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Research Article

Different types of Bernstein operators in inference of Gaussian graphical model

& | (Reviewing Editor)
Article: 1154706 | Received 23 Nov 2015, Accepted 11 Jan 2016, Published online: 25 Mar 2016

References

  • Atay-Kayis, A., & Massam, H. (2005). A Monte Carlo method for computing the marginal likelihood in nondecomposable gaussian graphical models. Biometrika, 92, 317–335.
  • Abel, U. (1995). The moments for the Meyer--Knig and Zeller operators. Journal of Approximation Theory, 82, 352–361.
  • Abel, U. (1996). On the asymptotic approximation with operators of Bleimann. Butzer and Hahn, Indagationes Mathematicae, 7(1), 1–9.
  • Agratini, O. (1996). A class of Bleimann, Butzer and Hahn type operators. Analele Universitătii Din Timişoara, 34, 173–180.
  • Barabasi, A. L., & Otiva, Z. N. (2004). Understanding the cell’s functional organization. Nature Reviews Genetics, 5, 101–113.
  • Bernstein, N. (1912). Démonstration du th\’{e}or\’{e}me de Weierstrass fond\’{e}e sur le calcul de probabilit\’{e}s. Community of Kharkov Mathematical Society, 13, 1–2.
  • Becker, M., Kucharski, D., & Nessel, R. J. (1977). Global approximation theorems for the Szasz--Mirakyanoperators in exponential weight spaces, in linear spaces and approximation. Proceeding of Conference of Oberwolfach, Basel ISNM, 40, 319–333.
  • Bleiman, G., Butzer, P. L., & Hahn, L. (1980). A Bernstein-type operator approximating continuous functions on the semi-axis. Indagationes Mathematicae, 42, 255–262.
  • Cheney, E. W., & Sharma, A. (1996). Bernstein power series. Canadian Journal of Mathematics, 16, 241–253.
  • Dauwels, J., Yu,, H., Xu, S., & Wang, X. (2013). Copula Gaussian graphical model for discrete data. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). Vancouver.
  • Dempster, A. (1972). Covariance selection. Biometrics, 28, 157–175.
  • Dobra, A., Lenkoski, A., & Rodriguez, A. (2011). Bayesian inference for general Gaussian graphical models with application to multivariate lattice data. Journal of the American Statistical Association, 106, 1418–1433.
  • Friedman, J. H., Hastie, T., & Tibshiran, R. (2007). Pathwise coordinate optimization. Annals of Applied Statistics, 2, 302–332.
  • Friedman, J. H., Hastie, T., & Tibshiran, R. (2008). Sparse inverse covariance estimation with the graphical lasso. Biostatistics, 9, 432–441.
  • Han, J.-D., Bertin, N., Hao, T., Goldberg, D. S., Berriz, G. F., Zhang, L. V., ... Vidal, M. (2004). Evidence for dynamically organized modularity in the yeast protein--protein interaction network. Letters to Nature, 430, 88–93.
  • Hermann, T., Szbados, J., & Tandori, K. (Eds.). (1991). Approximation Theory, Proceedings Conference Kecskemet/Hung., 1990, On the operator of Bleimann, Butzer and Hahn(58, pp. 355–360). North-Holland.
  • Jayasri, C., & Sitaraman, Y. (1993). On a Bernstein-type operator of Bleimann--Butzer and Hahn. Journal of Computational and Applied Mathematics, 47, 267–272.
  • Khan, K. A., Nevai, P., & Pinkus, A. (1991). Some properties of a Bernstein type pf operator of Bleimann, Butzer and Hahn. In P. Nevai & A. Pinkus (Eds.), Progress in approximation theory (pp. 497–504). New York, NY: Academic Press.
  • Khan, R. A. (1988). A note on a Bernstein-type operator of Bleimann. Butzer and Hahn, Journal of Approximation Theory, 53, 295–303.
  • Liu, H., Han, F., Yuan, M., Lafferty, J., & Wasserman, L. (2012). High dimensional semiparametric gaussian copula graphical models. The Annals of Statistics, 40, 2293–2326.
  • Lorentz, G. G. (1953). Bernstein polynomials (Mathematical Exposition No. 8). Toronto: University of Toronto.
  • Meinshaussen, N., & Bühlmann, P. (2006). High dimensional graphs and variable selection with the lasso. The Annals of Statistics, 34, 1436–1462.
  • Mercer, A. (1989). A Bernstein-type operator approximating continuous functions on the half-line. Bulletin of Calcutta Mathematical Societiy, 31, 133–137.
  • Muhammadi, A., & Wit, E. C. (2014). Bayesian structure learning in sparse Gaussian graphical models. Bayesian Analysis, 10, 109–138.
  • Purut\c{c}uo\u{g}lu, V., A\u{g}raz, M., & Wit, E. (2015). Bernstein approximations in glasso-based estimation of biological networks.
  • Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, Series B, 58, 267–288.
  • Tibshirani, R., Saunders, M., Rosset, S., Zhu, J., & Knight, K. (2005). Sparsity and smoothness via the fused lasso. Journal of the Royal Statistical Society, Series B, 67, 99–108.
  • Totik, V. (1984). Uniform approximation by Bernstein-type operators. Nederlandse Akademie van Wetenschappen (Indagationes Mathematicae), 50, 87–93.
  • Wang, H., & Li, S. Z. (2012). Efficient Gaussian graphical model determination under G-Wishart prior distributions. Electronic Journal of Statistics, 6, 168–198.
  • Witten, D. M., Frieman, J. H., & Simon, N. (2011). New insights and faster computations for the graphical lasso. Journal of Computational and Graphical Statistics, 20, 892–900.
  • Whittaker, J. (1990). Graphical models in applied multivariate statistics. New York, NY: Wiley.
  • Yuan, M., & Lin, Y. (2007). Model selection and estimation in the Gaussian graphical model. Biometrika, 94, 19–35.
  • Zou, H. (2006). The adaptive lasso and its oracle properties. Journal of the American Statistical Association, 101, 1418–1429.
  • Zou, H., & Hastie, T. (2005). Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society, Series B, 67, 91–108.