standard error of ratio
Dave Krantz
of A, B, and the ratio A/B. The answer to this question is not
distribution-free, and I'm not sure which of his packed-up books
would cover it. I always find it easier to derive something like
this by Taylor series than to dig out a book that discusses it.
The Taylor series for R = A/B, through its first two orders,
looks like this:
dR ~ (1/B)*dA - (A/B^2)*dB + (A/B^3)*dB^2 - (1/B^2)*dA*dB (EQ 1)
(there is no term for dA^2 since the second partial derivative of R
with respect to A is 0).
In the application here, dA and dB can be considered independent
random variables with mean 0 (departures of sample mean from population
mean for the two samples--it is important that the samples are
independent, not important whether they come from the same population).
If one squares EQ 1, takes expectations, uses independence and mean 0,
and simplifies, one gets the following second-order approximation,
in which cvA, cvB, and cvR denote the coefficients of variation of A, B, R
(i.e., cvA = SA/A, etc.), B3 denotes the normalized third central moment
of the denominator B (expectation of the third power of the deviation
from the mean, divided by the third power of the mean), and B4 denotes
the analogous normalized fourth central moment of B:
cvR ~ sqrt(cvA^2 + cvB^2 + (cvA*cvB)^2 - 2*B3 + B4) (EQ 2)
If the central limit theorem applies approximately to statistic B,
so its distribution is approximately Gaussian, EQ 2 can be simplified:
B3 is essentially 0 and B4 is approximately 3*(cvB^4). So for this case,
the second-order approximation becomes
cvR ~ sqrt(cvA^2 + cvB^2 + (cvA*cvB)^2 + 3*(cvB^4)) (EQ 3)
If cvA and cvB are less than 20%, this is a quite good approximation;
and if they are less than 10%, the latter two terms in it play little
role--the first order approximation, essentially Pythagorean combination
of coefficients of variation is pretty good. However, this second-order
approximation goes to pot rapidly for coefficients of variation
over 20%--one needs higher terms of the Taylor expansion and it
may no longer be worthwhile to look for any formula unless you have
particularly interesting special cases.
Dave Krantz (d...@stat.columbia.edu)