The Bivariate Normal Copula

Meyer, Christian

doi:10.1080/03610926.2011.611316

Cited by 65 publications

(38 citation statements)

References 48 publications

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“…In this section we are going to develop the algorithm. In order to keep the presentation lean we will often refer to the author's recent survey (Meyer 2009). For further background on normal distributions the reader is also referred to text books such as (Balakrishnan & Lai 2009), (Kotz, Balakrishnan & Johnson 2000) and (Patel & Read 1996).…”

Section: Theorymentioning

confidence: 99%

Recursive Numerical Evaluation of the Cumulative Bivariate Normal Distribution

Meyer¹

2013

J. Stat. Soft.

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Section: Theorymentioning

confidence: 99%

Recursive Numerical Evaluation of the Cumulative Bivariate Normal Distribution

Meyer¹

2013

J. Stat. Soft.

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“…For the particular case of a Gaussian copula Meyer [41] provides explicit analytical closed form expressions such that for the univariate case the cumulative distribution function F À1 (•) is defined as Fig. 5.…”

Section: Copula Namementioning

confidence: 99%

Numerical analysis of the accuracy of bivariate quantile distributions utilizing copulas compared to the GUM supplement 2 for oil pressure balance uncertainties

Ramnath

2017

Int. J. Metrol. Qual. Eng.

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Abstract. In the field of pressure metrology the effective area is A e = A 0 (1 + lP) where A 0 is the zero-pressure area and l is the distortion coefficient and the conventional practise is to construct univariate probability density functions (PDFs) for A 0 and l. As a result analytical generalized non-Gaussian bivariate joint PDFs has not featured prominently in pressure metrology. Recently extended lambda distribution based quantile functions have been successfully utilized for summarizing univariate arbitrary PDF distributions of gas pressure balances. Motivated by this development we investigate the feasibility and utility of extending and applying quantile functions to systems which naturally exhibit bivariate PDFs. Our approach is to utilize the GUM Supplement 1 methodology to solve and generate Monte Carlo based multivariate uncertainty data for an oil based pressure balance laboratory standard that is used to generate known high pressures, and which are in turn cross-floated against another pressure balance transfer standard in order to deduce the transfer standard's respective area. We then numerically analyse the uncertainty data by formulating and constructing an approximate bivariate quantile distribution that directly couples A 0 and l in order to compare and contrast its accuracy to an exact GUM Supplement 2 based uncertainty quantification analysis.

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“…Given the density and the distribution function of the univariate and bivariate standard normal distribution with correlation parameter ρ ∈ (−1, 1), the bivariate Gaussian copula function and density are expressed (Meyer 2013) as…”

Section: Bivariate Gaussian Copulamentioning

confidence: 99%

CopulaDTA: An R Package for Copula-Based Bivariate Beta-Binomial Models for Diagnostic Test Accuracy Studies in a Bayesian Framework

Nyaga¹,

Arbyn²,

Aerts³

2017

J. Stat. Soft.

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The current statistical procedures implemented in statistical software packages for pooling of diagnostic test accuracy data include HSROC regression (Rutter and Gatsonis 2001) and the bivariate random-effects meta-analysis model (BRMA; Reitsma et al. 2005;Arends et al. 2008;Chu and Cole 2006;Riley et al. 2007b). However, these models do not report the overall mean but rather the mean for a central study with random-effect equal to zero and have difficulties estimating the correlation between sensitivity and specificity when the number of studies in the meta-analysis is small and/or when the between-study variance is relatively large (Riley et al. 2007a).This tutorial on advanced statistical methods for meta-analysis of diagnostic accuracy studies discusses and demonstrates Bayesian modeling using the R package CopulaDTA (Nyaga 2017) to fit different models to obtain the meta-analytic parameter estimates. The focus is on the joint modeling of sensitivity and specificity using a copula based bivariate beta distribution. Essentially, we extend the work of Nikoloulopoulos (2015) by: (i) presenting the Bayesian approach which offers the flexibility and ability to perform complex statistical modeling even with small data sets and (ii) including covariate information, and (iii) providing an easy to use code. The statistical methods are illustrated by re-analyzing data of two published meta-analyses.Modeling sensitivity and specificity using the bivariate beta distribution provides marginal as well as study-specific parameter estimates as opposed to using the bivariate normal distribution (e.g., in BRMA) which only yields study-specific parameter estimates. Moreover, copula based models offer greater flexibility in modeling different correlation structures in contrast to the normal distribution which allows for only one correlation structure.

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The Bivariate Normal Copula

Cited by 65 publications

References 48 publications

Recursive Numerical Evaluation of the Cumulative Bivariate Normal Distribution

Recursive Numerical Evaluation of the Cumulative Bivariate Normal Distribution

Numerical analysis of the accuracy of bivariate quantile distributions utilizing copulas compared to the GUM supplement 2 for oil pressure balance uncertainties

CopulaDTA: An R Package for Copula-Based Bivariate Beta-Binomial Models for Diagnostic Test Accuracy Studies in a Bayesian Framework

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