Generating Functions and Short Recursions, With Applications to the Moments of Quadratic Forms in Noncentral Normal Vectors

Hillier, Grant; Kan, Raymond; Wang, Xiaoulu

doi:10.1017/s0266466613000364

Cited by 16 publications

(39 citation statements)

References 29 publications

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“…Ratios of quadratic forms, especially those in normal random variables, play a pivotal role in statistics, as many practically important test statistics can be written as a ratio of quadratic forms. Naturally, there is a vast body of literature regarding the distribution and moments of quadratic forms and ratios thereof (e.g., Magnus, 1986;Jones, 1986Jones, , 1987Smith, 1989Smith, , 1993Mathai & Provost, 1992;Hillier, 2001;Forchini, 2002;Hillier et al, 2009Hillier et al, , 2014Bao & Kan, 2013).…”

Section: Ratio Of Quadratic Formsmentioning

confidence: 99%

“…Another way is to expand a ratio into infinite series, which in turn is integrated using the zonal and invariant polynomials (e.g., Smith, 1989Smith, , 1993Hillier, 2001;Hillier et al, 2009Hillier et al, , 2014. (See Bao & Kan (2013) for connections between these two approaches.)…”

Section: Ratio Of Quadratic Formsmentioning

confidence: 99%

“…In order to make all above expressions available in the popular statistical software environment R, the present author has developed an independent implementation of the algorithms of Hillier et al (2014) and Bao & Kan (2013) and published it as the package qfratio on the GitHub repository (https://github.com/watanabe-j/qfratio). This package can be used to evaluate general moments of ratios of quadratic forms in normal variables with arbitrary mean and covariance (under certain constraints for singular cases as discussed in Appendix C.1).…”

Section: Software Implementationmentioning

confidence: 99%

“…Chikuse (1987) and Hillier et al (2009Hillier et al ( , 2014 presented a few recursive algorithms to evaluate d k 1 ,...,kr .…”

Section: A7 Example Application To Evolvability Measuresmentioning

confidence: 99%

“…Bao & Kan (2013) and Hillier et al (2014) presented further improvements for evaluating moments of ratios of quadratic forms in potentially noncentral normal variables, x ∼ N p (µ, I p ).…”

Section: A7 Example Application To Evolvability Measuresmentioning

confidence: 99%

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Exact expressions and numerical evaluation of average evolvability measures for characterizing and comparing G matrices

Watanabe

2022

Preprint

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Theory predicts that the additive genetic covariance (G) matrix determines a population's short-term (in)ability to respond to directional selection—evolvability in the Hansen–Houle sense—which is typically quantified and compared via certain scalar indices called evolvability measures. Often, interest is in obtaining the averages of these measures across all possible selection gradients, but explicit formulae for most of these average measures have not been known. Previous authors relied either on approximations by the delta method, whose accuracy is generally unknown, or Monte Carlo evaluations (including the random skewers analysis), which necessarily involve random fluctuations. This study presents new, exact expressions for the average conditional evolvability, average autonomy, average respondability, average flexibility, average response difference, and average response correlation, utilizing their mathematical structures as ratios of quadratic forms. The new expressions are infinite series involving top-order zonal and invariant polynomials of matrix arguments, and can be numerically evaluated as their partial sums with, for some measures, known error bounds. Whenever these partial sums numerically converge within reasonable computational time, they will replace the previous approximate methods. In addition, new expressions are derived for the average measures under a general normal distribution for the selection gradient, extending the applicability of these measures into a substantially broader class of selection regimes.

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Section: Ratio Of Quadratic Formsmentioning

confidence: 99%

Section: Ratio Of Quadratic Formsmentioning

confidence: 99%

Section: Software Implementationmentioning

confidence: 99%

“…Chikuse (1987) and Hillier et al (2009Hillier et al ( , 2014 presented a few recursive algorithms to evaluate d k 1 ,...,kr .…”

Section: A7 Example Application To Evolvability Measuresmentioning

confidence: 99%

Section: A7 Example Application To Evolvability Measuresmentioning

confidence: 99%

See 3 more Smart Citations

Exact expressions and numerical evaluation of average evolvability measures for characterizing and comparing G matrices

Watanabe

2022

Preprint

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Exact expressions and numerical evaluation of average evolvability measures for characterizing and comparing $$\textbf{G}$$ matrices

Watanabe

2023

J. Math. Biol.

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Theory predicts that the additive genetic covariance ($$\textbf{G}$$ G ) matrix determines a population’s short-term (in)ability to respond to directional selection—evolvability in the Hansen–Houle sense—which is typically quantified and compared via certain scalar indices called evolvability measures. Often, interest is in obtaining the averages of these measures across all possible selection gradients, but explicit formulae for most of these average measures have not been known. Previous authors relied either on approximations by the delta method, whose accuracy is generally unknown, or Monte Carlo evaluations (including the random skewers analysis), which necessarily involve random fluctuations. This study presents new, exact expressions for the average conditional evolvability, average autonomy, average respondability, average flexibility, average response difference, and average response correlation, utilizing their mathematical structures as ratios of quadratic forms. The new expressions are infinite series involving top-order zonal and invariant polynomials of matrix arguments, and can be numerically evaluated as their partial sums with, for some measures, known error bounds. Whenever these partial sums numerically converge within reasonable computational time and memory, they will replace the previous approximate methods. In addition, new expressions are derived for the average measures under a general normal distribution for the selection gradient, extending the applicability of these measures into a substantially broader class of selection regimes.

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Analytical Finite Sample Econometrics: From A. L. Nagar to Now

Bao

Ullah

2021

J. Quant. Econ.

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Generating Functions and Short Recursions, With Applications to the Moments of Quadratic Forms in Noncentral Normal Vectors

Abstract: Using generating functions, the top-order zonal polynomials that occur in much distribution theory under normality can be recursively related to other symmetric functions (power-sum and elementary symmetric functions, Ruben (1962), Hillier, Kan, and Wang (2009)

Cited by 16 publications

References 29 publications

Exact expressions and numerical evaluation of average evolvability measures for characterizing and comparing G matrices

Exact expressions and numerical evaluation of average evolvability measures for characterizing and comparing G matrices

Exact expressions and numerical evaluation of average evolvability measures for characterizing and comparing $$\textbf{G}$$ matrices

Analytical Finite Sample Econometrics: From A. L. Nagar to Now

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