{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T07:23:12Z","timestamp":1740122592505,"version":"3.37.3"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2020,8,27]],"date-time":"2020-08-27T00:00:00Z","timestamp":1598486400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,8,27]],"date-time":"2020-08-27T00:00:00Z","timestamp":1598486400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Wireless Pers Commun"],"published-print":{"date-parts":[[2021,2]]},"DOI":"10.1007\/s11277-020-07750-z","type":"journal-article","created":{"date-parts":[[2020,8,27]],"date-time":"2020-08-27T18:04:42Z","timestamp":1598551482000},"page":"1593-1612","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Exact Distribution of the Max\/Min of Two Correlated Random Variables"],"prefix":"10.1007","volume":"116","author":[{"given":"Y.","family":"Zhang","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0481-0372","authenticated-orcid":false,"given":"S.","family":"Nadarajah","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,8,27]]},"reference":[{"key":"7750_CR1","doi-asserted-by":"crossref","unstructured":"Abbaspour, S., Fatemi, H., & Pedram, M. (2006). Parameterized block-based non-Gaussian statistical gate timing analysis. In Proceedings of the Asia and South Pacific Conference on Design Automation (p. 6).","DOI":"10.1109\/DATE.2006.243891"},{"key":"7750_CR2","doi-asserted-by":"publisher","first-page":"1205","DOI":"10.1109\/TVLSI.2005.859469","volume":"13","author":"Y Abulafia","year":"2005","unstructured":"Abulafia, Y., & Kornfeld, A. (2005). Estimation of FMAX and ISB in microprocessors. IEEE Transactions on Very Large Scale Integration Systems, 13, 1205\u20131209.","journal-title":"IEEE Transactions on Very Large Scale Integration Systems"},{"key":"7750_CR3","volume-title":"Continuous bivariate distributions","author":"N Balakrishnan","year":"2009","unstructured":"Balakrishnan, N., & Lai, C. D. (2009). Continuous bivariate distributions. New York: Springer."},{"key":"7750_CR4","unstructured":"Baker, A.M. (2013). Max operation in statistical static timing analysis on the non-Gaussian variation sources for VLSI circuits. PhD Thesis, Department of Electrical and Computer Engineering, University of Nevada, Las Vegas, USA."},{"key":"7750_CR5","first-page":"1","volume":"3","author":"AM Baker","year":"2013","unstructured":"Baker, A. M., & Jiang, Y. (2013). Modeling and architectural simulations of the statistical static timing analysis of the non-Gaussian variation. International Journal of Scientific and Research Publications, 3, 1\u20137.","journal-title":"International Journal of Scientific and Research Publications"},{"key":"7750_CR6","doi-asserted-by":"publisher","first-page":"799","DOI":"10.1109\/TVLSI.2002.808426","volume":"10","author":"Y Cao","year":"2002","unstructured":"Cao, Y., Huang, X. J., Chang, N. H., et al. (2002). Effective on-chip inductance modeling for multiple signal lines and application to repeater insertion. IEEE Transactions on Very Large Scale Integration Systems, 10, 799\u2013805.","journal-title":"IEEE Transactions on Very Large Scale Integration Systems"},{"key":"7750_CR7","doi-asserted-by":"publisher","first-page":"1072","DOI":"10.1109\/TVLSI.2005.857175","volume":"13","author":"Y Cao","year":"2005","unstructured":"Cao, Y., Yang, X. D., Huang, X. J., et al. (2005). Switch-factor based loop RLC modeling for efficient timing analysis. IEEE Transactions on Very Large Scale Integration Systems, 13, 1072\u20131078.","journal-title":"IEEE Transactions on Very Large Scale Integration Systems"},{"key":"7750_CR8","doi-asserted-by":"crossref","unstructured":"Chang, H., Zolotov, V., Narayan, S., & Visweswariah, C. (2005). Parameterized block-based statistical timing analysis with non-Gaussian parameters, nonlinear delay functions. In Proceedings of the 42nd Annual Design Automation Conference (pp. 71\u201376).","DOI":"10.1145\/1065579.1065604"},{"key":"7750_CR9","doi-asserted-by":"publisher","first-page":"1383","DOI":"10.1109\/TVLSI.2011.2157843","volume":"20","author":"L Cheng","year":"2012","unstructured":"Cheng, L., Gong, F., Xu, W., Xiong, J., He, L., & Sarrafzadeh, M. (2012). Fourier series approximation for max operation in non-Gaussian and quadratic statistical static timing analysis. IEEE Transactions on Very Large Scale Integration Systems, 20, 1383\u20131391.","journal-title":"IEEE Transactions on Very Large Scale Integration Systems"},{"key":"7750_CR10","doi-asserted-by":"crossref","unstructured":"Cheng, L., Xiong, J., & He, L. (2007). Non-linear statistical static timing analysis for non-Gaussian variation sources. In Proceedings of the 44th Annual Design Automation Conference (pp. 