# Comparing the mean of several exponentially distributed variables

(Possibly related to Is there a nonparametric equivalent of Tukey HSD?)

Given a set of several exponentially distributed variables (representing data which is modeled this way), I'd like to check whether one of them has a significantly higher mean than the others. If the variables were normally distributed, I would have used Tukey's test; however, this is not the case, and I'm looking for as simple alternative as possible.

Some background on the data

Alternative methods and markets from which we buy goods are compared every day. Data is aggregated on a daily basis (i.e. I only have prices and number of transactions per day, which translate to a mean). The distribution of funds for the next day is based on previous' day analysis: best method gets most of the funds, and the other methods share the rest. Typically, 5-10 alternative methods are compared, with 10,000 and more transactions per method per day.

Current Alternatives

Based on my limited knowledge, I came up with two possible ways of comparing methods:

1. Based on the referenced question above, a nonparametric versions of Tukey's test can be used.

2. Since only the highest-mean variable is compared to the rest of the variables, I think it can be compared to the second highest with bonferroni correction. i.e. use effective alpha = 0.05 / number of variables. This might be very conservative, but since the numbers are high, the results are usable (i.e. most days have significent results).

My questions

1. Do the methods above make sense as solutions to the presented case? specifically, method (2) is simple and yields usable results, but I'm not sure it's statistically sound.

2. I'm interested in any other suggestions for performing this analysis. As noted, I'm more interested in simplicity and generalization, and less on getting the best significance values.

• Given you have specified an exponential distribution, the most obvious thing to do would be to work with GLMs. However, there's no particular reason you shouldn't use nonparametric procedures; they should work just fine (though it may help to remember you're dealing here with scale shifts in an exponential, since the mean is a scale parameter; if you work with logs you can use the nonparametric approach to estimate an interval for the location-shift in the logs, and hence an interval for the scale shift when transformed back) Commented Aug 23, 2013 at 0:39
• Thanks, GLMs sound like a good idea. However, I'm working with C#, which is not rich with good open-source statistical libraries; hence the need for as simple solutions as possible (to minimize the cost of implementation). Regarding the comment about nonparametric methods, I don't understand it at the moment, but I'll do some homework...
– etov
Commented Aug 25, 2013 at 9:31