How to parameterize the ratio of two normally distributed variables, or the inverse of one?

Question

Problem: I am parameterizing distributions for use as a priors and data in a Bayesian meta-analysis. The data are provided in the literature as summary statistics, almost exclusively assumed to be normally distributed (although none of the variables can be < 0, some are ratios, some are mass, and etc.).

I have come across two cases for which I have no solution. Sometimes the parameter of interest is the inverse of the data or the ratio of two variables.

Examples:

1. the ratio of two normally distributed variables:
   • data: mean and sd for percent nitrogen and percent carbon
   • parameter: ratio of carbon to nitrogen.
2. the inverse of a normally distributed variable:
   • data: mass/area
   • parameter: area/mass

My current approach is to use simulation:

e.g. for a set of percent carbon and nitrogen data with means: xbar.n,c, variance: se.n,c, and sample size: n.n, n.c:

set.seed(1)
per.c <- rnorm(100000, xbar.c, se.c*n.c) # percent C
per.n <- rnorm(100000, xbar.n, se.n*n.n) # percent N

I want to parameterize ratio.cn = perc.c/perc.n

# parameter of interest
ratio.cn <- perc.c / perc.n

Then choose the best fit distributions with range $0 \rightarrow \infty$ for my prior

library(MASS)
dist.fig <- list()
for(dist.i in c('gamma', 'lognormal', 'weibull')) {
    dist.fit[[dist.i]] <- fitdist(ratio.cn, dist.i)
}

Question: Is this a valid approach? Are there other / better approaches?

Thanks in advance!

Update: the Cauchy distribution, which is defined as the ratio of two normals with $\mu=0$, has limited utility since I would like to estimate variance. Perhaps I could calculate the variance of a simulation of n draws from a Cauchy?

I did find the following closed-form approximations but I haven't tested to see if they give the same results... Hayya et al, 1975 $$\hat{\mu}_{y:x} = \mu_y/mu_x + \sigma^2_x * \mu_y / \mu_x^3 + cov(x,y) * \sigma^2_x * \sigma^2_y / \mu_x^2$$ $$\hat{\sigma}^2_{y:x} = \sigma^2_x\times\mu_y / mu_x^4 + \sigma^2_y / mu_x^2 - 2 * cov(x,y) * \sigma^2_x * \sigma^2_y / mu_x^3$$

Hayya, J. and Armstrong, D. and Gressis, N., 1975. A note on the ratio of two normally distributed variables. Management Science 21: 1338--1341

Answer 1

You might want to look at some of the references under the Wikipedia article on Ratio Distribution. It's possible you'll find better approximations or distributions to use. Otherwise, your approach seems sound.

Update I think a better reference might be:

• Ratios of Normal Variables and Ratios of Sums of Uniform Variables (Marsaglia, 1965)

See formulas 2-4 on page 195.

Update 2

On your updated question regarding variance from a Cauchy -- as John Cook pointed out in the comments, the variance doesn't exist. So, taking a sample variance simply won't work as an "estimator". In fact, you'll find that your sample variance does not converge at all and fluctuates wildly as you keep taking samples.

Answer 2

Could you not assume that $y^{-1} \sim N(.,.)$ for the inverse of a normal random variable and do the necessary bayesian computation after identifying the appropriate parameters for the normal distribution.

My suggestion below to use the Cauchy does not work as pointed out in the comments by ars and John.

The ratio of two normally random variables follows the Cauchy distribution. You may want to use this idea to identify the parameters of the cauchy that most closely fits the data you have.