Algorithm AS 204: The Distribution of a Positive Linear Combination of χ 2 Random Variables

Farebrother, R. W.

doi:10.2307/2347721

Cited by 78 publications

(82 citation statements)

References 8 publications

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“…To compute c(p,Ĉ n ,L n ,V n ), we can numerically find the (1−α)-quantile of the distribution of the linear combination, evaluating the distribution function by using Farebrother's (1984) algorithm.…”

Section: Testing On the Regression Coefficients Under Possible Weak Imentioning

confidence: 99%

Instrumental Variables Estimation and Weak-Identification-Robust Inference Based on a Conditional Quantile Restriction

Marmer

Sakata

2011

SSRN Journal

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Extending the L1-IV estimator proposed by Sakata (1997Sakata ( , 2007, we develop a method, named the ρ τ -IV estimator, to estimate structural equations based on the conditional quantile restriction imposed on the error terms. We study its asymptotic behavior and show how to make statistical inferences on the regression parameters. Given practical importance of weak identification, a highlight of the paper is proposal of a test robust to the weak identification. The statistics used in our method can be viewed as a natural counterpart of the Anderson and Rubin's (1949) statistic in the ρτ -IV estimation. As is the case for the Anderson-Rubin test in the standard IV regression, our test can have a low power, when there many instruments excluded from the regression equation. We show how we could avoid such problem 1 by taking for instruments a small number of linear combinations of all available instruments that are suitably selected based on data.

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Section: Testing On the Regression Coefficients Under Possible Weak Imentioning

confidence: 99%

Instrumental Variables Estimation and Weak-Identification-Robust Inference Based on a Conditional Quantile Restriction

Marmer

Sakata

2011

SSRN Journal

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“…This is the (long) recursion given by Ruben [19]. Using Ruben's recursive algorithm for computingh k , Sheil and O'Muircheartaigh [23] and Farebrother [7] develop computer programs for approximating the pdf and cdf of q. However, it is often the case that a large number of terms is required to achieve a desirable accuracy, so it is important to have a faster method for computing theh k .…”

Section: Recursions For Theh Kmentioning

confidence: 99%

Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors

Hillier¹,

Kan²,

Wang³

2008

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Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in

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“…It can be considered exact (Solomon and Stephens 1977;Johnson et al 2002) since it provides error bounds and can be used to compute F Q N (x), for some quantile value x, to within a desired precision. Similar numerical methods such as Farebrother's method (Farebrother 1984) could also be used, but some (Sheil and O'Muircheartaigh 1977;Davis 1977;Davies 1980) lack the precision-bounding feature of Imhof's method. However, Imhof's method and Farebrother's method are both iterative, which affects their speed of computation, as shown in Sect.…”

Section: Introductionmentioning

confidence: 99%

A comparison of efficient approximations for a weighted sum of chi-squared random variables

Bodenham

Adams

2015

Stat Comput

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In many applications, the cumulative distribution function (cdf) F Q N of a positively weighted sum of N i.i.d. chi-squared random variables Q N is required. Although there is no known closed-form solution for F Q N , there are many good approximations. When computational efficiency is not an issue, Imhof's method provides a good solution. However, when both the accuracy of the approximation and the speed of its computation are a concern, there is no clear preferred choice. Previous comparisons between approximate methods could be considered insufficient. Furthermore, in streaming data applications where the computation needs to be both sequential and efficient, only a few of the available methods may be suitable. Streaming data problems are becoming ubiquitous and provide the motivation for this paper. We develop a framework to enable a much more extensive comparison between approximate methods for computing the cdf of weighted sums of an arbitrary random variable. Utilising this framework, a new and comprehensive analysis of four efficient approximate methods for computing F Q N is performed. This analysis procedure is much more thorough and statistically valid than previous approaches described in the literature. A surprising result of this analysis is that the accuracy of these approximate methods increases with N . Electronic supplementary materialThe online version of this article (

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Algorithm AS 204: The Distribution of a Positive Linear Combination of χ 2 Random Variables

Cited by 78 publications

References 8 publications

Instrumental Variables Estimation and Weak-Identification-Robust Inference Based on a Conditional Quantile Restriction

Instrumental Variables Estimation and Weak-Identification-Robust Inference Based on a Conditional Quantile Restriction

Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors

A comparison of efficient approximations for a weighted sum of chi-squared random variables

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