A collection of robust statistical tests in Julia



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This package contains a variety of functions from the field robust statistical methods. Many are estimators of location or dispersion; others estimate the standard error or the confidence intervals for the location or dispresion estimators, generally computed by the bootstrap method.

Many functions in this package are based on the R package WRS (an R-Forge repository) by Rand Wilcox. Others were contributed by users as needed. References to the statistics literature can be found below.

This package requires Compat, Rmath, Dataframes, and Distributions. They can be installed automatically, or by invoking Pkg.add("packagename").


Location estimators:

  • tmean(x, tr=0.2) - Trimmed mean: mean of data with the lowest and highest fraction tr of values omitted.
  • winmean(x, tr=0.2)- Winsorized mean: mean of data with the lowest and highest fraction tr of values squashed to the 20%ile or 80%ile value, respectively.
  • tauloc(x) - Tau measure of location by Yohai and Zamar.
  • onestep(x) - One-step M-estimator of location using Huber's ψ
  • mom(x) - Modified one-step M-estimator of location (MOM)
  • bisquareWM(x) - Mean with weights given by the bisquare rho function.
  • huberWM(x) - Mean with weights given by Huber's rho function.
  • trimean(x) - Tukey's trimean, the average of the median and the midhinge.

Dispersion estimators:

  • winvar(x, tr=0.2) - Winsorized variance.
  • wincov(x, y, tr=0.2) - Winsorized covariance.
  • pbvar(x) - Percentage bend midvariance.
  • bivar(x) - Biweight midvariance.
  • tauvar(x) - Tau measure of scale by Yohai and Zamar.
  • iqrn(x) - Normalized inter-quartile range (normalized to equal σ for Gaussians).
  • shorthrange(x) - Length of the shortest closed interval containing at least half the data.
  • scaleQ(x) - Normalized Rousseeuw & Croux Q statistic, from the 25%ile of all 2-point distances.
  • scaleS(x) - Normalized Rousseeuw & Croux S statistic, from the median of the median of all 2-point distances.
  • shorthrange!(x), scaleQ!(x), and scaleS!(x) are non-copying (that is, x-modifying) forms of the above.

Confidence interval or standard error estimates:

  • trimse(x) - Standard error of the trimmed mean.
  • trimci(x) - Confidence interval for the trimmed mean.
  • msmedse(x) - Standard error of the median.
  • binomci(s,n) - Binomial confidence interval (Pratt's method).
  • acbinomci(s,n) - Binomial confidence interval (Agresti-Coull method).
  • sint(x) - Confidence interval for the median (with optional p-value).
  • momci(x) - Confidence interval of the modified one-step M-estimator of location (MOM).
  • trimpb(x) - Confidence interval for trimmed mean.
  • pcorb(x) - Confidence intervale for Pearson's correlation coefficient.
  • yuend - Compare the trimmed means of two dependent random variables.
  • bootstrapci(x, est=f) - Compute a confidence interval for estimator f(x) by bootstrap methods.
  • bootstrapse(x, est=f) - Compute a standard error of estimator f(x) by bootstrap methods.

Utility functions:

  • winval(x, tr=0.2) - Return a Winsorized copy of the data.
  • idealf(x) - Ideal fourths, interpolated 1st and 3rd quartiles.
  • outbox(x) - Outlier detection.
  • hpsi(x) - Huber's ψ function.
  • contam_randn - Contaminated normal distribution (generates random deviates).
  • _weightedhighmedian(x) - Weighted median (breaks ties by rounding up). Used in scaleQ.


For location, consider the bisquareWM with k=3.9σ, if you can make any reasonable guess as to the "Gaussian-like width" σ (see dispersion estimators for this). If not, trimean is a good second choice, though less efficient. Also, though the author personally has no experience with them, tauloc, onestep, and mom might be useful.

For dispersion, the scaleS is a good general choice, though scaleQ is very efficient for nearly Gaussian data. The MAD is the most robust though less efficient. If scaleS doesn't work, then shorthrange is a good second choice.

