A New Result on the Distribution of Quadratic Forms

Ruben, Harold

doi:10.1214/aoms/1177703890

Cited by 36 publications

(23 citation statements)

References 1 publication

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“…When re=l, the p.d.f, of the univariate quadratic form are treated by many authors. We derive three representations of it by the use of P,(T, A), and give the relations between our results and the one of Kotz et al [9] and of Ruben [10], [11].…”

Section: The Pdf Of ~Xax' and The Relations With The Univariate Qumentioning

confidence: 70%

“…P, (T, A). We also derive :the p.d.f, of the trace of a multivariate non-central quadratic form by the use of P, (T, A) and compare the results of Ruben [10], [11] and Kotz et al [9] with our results. Finally, we also discuss the p.d.f.…”

Section: Introductionmentioning

confidence: 73%

“…To compare with the results of Kotz et al [9] and of Ruben [10], [11], we assume without loss of generality that A is a diagonal matrix where diagonal elements are a~, a2,'.., a, and a~_a~_ ---~a.>O. The p.d.f, of T=trXAX' is derived by the following multiple integral.…”

Section: ~ P(mentioning

confidence: 99%

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On the distribution of the multivariate quadratic form in multivariate normal samples

Hayakawa¹

1972

Ann Inst Stat Math

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Section: The Pdf Of ~Xax' and The Relations With The Univariate Qumentioning

confidence: 70%

Section: Introductionmentioning

confidence: 73%

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On the distribution of the multivariate quadratic form in multivariate normal samples

Hayakawa¹

1972

Ann Inst Stat Math

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“…We present a case study illustration of a general statistical model previously developed by others (Ruben 1962;Ruben 1963;Kotz et al 1967;Sheil and O'Muircheartaigh 1977). The model is applied here to the linearized solution method of the GPS equations in a new way.…”

Section: A Case Study Illustrationmentioning

confidence: 99%

“…Ruben derived an infinite linear combination representation for the probability density function of a quadratic form for a finite number of correlated normally distributed RVs (Ruben 1962(Ruben , 1963. He also provided estimates of error and a proof of convergence.…”

Section: Use Of Quadratic Form For K-dimensional Radial Protection Lementioning

confidence: 99%

Statistical characterization of composite protection levels for GPS

2010

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A statistical characterization is presented of Global Positioning System (GPS) user range error as a normally distributed random variable with non-zero mean over the length of the aircraft precision approach operation, correlated from one GPS measurement epoch to another and from one satellite to another. This leads directly to modeling GPS error in the position domain as multivariate normal with non-zero mean. Based on this model, a vertical composite protection level VPL c and a horizontal composite protection level HPL c are each calculated as scalar values from a univariate normal distribution displaced from the origin by the worst-case position domain bias combination possible, given the maximum possible individual satellite bias magnitudes and the satellite geometry. A method is then presented by which exact values-that is, values accurate to a user-defined error tolerance and consistent with statistical assumptions-of VPL c and HPL c are obtained, and by which computationally efficient approximations may be evaluated. A statistical quadratic form under the multivariate normal distribution is used to derive a new class of protection levels based on the probability enclosed within a radius defined in two or more dimensions. A central Chi-square representation of this quadratic form is also presented and is incorporated into a six-step computational procedure for the two-dimensional composite radial protection level RPL c . This procedure is extended to the spherical protection level (SPL c ) and the ellipsoidal protection level (EPL c ).

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