# MVN Rectangle Probabilities Approximation to multivariate normal rectangle probabilities using methods in Joe (1995, JASA) ```r mvnapp(lb,ub,mu,sigma,type=1,eps=1.e-05,nsim=0) ``` ## Arguments - `lb`: vector of lower limits of integral/probability - `ub`: vector of upper limits of integral/probability - `mu`: mean vector - `sigma`: covariance matrix, it is assumed to be positive-definite - `type`: indicator, type=1 refers to the first order approximation, type=2 is the second order approximation. - `eps`: accuracy/tolerance for bivariate marginal rectangle probabilities - `nsim`: an optional integer if random permutations are used in the approximation for dimension >=6; nsim=2000 recommended for dim>=6 ## Returns - **prob**: rectangle probability with approximation - **esterr**: indicator of accuracy in the approximation - **ifail**: = 0 if no problems >= 1 if problems from using Schervish's code in dimensions 2 to 4. ## References Joe, H (1995). Approximations to multivariate normal rectangle probabilities based on conditional expectations. **Journal of American Statistical Association**, 90 , 957--964. ## Author(s) Harry Joe harry.joe@ubc.ca ## Examples ```r m<-2 rh<-0.5 a<-c(-1,-1) b<-c(1,1) mu<-rep(0,m) s<-matrix(c(1,rh,rh,1),2,2) print(mvnapp(a,b,mu,s)) print(mvnapp(a,b,mu,s,type=2)) print(mvnapp(a,b,mu,s,type=2,nsim=3)) m<-3 rh<-0.3 a<-c(-1,-1,-2) b<-c(1,1,.5) mu<-rep(0,m) s<-matrix(c(1,.5,.6,.5,1,.7,.6,.7,1),3,3) print(mvnapp(a,b,mu,s)) print(mvnapp(a,b,mu,s,type=2)) print(mvnapp(a,b,mu,s,type=2,nsim=3)) m<-4 rh<- -0.1 a<-c(-1,-2.5,-2,-1.5) b<-c(1.68,1.11,.5,.25) mu<-rep(0,m) s<-matrix(c(1,.5,.3,.4,.5,1,.5,.4,.3,.5,1,.4,.4,.4,.4,1),m,m) print(mvnapp(a,b,mu,s)) print(mvnapp(a,b,mu,s,type=2)) print(mvnapp(a,b,mu,s,type=2,nsim=3)) m<-5 rh<-.4 a<-rep(-1,m) b<-rep(2,m) mu<-rep(0,m) s<-matrix(c(1,rh,rh,rh,rh,rh,1,rh,rh,rh,rh,rh,1,rh,rh,rh,rh,rh,1, rh,rh,rh,rh,rh,1),m,m) print(mvnapp(a,b,mu,s)) print(mvnapp(a,b,mu,s,type=2)) print(mvnapp(a,b,mu,s,type=2,nsim=3)) m<-6 a<-c(-1,-1,-1,-1.5,-1,-2) b<-rep(7,m) mu<-rep(0,m) s<-matrix(c(1,rh,rh,rh,rh,rh,rh,1,rh,rh,rh,rh,rh,rh,1,rh,rh,rh,rh,rh,rh,1, rh,rh,rh,rh,rh,rh,1,rh,rh,rh,rh,rh,rh,1),m,m) print(mvnapp(a,b,mu,s)) print(mvnapp(a,b,mu,s,type=2)) print(mvnapp(a,b,mu,s,type=2,nsim=3)) ```

mvnapp function