250\u2013255).","DOI":"10.1109\/DAC.2007.375162"},{"key":"7750_CR11","doi-asserted-by":"crossref","unstructured":"Chuang, C.Y., & Mak, W.K. (2009). Accurate closed-form parameterized block-based statistical timing analysis applying skew-normal distribution. In Proceedings of the 10th International Symposium on Quality Electronic Design (pp. 68\u201373).","DOI":"10.1109\/ISQED.2009.4810271"},{"key":"7750_CR12","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1080\/00031305.2016.1164756","volume":"70","author":"P Ding","year":"2016","unstructured":"Ding, P. (2016). On the conditional distribution of the multivariate $$t$$ distribution. The American Statistician, 70, 293\u2013295.","journal-title":"The American Statistician"},{"key":"7750_CR13","doi-asserted-by":"publisher","first-page":"370","DOI":"10.1109\/TVLSI.2006.874297","volume":"14","author":"H Eriksson","year":"2006","unstructured":"Eriksson, H., Larsson-Edefors, P., & Eckerbert, D. (2006). Toward architecture-based test-vector generation for timing verification of fast parallel multipliers. IEEE Transactions on Very Large Scale Integration Systems, 14, 370\u2013379.","journal-title":"IEEE Transactions on Very Large Scale Integration Systems"},{"key":"7750_CR14","volume-title":"Table of integrals, series, and products","author":"IS Gradshteyn","year":"2000","unstructured":"Gradshteyn, I. S., & Ryzhik, I. M. (2000). Table of integrals, series, and products (6th ed.). San Diego: Academic Press.","edition":"6"},{"key":"7750_CR15","doi-asserted-by":"crossref","unstructured":"Homma, K., Nitta, I., & Shibuya, T. (2008). Non-Gaussian statistical timing models of die-to-die and within-die parameter variations for full chip analysis. In Proceedings of the 2008 Asia and South Pacific Design Automation Conference (pp. 292\u2013297).","DOI":"10.1109\/ASPDAC.2008.4483961"},{"key":"7750_CR16","doi-asserted-by":"crossref","unstructured":"Imai, M., Sato, T., Nakayama, N., & Masu, K. (2008). Non-parametric statistical static timing analysis: An SSTA framework for arbitrary distribution. In Proceedings of the 45th Annual Design Automation Conference (pp. 698\u2013701).","DOI":"10.1145\/1391469.1391649"},{"key":"7750_CR17","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511550683","volume-title":"Multivariate t distributions and their applications","author":"S Kotz","year":"2004","unstructured":"Kotz, S., & Nadarajah, S. (2004). Multivariate t distributions and their applications. Cambridge: Cambridge University Press."},{"key":"7750_CR18","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1016\/j.spl.2015.02.005","volume":"100","author":"S Nadarajah","year":"2015","unstructured":"Nadarajah, S. (2015). Expansions for bivariate copulas. Statistics and Probability Letters, 100, 77\u201384.","journal-title":"Statistics and Probability Letters"},{"key":"7750_CR19","doi-asserted-by":"publisher","first-page":"210","DOI":"10.1109\/TVLSI.2007.912191","volume":"16","author":"S Nadarajah","year":"2008","unstructured":"Nadarajah, S., & Kotz, S. (2008). Exact distribution of the max\/min of two Gaussian random variables. IEEE Transactions on Very Large Scale Integration Systems, 16, 210\u2013212.","journal-title":"IEEE Transactions on Very Large Scale Integration Systems"},{"key":"7750_CR20","volume-title":"An introduction to copulas","author":"RB Nelsen","year":"2006","unstructured":"Nelsen, R. B. (2006). An introduction to copulas (2nd ed.). New York: Springer.","edition":"2"},{"key":"7750_CR21","doi-asserted-by":"publisher","first-page":"346","DOI":"10.1109\/92.532035","volume":"4","author":"CH Oh","year":"1996","unstructured":"Oh, C. H., & Mercer, M. R. (1996). Efficient logic-level timing analysis using constraint-guided critical path search. IEEE Transactions on Very Large Scale Integration Systems, 4, 346\u2013355.","journal-title":"IEEE Transactions on Very Large Scale Integration Systems"},{"key":"7750_CR22","doi-asserted-by":"crossref","unstructured":"Orshansky, M., & Bandyopadhyay, A. (2004). Fast statistical timing analysis handling arbitrary delay correlations. In Proceedings of the 41st Annual Design Automation Conference (pp. 337\u2013342).","DOI":"10.1145\/996566.996664"},{"key":"7750_CR23","volume-title":"Integrals and series, volumes 1, 2 and 3","author":"AP Prudnikov","year":"1986","unstructured":"Prudnikov, A. P., Brychkov, Y. A., & Marichev, O. I. (1986). Integrals and series, volumes 1, 2 and 3. Amsterdam: Gordon and Breach Science Publishers."