [2,] 10 401 82 103 [3,] 20 82 20 26 [4,] 30 103 26 1670 > set.seed(123) > fx <- dmvnorm(rbind(0, 1:4, matrix(rnorm(4*10), ncol=4)), sigma = Sig) > stopifnot(chk(fx, c(Inf, rep(0, 1+10)))) > ## gave NaN for a long time, then error, then NaN, now we have converged ;-) > > ### NaN spotted by David Charles Airey > ### data contains all input parameters and a special seed > ret <- + structure(list(N = 10L, NU = 25L, LOWER = c(-0.430060315238938, + -0.430060315238938, -0.430060315238938, -0.430060315238938, -0.430060315238938, + -0.430060315238938, -0.430060315238938, -0.430060315238938, -0.430060315238938, + -0.430060315238938), UPPER = c(0.430060315238938, 0.430060315238938, + 0.430060315238938, 0.430060315238938, 0.430060315238938, 0.430060315238938, + 0.430060315238938, 0.430060315238938, 0.430060315238938, 0.430060315238938 + ), INFIN = c(2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L), CORREL = c(0.5, + 0.5, 0.5, 0.5, 0.5, 0.5, -0.5, 0.5, -2.75206388903997e-16, -3.44007986129997e-16, + -0.5, -4.12809583355996e-16, 0.499999999999999, -6.19214375033994e-16, + 0.5, -0.5, -4.12809583355996e-16, -5.50412777807995e-16, 0.5, + 0.5, 0.5, -1.37603194451999e-16, -0.5, 0.5, -2.75206388903997e-16, + -0.5, 0.5, -1.37603194451999e-16, -6.88015972259993e-17, -0.5, + -2.75206388903997e-16, 0.5, -0.5, -2.06404791677998e-16, 0.5, + 0.5, 6.88015972259993e-17, 0, -0.5, 0.5, -6.88015972259993e-17, + -0.5, 0.5, -0.5, 0.5), DELTA = c(0, 0, 0, 0, 0, 0, 0, 0, 0, 0 + ), MAXPTS = 25000L, ABSEPS = 0.001, RELEPS = 0, error = NaN, + value = NaN, inform = 0L), .Names = c("N", "NU", "LOWER", + "UPPER", "INFIN", "CORREL", "DELTA", "MAXPTS", "ABSEPS", "RELEPS", + "error", "value", "inform")) > > RS <- + c(403L, 480L, 641015092L, 1848202935L, -2124158291L, -2116162620L, + 1818211306L, -796165035L, -1592745489L, -483415562L, -77025504L, + -1708531485L, 2015614337L, 1987179504L, 1442495118L, 792268281L, + -362319989L, 798403514L, 744985276L, -385857025L, -2143185067L, + -2123763604L, 1037640898L, -2059956451L, -1706850921L, 514864382L, + -1711546696L, -338706277L, -1586794759L, -132177848L, 1947022198L, + 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1751993866L + ) > > > f <- function() { + + error <- 0; value <- 0; inform <- 0 + ret <- .C(C_mvtdst, N = as.integer(ret$N), + NU = as.integer(ret$NU), + LOWER = as.double(ret$LOWER), + UPPER = as.double(ret$UPPER), + INFIN = as.integer(ret$INFIN), + CORREL = as.double(ret$CORREL), + DELTA = as.double(ret$DELTA), + MAXPTS = as.integer(ret$MAXPTS), + ABSEPS = as.double(ret$ABSEPS), + RELEPS = as.double(ret$RELEPS), + error = as.double(error), + value = as.double(value), + inform = as.integer(inform), RND = 1L) + ret + + } > > ### this special seed triggers the problem > ### error and value are NaN (already in FORTRAN) > .Random.seed <- RS > # stopifnot(!is.na(f()$value)) ### .C does not work here > > ### check tail with new quantile algorithm > p <- .95 > stopifnot(chk(round(qmvnorm(p, sigma = diag(3), tail = "upper")$quantile, 2), + round(qnorm(p^(1/3), lower = FALSE), 2))) > stopifnot(chk(round(qmvnorm(p, sigma = diag(3), tail = "lower")$quantile, 2), + round(qnorm(p^(1/3), lower = TRUE), 2))) > set.seed(29) > p <- .95 > d <- 4 > qmvnorm(p, sigma = diag(d), tail = "lower")$quantile [1] 2.23395 > qmvnorm(p, sigma = diag(d), tail = "upper")$quantile [1] -2.23395 > qmvnorm(p, sigma = diag(d), tail = "both")$quantile [1] 2.490844 > p <- 1 - .95 > d <- 4 > qmvnorm(p, sigma = diag(d), tail = "lower")$quantile [1] -0.06784195 > qmvnorm(p, sigma = diag(d), tail = "upper")$quantile [1] 0.06784195 > > ### package schwartz97 > qmvnorm(p = .5, tail = "lower", mean = c(6.75044368, 0.04996326), + sigma = rbind(c(0.10260550, 0.02096418), + c(0.02096418, 0.16049956)))$quantile [1] 6.750444 > stint <- c(6.75044332319072, 6.75044368) ## with very narrow start interval > qmvnorm(p = .5, tail = "lower", mean = c(6.75044368, 0.04996326), + sigma = rbind(c(0.10260550, 0.02096418), + c(0.02096418, 0.16049956)), interval=stint)$quantile [1] 6.750444 > > ### qmvnorm and qmvt should stop if supplied covariance matrix > ### is not positive semidefinite > > R2=matrix(c(0.7071068, 0.6924398, 0.7054602, 0.7054602, 0.6292745, + 0.6924398, 0.7071068, 0.6909812, 0.6909712, 0.6128670, + 0.7054602, 0.6909812, 0.7071068, 0.7071068, 0.6278091, + 0.7054602, 0.6909712, 0.7071068, 0.7071068, 0.6278091, + 0.6292745, 0.6128670, 0.6278091, 0.6278091, 0.7071068),ncol=5) > > call <- try(qmvnorm(p=1-0.0001726701, + mean=c(-0.8752332, -0.9487915, -0.9719237, + -0.5855204, -0.9046457), + sigma=R2,tail='lower.tail')$quantile, + silent=TRUE) > inherits(call, "try-error") [1] TRUE > grepl("Covariance matrix not positive semidefinite", geterrmessage()) [1] TRUE > > call <- try(qmvt(p=1-0.0001726701, + mean=c(-0.8752332, -0.9487915, -0.9719237, + -0.5855204, -0.9046457), + sigma=R2,tail='lower.tail')$quantile, + silent=TRUE) > inherits(call, "try-error") [1] TRUE > grepl("Covariance matrix not positive semidefinite", geterrmessage()) [1] TRUE > > ### qmvnorm was wrong for the univariate setting; reported by Chen-Wei > ### > > all.equal(qnorm(p = 0.2397501, mean = 1, sd = sqrt(2)), + qmvnorm(p=0.2397501 , mean = 1, sigma = 2)$quantile) [1] TRUE > > proc.time() user system elapsed 7.14 0.10 7.18