Robust Tests for Spherical Symmetry

Kariya, Takeaki; Eaton, Morris L.

doi:10.1214/aos/1176343755

Cited by 82 publications

(27 citation statements)

References 8 publications

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“…Then the uniform distribution is the unique distribution on C n that is invariant under O(n) (e.g., Kariya and Eaton, 1977).…”

Section: Resultsmentioning

confidence: 99%

Distribución de las transformaciones lineales de los residuos mínimos cuadrados studentizados internamente = Distribution of linear transformations of internally studentized least squares residuals

Pynnönem

2012

Pecvnia

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ResumenLos residuos de regresión por mínimos cuadrados ordinarios tienen una distribución que depende de un parámetro escalar. El término "Studentización" se utiliza comúnmente para describir una cantidad U dependiente de un parámetro de escala dividida por una estimación de escala S, de forma que el ratio resultante, U/S, sigue una distribución que no tiene el inconveniente del parámetro de escala desconocido. La Studentización externa hace referencia a un ratio en que el numerador y el denominador son independientes, mientras que la Studentización interna se refiere al ratio en que ambos son dependientes. La ventaja de la Studentización interna es que puede utilizarse cualquier estimador de escala común, mientras que en la Studentización externa, cada residuo es obtenido por un estimador de escala diferente, con el fin de alcanzar la independencia. Con errores de regresión normales, la distribución conjunta de un conjunto arbitrario (linealmente independiente) de residuos Studentizados internamente está bien documentada. Sin embargo, en algunas aplicaciones una combinación lineal de residuos internamente Studentizados puede resultar útil. Sus limitaciones han sido bien documentadas, pero la distribución no parece haberse derivado en la literatura. Este trabajo contribuye a la literatura existente, en el sentido de obtener la distribución conjunta de una transformación arbitraria lineal de residuos de regresión por mínimos cuadrados ordinarios internamente Studentizados con distribución esférica de error. Todas las principales versiones de los residuos de regresión internamente Studentizados que se han utilizado comúnmente en la literatura son casos especiales de la transformación lineal. Pecvnia, Monográfico (2012), 85-110 S. Pynnönen 86 AbstractOrdinary least squares regression residuals have a distribution that is dependent on a scale parameter. The term 'Studentization' is commonly used to describe a scale parameter dependent quantity U divided by a scale estimate S such that the resulting ratio, U/S, has a distribution that is free of from the nuisance unknown scale parameter. External Studentization refers to a ratio in which the nominator and denominator are independent, while internal Studentization refers to a ratio in which these are dependent. The advantage of the internal Studentization is that typically one can use a single common scale estimator, while in the external Studentization every single residual is scaled by different scale estimator to gain the independence. With normal regression errors the joint distribution of an arbitrary (linearly independent) subset of internally Studentized residuals is well documented. However, in some applications a linear combination of internally Studentized residuals may be useful. The boundedness of them is well documented, but the distribution seems not be derived in the literature. This paper contributes to the existing literature by deriving the joint distribution of an arbitrary linear transformation of internally Studentized residuals from ordinary lea...

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“…Then the uniform distribution is the unique distribution on C n that is invariant under O(n) (e.g., Kariya and Eaton, 1977).…”

Section: Resultsmentioning

confidence: 99%

Distribución de las transformaciones lineales de los residuos mínimos cuadrados studentizados internamente = Distribution of linear transformations of internally studentized least squares residuals

Pynnönem

2012

Pecvnia

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“…Since h(gr)/k is a spherical density of go's i<j, L and e=(el,..., era(re+l)~2) are independent and e obeys Dirichlet distribution D(1/2, .... 1/2) (Kariya and Eaton (1977)). Hence, with N1=nl+n3-p3, N2=n+p3, Now consider the integral of the fourth term in (3.15).…”

Section: R~ = T X(12)(x(12)x(12) Q-(22)) A(12) ~ ¢ •mentioning

confidence: 99%

“…381 (1969), Kariya and Eaton (1977), Dawid (1977), Chielewski (1980), Jensen and Good (1981) and Eaton and Kariya (1981), among others. The optimality robustness for multivariate distribution concerning UMPI and LBI properties has been treated by Kariya and Eaton (1977), Kariya (1981) and Kariya and Sinha (1984). In this paper we develop simple techniques for proving the robustness of the local minimax property of some tests in some complex problems concerning the family of elliptically symmetric distributions.…”

Section: Introductionmentioning

confidence: 99%

Locally minimax tests in symmetrical distributions

Giri

1988

Ann Inst Stat Math

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“…The problem when the SSD can be considered as the underlying distribution of the sampled data has been the long lasting interest to statisticians in the study of goodness-of-fit techniques. For example, Kariya and Eaton (1977) and Gupta and Kabe (1993) proposed some robust tests for spherical symmetry based on non-independent samples.…”

Section: Introductionmentioning

confidence: 99%