},{"key":"7750_CR24","doi-asserted-by":"publisher","first-page":"2359","DOI":"10.1109\/TVLSI.2015.2501370","volume":"24","author":"S Ramprasath","year":"2016","unstructured":"Ramprasath, S., Vijaykumar, M., & Vasudevan, V. (2016). A skew-normal canonical model for statistical static timing analysis. IEEE Transactions on Very Large Scale Integration Systems, 24, 2359\u20132368.","journal-title":"IEEE Transactions on Very Large Scale Integration Systems"},{"key":"7750_CR25","unstructured":"Shaw, W. T., & Buckley I. R. C. (2009). The alchemy of probability distributions: Beyond Gram-Charlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map. arXiv:0901.0434v1"},{"key":"7750_CR26","doi-asserted-by":"crossref","unstructured":"Singh, J., & Sapatnekar, S. (2006). Statistical timing analysis with correlated non-Gaussian parameters using independent component analysis. In Proceedings of the 43rd ACM\/IEEE Design Automation Conference (pp. 155\u2013160).","DOI":"10.1109\/DAC.2006.229199"},{"key":"7750_CR27","doi-asserted-by":"publisher","first-page":"12","DOI":"10.1109\/TVLSI.2003.820525","volume":"12","author":"B Taskin","year":"2004","unstructured":"Taskin, B., & Kourtev, I. S. (2004). Linearization of the timing analysis and optimization of level-sensitive digital synchronous circuits. IEEE Transactions on Very Large Scale Integration Systems, 12, 12\u201327.","journal-title":"IEEE Transactions on Very Large Scale Integration Systems"},{"key":"7750_CR28","doi-asserted-by":"publisher","first-page":"2542","DOI":"10.1587\/transfun.E96.A.2542","volume":"E96\u2013A","author":"S Tsukiyama","year":"2013","unstructured":"Tsukiyama, S., & Fukui, M. (2013). A new delay distribution model with a half triangular distribution for statistical static timing analysis. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, E96\u2013A, 2542\u20132552.","journal-title":"IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences"},{"key":"7750_CR29","doi-asserted-by":"publisher","first-page":"662","DOI":"10.1109\/TVLSI.2004.827602","volume":"12","author":"A Valentian","year":"2004","unstructured":"Valentian, A., Thomas, O., Vladimirescu, A., et al. (2004). Modeling subthreshold SOI logic for static timing analysis. IEEE Transactions on Very Large Scale Integration Systems, 12, 662\u2013668.","journal-title":"IEEE Transactions on Very Large Scale Integration Systems"},{"key":"7750_CR30","first-page":"53","volume":"2","author":"V Vijayabhasker","year":"2014","unstructured":"Vijayabhasker, V., & Purushotham Naik, R. (2014). Static timing analysis of the non-Gaussian variation sources for VLSI circuits. International Journal of Computer Science and Mobile Applications, 2, 53\u201357.","journal-title":"International Journal of Computer Science and Mobile Applications"},{"key":"7750_CR31","doi-asserted-by":"crossref","unstructured":"Vijaykumar, M., & Vasudevan, V. (2014). Statistical static timing analysis using a skew-normal canonical delay model. In Proceedings of the Conference on Design, Automation and Test in Europe, Article No. 258.","DOI":"10.7873\/DATE.2014.271"},{"key":"7750_CR32","doi-asserted-by":"crossref","unstructured":"Zhan, Y., Strojwas, A.J., Li, X., Pileggi, L.T., Newmark, D., & Sharma, M. (2005). Correlation-aware statistical timing analysis with non-Gaussian delay distributions. In Proceedings of the 42nd Design Automation Conference (pp. 77\u201382).","DOI":"10.1109\/DAC.2005.193777"},{"key":"7750_CR33","doi-asserted-by":"crossref","unstructured":"Zhang, L., Chen, W., Hu, Y., Gubner, J.A., & Chen, C.C.P. (2005). Correlation-preserved non-Gaussian statistical timing analysis with quadratic timing model. In Proceedings of the 42nd Annual Design Automation Conference (pp. 83\u201388).","DOI":"10.1145\/1065579.1065606"},{"key":"7750_CR34","doi-asserted-by":"crossref","unstructured":"Zjajo, A., Tang, Q., Berkelaar, M., & van der Meijs, N. (2011). Accuracy consideration of a non-Gaussian interconnect delay model for submicron CMOS statistical static timing analysis. In Proceedings of the IEEE 4th International Nanoelectronics Conference (pp. 1\u20132).","DOI":"10.1109\/INEC.2011.5991680"}],"container-title":["Wireless Personal Communications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11277-020-07750-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11277-020-07750-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11277-020-07750-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,8,26]],"date-time":"2021-08-26T23:25:41Z","timestamp":1630020341000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11277-020-07750-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,27]]},"references-count":34,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,2]]}},"alternative-id":["7750"],"URL":"https:\/\/doi.org\/10.1007\/s11277-020-07750-z","relation":{},"ISSN":["0929-6212","1572-834X"],"issn-type":[{"type":"print","value":"0929-6212"},{"type":"electronic","value":"1572-834X"}],"subject":[],"published":{"date-parts":[[2020,8,27]]},"assertion":[{"value":"27 August 2020","order":1,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}