Creating empirical distribution

[tomaskrehlik]

I am writing a small package that does unit root tests on a time series. I was always saddened by the inconsistent framework of various R statistical tests so I thought how to do it right in case of Julia. I am aware of the HypothesisTests package, but there seems to be no general interface to statistical tests. Given that the statistical tests have rather standard set of attributes I created a mock-up types that I think would work quite well.

       type StatisticalTest
        statistic::Float64
        # Probably some Univariate distribution maybe done from empirical dist.
        distribution:: ...
        pvalue::Float64
        description::ASCIIString
        nullHypothesis::ASCIIString
        function StatisticalTest(statistic, distribution, pvalue)
            new(statistic, distribution, pvalue, description, nullHypothesis)
        end
    end

However, I got to a dead-end with this type in case of the unit root tests, because there is no standard distribution and the critical values are tabulated. (It is also possible to simulate them quite easily.) Is there a way how to specify some empirical distribution within the Distributions package? Ideally, I would like the distribution to be a type from Distributions also taking all the other facilities from there.

[andreasnoack]

There is EmpiricalUnivariateDistribution in this package so if you have simulated the test statistic then just input the values in a vector to the constructor and you'll have an approximate cdf and quantile function. It appears that the show methods need a fix for this type, but besides of that, I think it works.

What kind of unit root tests are you considering? If you are considering tests with a Dickey-Fuller type distribution, the squared or two-sided version is very well approximated by the Γ distribution matched on the shape and scale parameters.

[tomaskrehlik]

Thanks a lot, maybe it would deserve a mention in the documentation. Search for empirical does not give anything.

Yes, Dickey-Fuller is one of those I am considering. The first batch would be something like DF, ADF, KPSS, Phillips-Peron as a good start. I honestly do no know much about the Gamma approximation, so I was planning so far to stick with the empirical simulation. Maybe somebody more knowledgeable will do a pull request. :)

[andreasnoack]

I'll might do someday. I've been through that process before. A reference for the Γ approximations is

Doornik, J, Approximations to the Asymptotic Distribution of Cointegration Tests, Journal of Economic Surveys 1998

[johnmyleswhite]

Should we relabel this issue?

[hdavid16]

There is EmpiricalUnivariateDistribution in this package so if you have simulated the test statistic then just input the values in a vector to the constructor and you'll have an approximate cdf and quantile function. It appears that the show methods need a fix for this type, but besides of that, I think it works.

What kind of unit root tests are you considering? If you are considering tests with a Dickey-Fuller type distribution, the squared or two-sided version is very well approximated by the Γ distribution matched on the shape and scale parameters.

I believe that the EmpiricalUnivariateDistribution no longer exists in the package. FYI, a workaround for creating an empirical discrete distribution is given in https://discourse.julialang.org/t/empirical-distribution-type-for-continuous-variables/5676/15?u=hdavid16