using QDistributions, Plots

QDistributions.jl

A Julia package for (continuous) quantile distributions and associated functions. Particularly QDistributions implements:

• Quantile function (quantile) and complimentary (non-increasing) quantile function (cquantile).
• Quantile-based statistics: median (median), inter-quartile range (iqr), robust (Galton-Bowley) skewness (rskewness), robust (Moors’) kurtosis (rkurtosis).
• Quantile density function (qdf), density quantile function (dqf) and its logarithm (logdqf).
• Inverse of quantile function (cdf): close-form, where available, or approximated by a bracketed rootfinder, otherwise. Complimentary CDF (survival function) ccdf
• Probability density function (pdf), as a thin wrapper over dqf and cdf; also logpdf.
• Sampling from a distribution (rand and sampler functions)

Where possible, the following functions are also implemented:

• Pearson moments: mean, var, skewness, kurtosis
• L-moments: lmoment
• L-coefficient of variance (lvar), L-skewness ratio (lskewness), L-kurtosis ratio (lkurtosis).

Generalized Lambda Distribution (CSW parameterization)

Generalilzed Lambda Distribution reparameterized by Chalabi, Scott, and Wuertz (2012) (GLD CSW) is a flexible quantile-based distribution with two shape parameters: asymmetry $\chi$ and steepness $\xi$. Besides, it is an IQR-scaled distribution, meaning that its scale parameter $\sigma$ corresponds to the inter-quartile range (IQR).

d = GLDcsw(1, 5, -0.5, 0.4)

# corner cases
#d = GLDcsw(1, 5, 0, 0.5) 
#d = GLDcsw(1, 5, -0.6, (1-0.6)/2)
#d = GLDcsw(1, 5, -0.6, (1+0.6)/2)


#p_grd = ppoints(100)
p_grd = 0:0.01:1

q_grd = quantile(d, p_grd)
f_grd = map(Base.Fix1(qdf,d), p_grd)
dq_grd = map(Base.Fix1(dqf,d), p_grd)

plot(p_grd, q_grd, title="Quantile function", label=string(d))

p_grd_aprx = cdf(d, q_grd)

plot(q_grd, p_grd_aprx, title="Inverse quantile function (CDF)", label=string(d))

plot(p_grd, f_grd, title="Quantile density function", label=string(d))

plot(q_grd, dq_grd, title="Density Quantile Function", label=string(d))

Flattened Skew-Logistic Distribution (FSLD)

One of the most iconic, yet simple quantile-based distributions introduced by Chakrabarty and Sharma (2018) and Chakrabarty and Sharma (2021), combining ideas presented in Gilchrist (2000).

using QDistributions, Plots
p_grd = 0:0.01:1

dl = FGLD(1, 5, 0.2, 3)

q_grd = quantile(dl, p_grd)
f_grd = map(Base.Fix1(qdf,dl), p_grd)
dq_grd = map(Base.Fix1(dqf,dl), p_grd)

plot(p_grd, q_grd, title="Quantile function", label=string(dl))

p_grd_aprx = cdf(dl, q_grd)

plot(q_grd, p_grd_aprx, title="Inverse quantile function (CDF)", label=string(dl))

plot(p_grd, f_grd, title="Quantile density function", label=string(dl))

plot(q_grd, dq_grd, title="Density Quantile Function", label=string(dl))

Contributing

Citing

References

Chakrabarty, Tapan Kumar, and Dreamlee Sharma. 2018. “The Quantile-Based Skew Logistic Distribution with Applications.” In Statistics and Its Applications, edited by Asis Kumar Chattopadhyay and Gaurangadeb Chattopadhyay, 51–73. Springer Proceedings in Mathematics & Statistics. Singapore: Springer. https://doi.org/10.1007/978-981-13-1223-6_6.

———. 2021. “A Generalization of the Quantile-Based Flattened Logistic Distribution.” Annals of Data Science 8 (3): 603–27. https://doi.org/10.1007/s40745-021-00322-3.

Chalabi, Yohan, David J Scott, and Diethelm Wuertz. 2012. “Flexible Distribution Modeling with the Generalized Lambda Distribution.” Working paper MPRA Paper No. 43333,. Zurich, Switzerland: ETH. https://mpra.ub.uni-muenchen.de/43333/.

Gilchrist, Warren. 2000. Statistical Modelling with Quantile Functions. Boca Raton: Chapman & Hall/CRC. https://doi.org/10.1201/9781420035919.