Test whether or not quantile values belong to a distribution

Question

I have a list of 23 values for the following quantiles:

[1] 0.010 0.025 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600 0.650 0.700 0.750 0.800 0.850
[20] 0.900 0.950 0.975 0.990

I want to test whether or not these quantiles belong to a particular known distribution. My thought was to use calculate a Kolmogorov-Smirnov test statistic, but since I have quantile values and not data I don't know how to find the right p-value.

Any thoughts on how to get an accurate p-value or of another test I can use?

Answer 1

You can plug these probabilities into a friendly neighbourhood theoretical quantile function and then plot observed quantiles versus expected quantiles.

1. Good news: With those 23 quantiles you have much of the information about the distribution that is available.
2. Good news: A graph based on those 23 data points might tell you most of what you need to know. Often it's a matter of informal indication such as "The normal is clearly a poor fit, so what to do next?".
3. Bad news: Whatever lies beyond the 1% and 99% points might be extraordinarily important detail, but if you don't have it, you won't know. (Yet how often are the raw data hidden, but only certain quantiles available?)
4. Bad news: It will be uphill work turning that into a significance test yielding a P-value, if that is your goal, although many posts here, and many expert opinions elsewhere, imply that you aren't missing much of value.

Kolmogorov-Smirnov is beautiful mathematics, but often oversold statistically.

More crucially, the form of marginal distributions is less important often than variation around systematic structure as approximated by a model.