APPENDIX A: ROSENBLEUTH'S POINT ESTIMATE METHOD
The PEM of Rosenbleuth (1975) is described here in detail for a function of two random
variables y(x1,x2). The correlation coefficient of random variables x1 and x2 is defined as
Cov( x1, x2 )
ρ=
(A1)
Sx1 , Sx2
where Sx is the standard deviation of x and Cov(x1,x2) is the covariance. A correlation
coefficient of 1 indicates a perfect linear correlation between the variables, and a coeffi-
cient of 0 indicates perfectly uncorrelated variables. Numerical experiments have shown
that if the relationship between random variables over a limited range can be written as
b
x1 = ax2 ,
(A2)
then the magnitude of ρ will be approximately 1.
To simplify this discussion, we will assume that the distribution of each random vari-
able is symmetric about the mean. The point estimates that represent the distribution of
each random variable are then
1+ ρ
P+ + = P =
(A3)
4
1 ρ
P+ = P + =
4
These point estimates are weighting factors that sum to 1 and, in the case of uncorrelated
random variables, are each 1/4. The function y(x1,x2) is evaluated at points that are a
standard deviation from the mean of each random variable :
y+ + = y (x1 + Sx1 , x2 + Sx2 )
(A4)
y+ = y (x1 + Sx1 , x2 Sx2 )
y + = y (x1 Sx1 , x2 + Sx2 )
y = y (x1 Sx1 , x2 Sx2 )
Then, the expected values or moments of the function can be obtained as
E(yn ) = P+ + y+ + + P+ y+ + P + y + + P y .
n
n
n
n
(A5)
The expected value or mean of y is found by setting n = 1. The variance of y is readily
obtained as
V(y) = E(y 2 ) [E(y)] = Sy .
2
2
(A6)
The minimum and maximum values of y fall outside the limits of the values obtained in eq
A4. Rosenbleuth (1975) discussed the generalization of this method to functions of any
number of random variables and random variables with asymmetric distributions.
In summary, the PEM provides estimates of the mean, variance, and limits of the
distribution of a function of random variables. These four parameters uniquely specify a
17
Previous Page
