I can't seem to get computationally stable estimates for the following system:
Y=a+bX+cX^2+dX^3, where X~N(0,1). (Y is expressed as a linear combination of
the first three powers of a standard normal variable.) Assuming that E(Y)=0 and
Var(Y)=1, one can obtain the following equations after tedious algebraic
calculations:
1) b^2+6bd+2c^2+15d^2=1
2) 2c(b^2+24bd+105d^2+2)=E(Y^3)
3) 24[bd+c^2(1+b^2+28bd)+d^2(12+48bd+141c^2+225d^2)]=E(Y^4)-3
Obviously, a=-c. Suppose that distributional form of Y is given so we know
E(Y^3) and E(Y^4). In other words, we have access to the third and fourth raw
moments. How do we solve for these four coefficients? I reduced the number of
unknowns/equations to two, and subsequently used a grid approach. It works well
when I am close to the center of the support, but fails miserably at the tails.
Any ideas? Hopefully, there is a nice R function that does this.
Hakan Demirtas
[[alternative HTML version deleted]]
______________________________________________
R-devel@r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-devel
