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Research Article

Universal Inference Meets Random Projections: A Scalable Test for Log-Concavity

ORCID Icon, , ORCID Icon & ORCID Icon
Received 13 May 2023, Accepted 16 Apr 2024, Published online: 31 May 2024

References

  • An, M. Y. (1997), “Log-Concave Probability Distributions: Theory and Statistical Testing,” Duke University Dept of Economics Working Paper.
  • Bagnoli, M., and Bergstrom, T. (2005), “Log-Concave Probability and its Applications,” Economic Theory, 26, 445–469. DOI: 10.1007/s00199-004-0514-4.
  • Barber, R. F., and Samworth, R. J. (2021), “Local Continuity of Log-Concave Projection, with Applications to Estimation under Model Misspecification,” Bernoulli, 27, 2437–2472. DOI: 10.3150/20-BEJ1316.
  • Bowman, A. W. (1984), “An Alternative Method of Cross-Validation for the Smoothing of Density Estimates,” Biometrika, 71, 353–360. DOI: 10.1093/biomet/71.2.353.
  • Carroll, R. J., Delaigle, A., and Hall, P. (2011), “Testing and Estimating Shape-Constrained Nonparametric Density and Regression in the Presence of Measurement Error,” Journal of the American Statistical Association, 106, 191–202. DOI: 10.1198/jasa.2011.tm10355.
  • Chacón, J. E., and Duong, T. (2010), “Multivariate Plug-In Bandwidth Selection with Unconstrained Pilot Bandwidth Matrices,” Test, 19, 375–398. DOI: 10.1007/s11749-009-0168-4.
  • Chen, W., Mazumder, R., and Samworth, R. (2021), “A New Computational Framework for Log-Concave Density Estimation,” arXiv preprint arXiv:2105.11387.
  • Chen, Y., and Samworth, R. J. (2013), “Smoothed Log-Concave Maximum Likelihood Estimation with Applications,” Statistica Sinica, 23, 1373–1398. DOI: 10.5705/ss.2011.224.
  • Cule, M., Gramacy, R., and Samworth, R. (2009), “LogConcDEAD: An R Package for Maximum Likelihood Estimation of a Multivariate Log-Concave Density,” Journal of Statistical Software, 29, 1–20. DOI: 10.18637/jss.v029.i02.
  • Cule, M., and Samworth, R. (2010), “Theoretical Properties of the Log-Concave Maximum Likelihood Estimator of a Multidimensional Density,” Electronic Journal of Statistics, 4, 254–270. DOI: 10.1214/09-EJS505.
  • Cule, M., Samworth, R., and Stewart, M. (2010), “Maximum Likelihood Estimation of a Multi-Dimensional Log-Concave Density,” Journal of the Royal Statistical Society, Series B, 72, 545–607. DOI: 10.1111/j.1467-9868.2010.00753.x.
  • Delignette-Muller, M. L., and Dutang, C. (2015), “fitdistrplus: An R Package for Fitting Distributions,” Journal of Statistical Software, 64, 1–34. DOI: 10.18637/jss.v064.i04.
  • Dowle, M., and Srinivasan, A. (2021), data.table: Extension of ‘data.frame’. R package version 1.14.2.
  • Dümbgen, L., Hüsler, A., and Rufibach, K. (2007), “Active Set and EM Algorithms for Log-Concave Densities Based on Complete and Censored Data,” arXiv preprint arXiv:0707.4643.
  • Dümbgen, L., and Rufibach, K. (2011), “logcondens: Computations Related to Univariate Log-Concave Density Estimation,” Journal of Statistical Software, 39, 1–28. DOI: 10.18637/jss.v039.i06.
  • Duong, T. (2021), ks: Kernel Smoothing. R package version 1.13.2.
  • Duong, T., and Hazelton, M. L. (2003), “Plug-In Bandwidth Matrices for Bivariate Kernel Density Estimation,” Journal of Nonparametric Statistics, 15, 17–30. DOI: 10.1080/10485250306039.
  • ———(2005), “Cross-Validation Bandwidth Matrices for Multivariate Kernel Density Estimation,” Scandinavian Journal of Statistics, 32, 485–506.
  • Genz, A., and Bretz, F. (2009), Computation of Multivariate Normal and t Probabilities, Lecture Notes in Statistics, Heidelberg: Springer-Verlag.
  • Genz, A., Bretz, F., Miwa, T., Mi, X., Leisch, F., Scheipl, F., and Hothorn, T. (2021), mvtnorm: Multivariate Normal and t Distributions. R package version 1.1-3.