The first reference on scaleQ and scaleS (below) is a lengthy discussion of the tradeoffs among scaleQ, scaleS, shortest half, and median absolute deviation (MAD, see BaseStats.mad for Julia implementation). All four have the virtue of having the maximum possible breakdown point, 50%. This means that replacing up to 50% of the data with unbounded bad values leaves the statistic still bounded. The efficiency of Q is better than S and S is better than MAD (for Gaussian distributions), and the influence of a single bad point and the bias due to a fraction of bad points is only slightly larger on Q or S than on MAD. Unlike MAD, the other three do not implicitly assume a symmetric distribution.

To choose between Q and S, the authors note that Q has higher statistical efficiency, but S is typically twice as fast to compute and has lower gross-error sensitivity. An interesting advantage of Q over the others is that its influence function is continuous. For a rough idea about the efficiency, the large-N limit of the standardized variance of each quantity is 2.722 for MAD, 1.714 for S, and 1.216 for Q, relative to 1.000 for the standard deviation (given Gaussian data). The paper gives the ratios for Cauchy and exponential distributions, too; the efficiency advantages of Q are less for Cauchy than for the other distributions.


#Set up a sample dataset:
x=[1.672064, 0.7876588, 0.317322, 0.9721646, 0.4004206, 1.665123, 3.059971, 0.09459603, 1.27424, 3.522148,
   0.8211308, 1.328767, 2.825956, 0.1102891, 0.06314285, 2.59152, 8.624108, 0.6516885, 5.770285, 0.5154299]

julia> mean(x)     #the mean of this dataset

tmean: trimmed mean

julia> tmean(x)            #20% trimming by default

julia> tmean(x, tr=0)      #no trimming; the same as the output of mean()

julia> tmean(x, tr=0.3)    #30% trimming

julia> tmean(x, tr=0.5)    #50% trimming, which gives you the median of the dataset.

winval: winsorize data

That is, return a copy of the input array, with the extreme low or high values replaced by the lowest or highest non-extreme value, repectively. The fraction considered extreme can be between 0 and 0.5, with 0.2 as the default.

julia> winval(x)           #20% winsorization; can be changed via the named argument `tr`.
20-element Any Array:

winmean, winvar, wincov: winsorized mean, variance, and covariance

julia> winmean(x)          #20% winsorization; can be changed via the named argument `tr`.
julia> winvar(x)
julia> wincov(x, x)
julia> wincov(x, x.^2)

trimse: estimated standard error of the trimmed mean

julia> trimse(x)           #20% winsorization; can be changed via the named argument `tr`.

trimci: (1-α) confidence interval for the trimmed mean

Can be used for paired groups if x consists of the difference scores of two paired groups.

julia> trimci(x)                 #20% winsorization; can be changed via the named argument `tr`.
(1-α) confidence interval for the trimmed mean

Degrees of freedom:   11
Estimate:             1.292180
Statistic:            3.469611
Confidence interval:  0.472472       2.111889
p value:              0.005244

idealf: the ideal fourths:

Returns (q1,q3), the 1st and 3rd quartiles. These will be a weighted sum of the values that bracket the exact quartiles, analogous to how we handle the median of an even-length array.

julia> idealf(x)

pbvar: percentage bend midvariance

A robust estimator of scale (dispersion). See NIST ITL webpage for more.

julia> pbvar(x)

bivar: biweight midvariance

A robust estimator of scale (dispersion). See NIST ITL webpage for more.

julia> bivar(x)

tauloc, tauvar: tau measure of location and scale

Robust estimators of location and scale, with breakdown points of 50%.

