OrderStatistics

A package for working with order statistics of random variables.

Installation

This module is not yet registered. To install run

Pkg.clone("https://github.com/gajomi/OrderStatistics.jl")

##Basic usage

High level api

The high level api is defined by a collection of methods for computing order statistics of different kinds of random variables. The basic breakdown is

• orderstatistic- returns a univariate marginal order statistic
• min/max returns the minimum or maximum order statistic
• range returns the difference between the minimum and maximum
• spacing returns the marginal spacing between adjacent order statistics, which includes the maximum/minimum of the support in the case that they are finite

Some of these methods have analogs that return the joint distributions

• jointorderstatistic - returns a multivariate joint order statistic (full or marginal)
• jointspacing - returns a multivariate joint order statistic (full or marginal)

There are also plural analogs that serve as convenience methods for iterating over collections of order statistics

• orderstatistics,jointorderstatistics, spacings,jointspacings

At present only independent random sequences (IID/INID) from IndependentRandomSequences.jl are supported as the distribution type. Most of these methods support a variety of signatures. See method documentation for usage.

Example

using Distributions
using IndependentRandomSequences
using OrderStatistics

U,N = Uniform(),3
sequence = IIDRandomSequence(U,N)
Vs = orderstatistics(sequence)

>>>3-element Array{Distributions.Beta,1}:
    Distributions.Beta(α=1.0, β=3.0)
    Distributions.Beta(α=2.0, β=2.0)
    Distributions.Beta(α=3.0, β=1.0)

jointspacing(sequence)

>>>Distributions.Dirichlet(alpha=[1.0,1.0,1.0,1.0])

X,N = Bernoulli(),20
sequence = IIDRandomSequence(X,N)
Zs = orderstatistics(sequence)

>>1x20 Array{Int64,2}:
 0  0  0  0  0  0  0  1  0  0  1  1  1  1  1  1  1  1  1  1

Es = [Exponential(1./rate) for rate=1:3]
sequence = INIDRandomSequence(Es)
minE = min(sequence)

>>>Distributions.Exponential(θ=0.16666666666666666)

Colorful Example

X,N = Normal(),3
sequence = IIDRandomSequence(X,N)
Zs = orderstatistics(sequence);

trials = 2^15
samples = [rand(Z,trials) for Z in Zs]
nedges = 64
edges = linspace(-3,3,nedges)
xx = linspace(-3,3,256)
oshists = [hist(samples[k],edges)[2]/trials for k= 1:N]
empiricalpdfs=[layer(x=midpoints(edges),y=nedges*oshist/(edges[end]-edges[1]),
                Geom.point,Theme(default_color=c)) for (oshist,c) in zip(oshists,colors)]
theorypdfs = [layer(x=xx,y = [pdf(Z,x) for x in xx],
                    Geom.line,Theme(default_color=c)) for (Z,c) in zip(Zs,colors)]

empiricalcdfs=[layer(x=midpoints(edges),y=cumsum(oshist),
                Geom.point,Theme(default_color=c)) for (oshist,c) in zip(oshists,colors)]
theorycdfs = [layer(x=xx,y = [cdf(Z,x) for x in xx],
                    Geom.line,Theme(default_color=c)) for (Z,c) in zip(Zs,colors)]

set_default_plot_size(20cm, 8cm)
hstack(plot(empiricalpdfs...,theorypdfs...,
        Guide.XLabel("x"), Guide.YLabel("probablity density"),gajomitheme),
plot(empiricalcdfs...,theorycdfs...,
        Guide.XLabel("x"), Guide.YLabel("cumulative probability"),gajomitheme))

Low level API (subject to change)

The high level api make it possible to return order statistic quantities in the form of types "known" to Julia through the Distributions package (and perhaps others). When such a "known" type is no available, a call is made to the low level api, which makes a call a generic fallback types: ScalarOrderStatistic and JointOrderStatistic.

#Contributing

Know an analytical result not included here?

The easy way to contribute to the module is to write instances of the high level api methods that return (correct) results as ordinary Distribution types. These methods should be type stable (i.e. a change in the distribution parameters or order shouldn't change the returned type). Correct results might not be obvious to the casual observer, so if you can include a citation that outlines a derivation that would be appreciated.

Roadmap to 0.1

There are currently many holes in the basic fallback functionality. Check out the roadmap to 0.1 to get a sense for this.

Inferential statistics

At present the focus of this package is entirely on the probabilistic aspects of order statistics. However, as the name suggests, they are an important part of non-parametric statistics. For those with a vision of what this would looking like, please fork the repo! If you are interested in a specific limited bit of functionality related to inference please open an issue with your feature request.