  • Hazelton, M. L. (2011), “Assessing Log-Concavity of Multivariate Densities,” Statistics & Probability Letters, 81, 121–125. DOI: 10.1016/j.spl.2010.10.001.
  • Jones, M., Marron, J. S., and Park, B. U. (1991), “A Simple Root n Bandwidth Selector,” The Annals of Statistics, 19, 1919–1932. DOI: 10.1214/aos/1176348378.
  • Kappel, F., and Kuntsevich, A. V. (2000), “An Implementation of Shor’s r-Algorithm,” Computational Optimization and Applications, 15, 193–205. DOI: 10.1023/A:1008739111712.
  • Kim, A. K., and Samworth, R. J. (2016), “Global Rates of Convergence in Log-Concave Density Estimation,” Annals of Statistics, 44, 2756–2779.
  • Koenker, R., and Mizera, I. (2018), “Shape Constrained Density Estimation via Penalized Rényi Divergence,” Statistical Science, 33, 510–526. DOI: 10.1214/18-STS658.
  • Kur, G., Dagan, Y., and Rakhlin, A. (2019), “Optimality of Maximum Likelihood for Log-Concave Density Estimation and Bounded Convex Regression,” arXiv preprint arXiv:1903.05315.
  • Nagler, T., and Vatter, T. (2020), kde1d: Univariate Kernel Density Estimation. R package version 1.13.2.
  • Nystrom, N. A., Levine, M. J., Roskies, R. Z., and Scott, J. R. (2015), “Bridges: A Uniquely Flexible HPC Resource for New Communities and Data Analytics,” in Proceedings of the 2015 XSEDE Conference: Scientific Advancements Enabled by Enhanced Cyberinfrastructure, XSEDE ’15, New York, NY, USA: Association for Computing Machinery. DOI: 10.1145/2792745.2792775.
  • Prékopa, A. (1973), “On Logarithmic Concave Measures and Functions,” Acta Scientiarum Mathematicarum, 34, 335–343.
  • R Core Team. (2021), R: A Language and Environment for Statistical Computing, Vienna, Austria: R Foundation for Statistical Computing.
  • Rudemo, M. (1982), “Empirical Choice of Histograms and Kernel Density Estimators,” Scandinavian Journal of Statistics, 9, 65–78.
  • Samworth, R. J. (2018), “Recent Progress in Log-Concave Density Estimation,” Statistical Science, 33, 493–509. DOI: 10.1214/18-STS666.
  • Schubert, M. (2019), “clustermq Enables Efficient Parallelisation of Genomic Analyses,” Bioinformatics, 35, 4493–4495. DOI: 10.1093/bioinformatics/btz284.
  • Scrucca, L., Fop, M., Murphy, T. B., and Raftery, A. E. (2016), “mclust 5: Clustering, Classification and Density Estimation Using Gaussian Finite Mixture Models,” The R Journal, 8, 289–317.
  • Shor, N. Z. (2012), Minimization Methods for Non-Differentiable Functions (Vol. 3), Berlin: Springer.
  • Towns, J., Cockerill, T., Dahan, M., Foster, I., Gaither, K., Grimshaw, A., Hazlewood, V., Lathrop, S., Lifka, D., Peterson, G. D., Roskies, R., Scott, J. R., and Wilkins-Diehr, N. (2014), “XSEDE: Accelerating Scientific Discovery,” Computing in Science & Engineering, 16, 62–74. DOI: 10.1109/MCSE.2014.80.
  • Venables, W. N., and Ripley, B. D. (2002), Modern Applied Statistics with S (4th ed.), New York: Springer.
  • Wand, M. P., and Jones, M. C. (1994), “Multivariate Plug-In Bandwidth Selection,” Computational Statistics, 9, 97–116.
  • Wasserman, L., Ramdas, A., and Balakrishnan, S. (2020), “Universal Inference,” Proceedings of the National Academy of Sciences, 117, 16880–16890. DOI: 10.1073/pnas.1922664117.
  • Wickham, H., Averick, M., Bryan, J., Chang, W., McGowan, L. D., François, R., Grolemund, G., Hayes, A., Henry, L., Hester, J., Kuhn, M., Pedersen, T. L., Miller, E., Bache, S. M., Müller, K., Ooms, J., Robinson, D., Seidel, D. P., Spinu, V., Takahashi, K., Vaughan, D., Wilke, C., Woo, K., and Yutani, H. (2019), “Welcome to the tidyverse,” Journal of Open Source Software, 4, 1686. DOI: 10.21105/joss.01686.
  • Wong, W. H., and Shen, X. (1995), “Probability Inequalities for Likelihood Ratios and Convergence Rates of Sieve MLEs,” The Annals of Statistics, 23, 339–362. DOI: 10.1214/aos/1176324524.

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