scholarly journals Distribution of an arbitrary linear transformation of internally Studentized residuals of multivariate regression with elliptical errors

2012 ◽  
Vol 107 ◽  
pp. 40-52 ◽  
Author(s):  
Seppo Pynnönen
Author(s):  
Seppo Pynnönem

<p>Los residuos de regresión por mínimos cuadrados ordinarios tienen una distribución que depende de un parámetro escalar. El término “<em>Studentización</em>” se utiliza comúnmente para describir una cantidad <em>U</em> dependiente de un parámetro de escala dividida por una estimación de escala <em>S</em>, de forma que el ratio resultante,<em> </em><em>U</em>/<em>S</em>, sigue una distribución que no tiene el inconveniente del parámetro de escala desconocido. La <em>Studentización</em> externa hace referencia a un ratio en que el numerador y el denominador son independientes, mientras que la <em>Studentización</em> interna se refiere al ratio en que ambos son dependientes. La ventaja de la <em>Studentización</em> interna es que puede utilizarse cualquier estimador de escala común, mientras que en la <em>Studentización</em> 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 <em>Studentizados</em> internamente está bien documentada. Sin embargo, en algunas aplicaciones una combinación lineal de residuos internamente <em>Studentizados</em> 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 <em>Studentizados</em> con distribución esférica de error. Todas las principales versiones de los residuos de regresión internamente <em>Studentizados</em> que se han utilizado comúnmente en la literatura son casos especiales de la transformación lineal.</p><p>Ordinary 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 <em>U</em> divided by a scale estimate <em>S</em> such that the resulting ratio, <em>U</em>/<em>S</em>, has a distribution that is free of from the nuisance unknown scale parameter. <em>External</em> Studentization refers to a ratio in which the nominator and denominator are independent, while <em>internal</em> 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 least squares regression with spherical error distribution. All major versions of commonly utilized internally Studentized regression residuals in literature are obtained as special cases of the linear transformation</p>


Author(s):  
Seppo Pynnönem

<p>Los residuos de regresión por mínimos cuadrados ordinarios tienen una distribución que depende de un parámetro escalar. El término “<em>Studentización</em>” se utiliza comúnmente para describir una cantidad <em>U</em> dependiente de un parámetro de escala dividida por una estimación de escala <em>S</em>, de forma que el ratio resultante,<em> </em><em>U</em>/<em>S</em>, sigue una distribución que no tiene el inconveniente del parámetro de escala desconocido. La <em>Studentización</em> externa hace referencia a un ratio en que el numerador y el denominador son independientes, mientras que la <em>Studentización</em> interna se refiere al ratio en que ambos son dependientes. La ventaja de la <em>Studentización</em> interna es que puede utilizarse cualquier estimador de escala común, mientras que en la <em>Studentización</em> 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 <em>Studentizados</em> internamente está bien documentada. Sin embargo, en algunas aplicaciones una combinación lineal de residuos internamente <em>Studentizados</em> 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 <em>Studentizados</em> con distribución esférica de error. Todas las principales versiones de los residuos de regresión internamente <em>Studentizados</em> que se han utilizado comúnmente en la literatura son casos especiales de la transformación lineal.</p><p>Ordinary 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 <em>U</em> divided by a scale estimate <em>S</em> such that the resulting ratio, <em>U</em>/<em>S</em>, has a distribution that is free of from the nuisance unknown scale parameter. <em>External</em> Studentization refers to a ratio in which the nominator and denominator are independent, while <em>internal</em> 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 least squares regression with spherical error distribution. All major versions of commonly utilized internally Studentized regression residuals in literature are obtained as special cases of the linear transformation</p>


2019 ◽  
Vol 168 ◽  
pp. 108944 ◽  
Author(s):  
Gianluca Pastorelli ◽  
Shuo Cao ◽  
Irena Kralj Cigić ◽  
Costanza Cucci ◽  
Abdelrazek Elnaggar ◽  
...  

2013 ◽  
Vol 79 (8) ◽  
pp. 747-753 ◽  
Author(s):  
Benjamin Bograd ◽  
Carlos Rodriguez ◽  
Richard Amdur ◽  
Fred Gage ◽  
Eric Elster ◽  
...  

Despite the well-documented use of damage control laparotomy (DCL) in civilian trauma, its use has not been well described in the combat setting. Therefore, we sought to document the use of DCL and to investigate its effect on patient outcome. Prospective data were collected on 1603 combat casualties injured between April 2003 and January 2009. One hundred seventy patients (11%) underwent an exploratory laparotomy (ex lap) in theater and comprised the study cohort. DCL was defined as an abbreviated ex lap resulting in an open abdomen. Patients were stratified by age, Injury Severity Score (ISS), Glasgow Coma Score (GCS), mechanism of injury, and blood product administration. Multivariate regression analyses were used to determine risks factors for intensive care unit length of stay (ICU LOS), hospital length of stay (HLOS), and the need for DCL. Mean age of the cohort was 24 ± 5 years, ISS was 21 ± 11, and 94 per cent sustained penetrating injury. Patients with DCL comprised 50.6 per cent (n = 86) of the study cohort and had significant increases in ICU admission ( P < 0.001), ICU LOS ( P < 0.001), HLOS ( P < 0.05), ventilator days ( P < 0.001), abdominal complications ( P < 0.05), but not mortality ( P = 0.65) compared with patients without DCL. When compared with the non-DCL group, patients undergoing DCL required significantly more blood products (packed red blood cells, fresh-frozen plasma, platelets, and cryoprecipitate; P < 0.001). Multivariate regression analyses revealed blood transfusion and GCS as significant risk factors for DCL ( P < 0.05). Patients undergoing DCL had increased complications and resource use but not mortality compared with patients not undergoing DCL. The need for combat DCL may be different compared with civilian use. Prospective studies to evaluate outcomes of DCL are warranted.


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