Copula methods have been used for more than a decade in finance, actuarial science and earth sciences. There is a large literature on copulas. A good place to start is the book by Roger Nelsen (“An Introduction to Copulas”) and the review by Trivedi and Zimmer (“Copula Modeling: An Introduction for Practitioners”). Although copula methods are clearly an important generalization of the multivariate normal modeling approach used to model semiconductor yield and quality and reliability, the existing copula literature gives no guidance on how to use copula methods to model semiconductor manufacturing and use. So, the Integrated Circuit Design and Test (ICDT) Laboratory at PSU used DRAM variable retention time data acquired in ICDT as a case study to develop the essential elements of copula-based modeling of a semiconductor product. Generalization to any kind of semiconductor product from the methods described here for the relatively simple DRAM case will be clear to practitioners.
Offered here are documents which describe the method, expand upon various aspects, and document some of the things we learned about copula methods.
Copula Methods slide presentation, including DRAM case study. Many details are only sketched. (Presented at Micron on 3/22/2013.)
Copula Methods slide presentation, including DRAM case study. Has more detail about mathematics of copulas. (Presented at Intel on 11/30/2012.)
Prepublication copy of DRAM paper. Certain aspects such as development of Test/Use model, and of scaling from bit to array including fault tolerance are only sketched. Update: This paper has been accepted for publication in IEEE Transactions on Computers.
Supplemental Material for DRAM paper. Explains in detail how the Test/Use model Eq. (15) in the prepublication DRAM paper was derived, and gives details of statistical models of arrays and fault tolerance that could only be sketched in the prepublication DRAM paper.
RTN Time-In State. Shows how to extend the time-in-state Test/Use model beyond that described in the prepublication DRAM paper.
Clayton Copula. Describes properties of the Clayton copula, including the interesting truncation-invariance pointed out by Oakes, and used in the DRAM model.
Wedge Copula. An interesting geometrical copula which fitted the DRAM data quite well, but was superseded by the Clayton copula.
Stripe Copula. Another interesting geometrical copula which did not pan out for the DRAM study.
Frank Copula Synthesis. A correction of a formula given in M. Armstrong’s catalog of copulas for synthesizing from the Frank copula.
Copula Manipulations using Generalized Functions. Guidance and demonstration of a way to manipulate copulas with singularities using the Dirac delta function and the Heaviside function. Inspired by an article by Rodney E. Kreps (“A Partially Comonotonic Algorithm for Loss Generation”, ASTIN Colloquium Papers 2000, p165).