On Oct 15, 2013, at 17:18 , Berend Hasselman wrote: > > On 15-10-2013, at 15:24, Ron Michael <ron_michae...@yahoo.com> wrote: > >> Hi, >> >> I need to solve following simultaneous equations for A, B, Y1, Y2: >> >> B * Phi(Y1 - A) + (1-B) * Phi(Y1 + A) = 0.05 >> B * Phi(Y2 - A) + (1-B) * Phi(Y2 + A) = 0.01 >> >> Y1 <= -1.65 >> Y2 >= -2.33 >> >> 0 <= B <=1 >> >> Phi is CDF for standard normal >> >> If there is no unique solution, then I should be able to get some feassible >> solution(s) >> >> Is there any way that using R I can achieve that? > > > You cannot solve a system of 2 equations with 4 unknowns (variables). > You can only try to find 4 values that get as close as possible (in whatever > sense) to solving the system.
Think again.... A=0, B=whatever, Y1=qnorm(.05), Y2=qnorm(.01) (4 equations in 2 unknowns is generally harder.) There are also solutions for A+Y1 = qnorm(.05), A+Y2 = qnorm(.01), B=0, etc. --- except that they will violate the constraints on at least one of Y1, Y2 (assuming that the constraints are actually the two qnorm()s rounded to two decimal places)! The 65536$ question is whether there are nontrivial solutions, with Y1+/-A on different sides of qnorm(.05), and consequently 0 < B < 1. I kind of suspect that there aren't any; for instance > Y1 <- qnorm(.05) > Y2 <- qnorm(.01) > A <- .01 > B <- (pnorm(Y1+A)-0.05)/(pnorm(Y1+A)-pnorm(Y1-A)) > B*(pnorm(Y1-A))+(1-B)*pnorm(Y1+A) [1] 0.05 > B*(pnorm(Y2-A))+(1-B)*pnorm(Y2+A) [1] 0.01000091 > B*(pnorm(Y2+.001-A))+(1-B)*pnorm(Y2+.001+A) [1] 0.01002759 Notice that we can get the first equation satisfied exactly, but then the 2nd one overshoots by an amount which is increasing in Y2. We could fix that by decreasing Y2 but that violates the constraint. -- Peter Dalgaard, Professor Center for Statistics, Copenhagen Business School Solbjerg Plads 3, 2000 Frederiksberg, Denmark Phone: (+45)38153501 Email: pd....@cbs.dk Priv: pda...@gmail.com ______________________________________________ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
