momentchi2

A Python implementation of four approximate methods for computing the cumulative distribution function of a weighted sum of chi-squared random variables. All the methods are based on moment-matching techniques.

Methods

Based on the R package momentchi2, this Python version contains the following methods:

• Hall-Buckley-Eagleson (function hbe)
• Satterthwaite-Welch (function sw)
• Wood's F method (function wf)
• Lindsay-Pilla-Basak method (function lpb4)

Installation instructions

Install using pip:

python3 -m pip install momentchi2

Package dependencies

The packages numpy and scipy are required to be installed.

Which method should I use?

All four methods (sw, hbe, wf and lpb4) are good, but the Hall-Buckley-Eagleson method is recommended for situations where the number of coefficients is moderately large (say, greater than 100). For a smaller number of coefficients (e.g. up to 10), the Lindsay-Pilla-Basak method is recommended. See Bodenham and Adams (2016) for a detailed analysis.

Examples:

## Hall-Buckley-Eagleson method
from momentchi2 import hbe

# should give value close to 0.95, actually 0.94908
hbe(coeff=[1.5, 1.5, 0.5, 0.5], x=10.203)            

# x is a list, output approx. 0.05, 0.95
hbe([1.5, 1.5, 0.5, 0.5], [0.627, 10.203])  

# x is a numpy array - preferred approach for speed
import numpy as np
from momentchi2 import hbe
hbe( np.array([1.5, 1.5, 0.5, 0.5]), np.array([0.627, 10.203]) )  

# Other methods, e.g. sw, wf or lpb4
# All methods called: methodname(coeff, x)
from momentchi2 import sw
sw([1.5, 1.5, 0.5, 0.5], [0.627, 10.203])  

from momentchi2 import wf
wf([1.5, 1.5, 0.5, 0.5], [0.627, 10.203])  

from momentchi2 import lpb4
lpb4([1.5, 1.5, 0.5, 0.5], [0.627, 10.203])  

# for a larger number of coefficients in coeff vector, 
# can increase the number of moments p for improved accuracy.
# NOTE: we need len(coeff) >= p. Default value of p is p=4.
lpb4([0.1, 2.3, 3.4, 5.6, 7.8, 8.9, 9.1], [9.366844, 82.0018], p=6)  

Details

All methods take two input arguments:

• coeff: a list of the coefficients of the weighted sum (where all values must be strictly greater than 0), and
• x: the quantile value(s) at which point(s) the cumulative distribution function is computed.

So calling a method is: methodname(coeff, x), where e.g. methodname is hbe.

Input for quantile vector x can be a float (single value) or a list of values, or a numpy array. Internally, lists are converted to numpy arrays (and then back to lists), so that the output format of x is the same as the input format.

The Lindsay-Pilla-Basak (lpb4) method has a parameter p which is set to 4 by default and this is sufficient in most cases. If the number of coefficients is larger (e.g. greater than 8), then the lpb4 method can be used for larger . Of course, the increased accuracy comes at an increased computational cost.

There are a few pathological cases where Wood's F method or the Lindsay-Pilla-Basak method can fail (e.g. number of coefficients < p), in which case the hbe method will be called.

Package references

1. D. A. Bodenham and N. M. Adams. A comparison of efficient approximations for a weighted sum of chi-squared random variables. Statistics and Computing, 26(4):917-928, 2016.

2. D. A. Bodenham (2016). momentchi2: Moment-Matching Methods for Weighted Sums of Chi-Squared Random Variables, https://cran.r-project.org/package=momentchi2

Method references

Satterthwaite-Welch

3. B. L.Welch. The significance of the difference between two means when the population variances are unequal. Biometrika, 29(3/4):350-362, 1938.

4. F. E. Satterthwaite. An approximate distribution of estimates of variance components. Biometrics Bulletin, 2(6):110-114,

5. G. E. P. Box Some theorems on quadratic forms applied in the study of analysis of variance problems, I. Effects of inequality of variance in the one-way classification. The Annals of Mathematical Statistics, 25(2):290-302, 1954.

Hall-Buckley-Eagleson

6. P. Hall. Chi squared approximations to the distribution of a sum of independent random variables. The Annals of Probability, 11(4):1028-1036, 1983.

7. M. J. Buckley and G. K. Eagleson. An approximation to the distribution of quadratic forms in normal random variables. Australian Journal of Statistics, 30(1):150-159, 1988.

Wood's F method

8. A. T. A. Wood. An F approximation to the distribution of a linear combination of chi-squared variables. Communications in Statistics-Simulation and Computation, 18(4):1439-1456, 1989.

Lindsay-Pilla-Basak method

9. B. G. Lindsay, R. S. Pilla, and P. Basak. Moment-based approximations of distributions using mixtures: Theory and applications. Annals of the Institute of Statistical Mathematics, 52(2):215-230, 2000.

An exact solution: Imhof's method

Note that while these methods are all approximate, they are very fast and are accurate to two or three decimal places. If an exact answer is required to arbitrary accuracy, consider Imhof's method, which is implemented in the R package CompQuadForm.

10. J. P. Imhof. Computing the distribution of quadratic forms in normal variables. Biometrika 48(3/4): 419-426, 1961.

11. P. Lafaye de Micheaux (2010). Computes the distribution function of quadratic forms in normal variables using Imhof's method, Davies's algorithm, Farebrother's algorithm or Liu et al.'s algorithm https://cran.r-project.org/web/packages/CompQuadForm/index.html

12. P. Duchesne and P. Lafaye de Micheaux. Computing the distribution of quadratic forms: Further comparisons between the Liu-Tang-Zhang approximation and exact methods. Computational Statistics and Data Analysis 54(4):858-862, 2010