See Yohai and Zamar JASA, vol 83 (1988), pp 406-413 and Maronna and Zamar Technometrics, vol 44 (2002), pp. 307-317.

julia> tauloc(x)       #the named argument `cval` is 4.5 by default.
julia> tauvar(x)

outbox: outlier detection

Use a modified boxplot rule based on the ideal fourths; when the named argument mbox is set to true, a modification of the boxplot rule suggested by Carling (2000) is used.

julia> outbox(x)
Outlier detection method using
the ideal-fourths based boxplot rule

Outlier ID:         17
Outlier value:      8.62411
Number of outliers: 1
Non-outlier ID:     1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20

msmedse: Standard error of the median

Return the standard error of the median, computed through the method recommended by McKean and Schrader (1984).

julia> msmedse(x)

binomci(), acbinomci(): Binomial confidence interval

Compute the (1-α) confidence interval for p, the binomial probability of success, given s successes in n trials. Instead of s and n, can use a vector x whose values are all 0 and 1, recording failure/success one trial at a time. Returns an object.

binomci uses Pratt's method; acbinomci uses a generalization of the Agresti-Coull method that was studied by Brown, Cai, & DasGupta.

julia> binomci(2, 10)           # # of success and # of total trials are provided. By default alpha=.05
p_hat:               0.2000
confidence interval: 0.0274   0.5562
Sample size          10

julia> trials=[1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0]
julia> binomci(trials, alpha=0.01)    #trial results are provided in array form consisting of 1's and 0's.
 p_hat:               0.5000
 confidence interval: 0.1768   0.8495
 Sample size          12

julia> acbinomci(2, 10)           # # of success and # of total trials are provided. By default alpha=.05
p_hat:               0.2000
confidence interval: 0.0459   0.5206
Sample size          10


Compute the confidence interval for the median. Optionally, uses the Hettmansperger-Sheather interpolation method to also estimate a p-value.

julia> sint(x)
Confidence interval for the median

 Confidence interval:  0.547483       2.375232

julia> sint(x, 0.6)
Confidence interval for the median with p-val

 Confidence interval:  0.547483       2.375232
 p value:              0.071861


Compute Huber's ψ. The default bending constant is 1.28.

julia> hpsi(x)
20-element Array{Float64,1}:


Compute one-step M-estimator of location using Huber's ψ. The default bending constant is 1.28.

julia> onestep(x)

bootstrapci, bootstrapse

Compute a bootstrap, (1-α) confidence interval (bootstrapci) or a standard error (bootstrapse) for the measure of location corresponding to the argument est. By default, the median is used. Default α=0.05.

julia> ci = bootstrapci(x, est=onestep, nullvalue=0.6)
 Estimate:             1.305811
 Confidence interval:  0.687723       2.259071
 p value:              0.026000

julia> se = bootstrapse(x, est=onestep)

mom and mom!

Returns a modified one-step M-estimator of location (MOM), which is the unweighted mean of all values not more than (bend times the mad(x)) away from the data median.

julia> mom(x)


Compute the bootstrap (1-α) confidence interval for the MOM-estimator of location based on Huber's ψ. Default α=0.05.

julia> momci(x, seed=2, nboot=2000, nullvalue=0.6)
Estimate:             1.259646
Confidence interval:  0.504223       2.120979
p value:              0.131000


Create contaminated normal distributions. Most values will by from a N(0,1) zero-mean unit-variance normal distribution. A fraction epsilon of all values will have k times the standard devation of the others. Default: epsilon=0.1 and k=10.

julia> srand(1);
julia> std(contam_randn(2000))


Compute a (1-α) confidence interval for a trimmed mean by bootstrap methods.

julia> trimpb(x, nullvalue=0.75)
 Estimate:             1.292180
 Confidence interval:  0.690539       2.196381
 p value:              0.086000


Compute a .95 confidence interval for Pearson's correlation coefficient. This function uses an adjusted percentile bootstrap method that gives good results when the error term is heteroscedastic.

julia> pcorb(x, x.^5)
 Estimate:             0.802639
 Confidence interval:  0.683700       0.963478


Compare the trimmed means of two dependent random variables using the data in x and y. The default amount of trimming is 20%.

julia> srand(3)
julia> y2 = randn(20)+3;
julia> yuend(x, y2)

Comparing the trimmed means of two dependent variables.

Sample size:          20
Degrees of freedom:   11
Estimate:            -1.547776
Standard error:       0.460304
Statistic:           -3.362507
Confidence interval: -2.560898      -0.534653
p value:              0.006336

Unmaintained functions

See UNMAINTAINED.md for information about functions that the maintainers have not yet understood but also not yet deleted entirely.


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