Significance of difference between 2 variables of Cauchy distribution

Question

I've got to verify whether there is a significant difference in value market shares between 2 versions of packaging (A and B) of a given beverage:

For the 'Packaging A' I've got a sample of 600 purchases on which I compute average 'Packaging A' purchase value as well as the average value of entire category purchase. Both outcomes are of normal distribution so their ratio - value share - is of the Cauchy distribition.

Could anyone advice me, how to find a Cauchy distribution statistic by which I could verify the hypothesis of difference between the 2 packaging versions in value shares? (for the Packaging B the sample is 650).

Answer 1

1) The two distributions aren't Normal, since there's no chance of observing a 0 or negative value. That doesn't mean the Normal isn't a useful approximation, but when taking the ratio and trying to work with the functional form of the Normal, you're making life a little more difficult than it has to be.

2) I'd suggest using the bootstrap to build your confidence interval on the ratio. Specifically, draw, say, 1000 samples of size 600 with replacement from the "Packaging A" results, and an equal number of samples of size 650 with replacement from the "Packaging B" results. For a simple confidence interval, form the 1000 ratios, sort, and just pick off the 25th and 975th largest numbers.

For (typically) better confidence intervals, although possibly not much better, use the "boot" package in R. An example with a little cheating (padding the shorter series with NA means the bootstrap might select samples a little smaller or larger than 600 from the shorter series, although with such a large sample this will have little effect on the results) is below:

# Create random purchase values; pad shorter series with NA 
PurchaseValueA <- c(rgamma(600, 5, 1), rep(NA, 50)) 
PurchaseValueB <- rgamma(650, 4.75, 1.1)
df <- data.frame(PVA=PurchaseValueA, PVB=PurchaseValueB)

# Bootstrap statistic function
foo <- function(data, i) {
  mean(data$PVA[i], na.rm=TRUE) / mean(data$PVB[i], na.rm=TRUE)
}

# Run the bootstrap, calculate confidence intervals 
boot.foo <- boot(df, foo, 1000)
boot.ci(boot.foo)

BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
Based on 10000 bootstrap replicates

CALL : 
boot.ci(boot.out = boot.foo)

Intervals : 
Level      Normal              Basic         
95%   ( 1.151,  1.273 )   ( 1.150,  1.272 )  

Level     Percentile            BCa          
95%   ( 1.153,  1.275 )   ( 1.152,  1.274 )  
Calculations and Intervals on Original Scale
Warning message:
In boot.ci(boot.foo) : bootstrap variances needed for studentized intervals

As we can see, all the CIs are essentially the same.