Computing the noncentral gamma distribution, its inverse and the noncentrality parameter

Oliveira, Izabela Regina Cardoso de; Ferreira, Daniel Furtado

doi:10.1007/s00180-012-0371-8

Cited by 5 publications

(7 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…-Under-and overflow problems were reported by Benton and Krishnamoorthy (2003), Ding (1997), Helstrom and Ritcey (1985) independently of us; -we also reveal in Baharev and Kemény 2008 that the algorithms of Norton (1983) and Lenth (1987) are exposed to over-and underflow issues, and that the Appendix 12 of Lorenzen and Anderson (1993, p. 374) is most likely bogus due to overflow; -catastrophic round-off errors were reported by Frick (1990); -drastically increasing computation time and hang-ups were observed by Chattamvelli (1995), Benton and Krishnamoorthy (2003); -other noncentral distributions are similarly challenging, see for example Oliveira and Ferreira (2012). If some of the intermediate computations suffer catastrophic loss of precision, the root-finding method can still succeed, and the final result presented to the user may nevertheless seem plausible.…”

Section: Introductionmentioning

confidence: 75%

Computing the noncentral-F distribution and the power of the F-test with guaranteed accuracy

2016

View full text Add to dashboard Cite

The computations involving the noncentral-F distribution are notoriously difficult to implement properly in floating-point arithmetic: Catastrophic loss of precision, floating-point underflow and overflow, drastically increasing computation time and program hang-ups, and instability due to numerical cancellation have all been reported. It is therefore recommended that existing statistical packages are crosschecked, and the present paper proposes a numerical algorithm precisely for this purpose. To the best of our knowledge, the proposed method is the first method that can compute the noncentrality parameter of the noncentral-F distribution with guaranteed accuracy over a wide parameter range that spans the range relevant for practical applications. Although the proposed method is limited to cases where the the degree of freedom of the denominator of the F test statistic is even, it does not affect its usefulness significantly: All of those algorithmic failures and inaccuracies that we can still reproduce today could have been prevented by simply cross-checking against the proposed method. Two numerical examples are presented where the intermediate computations went wrong silently, but the final result of the computations seemed nevertheless plausible, and eventually erroneous results were published. Cross-checking against the proposed method would have caught the numerical errors in both cases. The source code of the algorithm is available on GitHub, together with self-contained command-line executables. These executables can read the data to be cross-checked from plain text files, making it easy to cross-check any statistical software in an automated fashion.

show abstract

Section: Introductionmentioning

confidence: 75%

Computing the noncentral-F distribution and the power of the F-test with guaranteed accuracy

2016

View full text Add to dashboard Cite

show abstract

“…The difference between our approximation and R function qchisq for the (central) chi-square quantiles was similarly small as for the non-central chi-square quantiles. A referee brought the R package “ncg” (Oliveira and Ferreira, 2013; Ferreira, Oliveira and Toledo, 2015) for computing the noncentral gamma distribution-related quantities to our attention. Non-central chi-square variable (X) is a scaled non-central gamma variable (Z) through a transformation Z = X/ 2.…”

Section: Numerical Examplesmentioning

confidence: 99%

A general approximation to quantiles

Zelterman

2016

Communications in Statistics - Theory and Methods

View full text Add to dashboard Cite

For many continuous distributions, a closed-form expression for their quantiles does not exist. Numerical approximations for their quantiles are developed on a distribution-by-distribution basis. This work develops a general approximation for quantiles using the Taylor expansion. Our method only requires that the distribution has a continuous probability density function and its derivatives can be derived to a certain order (usually 3 or 4). We demonstrate our unified approach by approximating the quantiles of the normal, exponential, and chi-square distributions. The approximation works well for these distributions.

show abstract

“…which is called the reflection formula. This formula is important for range reduction and it is easily proved by using (19). For computing the gamma function, we use recursion for small values of x and the relation given in (22) for larger values.…”

Section: Gamma Functionmentioning

confidence: 99%

“…which is called the reflection formula. This formula is important for range reduction and it is easily proved by using (19).…”

Section: Gamma Functionmentioning

confidence: 99%

See 1 more Smart Citation

GammaCHI: A package for the inversion and computation of the gamma and chi-square cumulative distribution functions (central and noncentral)

Gil

Segura

Temme³

2015

Computer Physics Communications

View full text Add to dashboard Cite

A Fortran 90 module GammaCHI for computing and inverting the gamma and chi-square cumulative distribution functions (central and noncentral) is presented. The main novelty of this package are the reliable and accurate inversion routines for the noncentral cumulative distribution functions. Additionally, the package also provides routines for computing the gamma function, the error function and other functions related to the gamma function. The module includes the routines cdfgamC, invcdfgamC, cdfgamNC, invcdfgamNC, errorfunction, inverfc, gamma, loggam, gamstar and quotgamm for the computation of the central gamma distribution function (and its complementary function), the inversion of the central gamma distribution function, the computation of the noncentral gamma distribution function (and its complementary function), the inversion of the noncentral gamma distribution function, the computation of the error function and its complementary function, the inversion of the complementary error function, the computation of: the gamma function, the logarithm of the gamma function, the regulated gamma function and the ratio of two gamma functions, respectively. The computation and inversion of gamma and chi-square cumulative distribution functions (central and noncentral) as well as the computation of the error and gamma functions is needed in many problems of applied and mathematical physics. PROGRAM SUMMARY Solution method:The algorithms use different methods of computation depending on the range of parameters: asymptotic expansions, quadrature methods, etc. Restrictions:In the inversion of the central gamma/chi-square distribution functions, very

show abstract

Computing the noncentral gamma distribution, its inverse and the noncentrality parameter

Cited by 5 publications

References 7 publications

Computing the noncentral-F distribution and the power of the F-test with guaranteed accuracy

Computing the noncentral-F distribution and the power of the F-test with guaranteed accuracy

A general approximation to quantiles

GammaCHI: A package for the inversion and computation of the gamma and chi-square cumulative distribution functions (central and noncentral)

Contact Info

Product

Resources

About