Weighted bootstrapped kernel density estimators in two-sample problems

References

• Alvarez-Andrade, S., and Bouzebda, S. (2013), ‘Strong Approximations for Weighted Bootstrap of Empirical and Quantile Processes with Applications’, Statistical Methodology, 11, 36–52. doi: 10.1016/j.stamet.2012.09.001
   Web of Science ®Google Scholar
• Anderson, N.H., Hall, P., and Titterington, D.M. (1994), ‘Two-Sample Test Statistics for Measuring Discrepancies Between two Multivariate Probability Density Functions Using Kernel-Based Density Estimates’, Journal of Multivariate Analysis, 50, 41–54. doi: 10.1006/jmva.1994.1033
   Web of Science ®Google Scholar
• Barbe, P., and Bertail, P. (1995), The Weighted Bootstrap. Lecture Notes in Statistics, Heidelberg: Springer.
   Google Scholar
• Beirlant, J., and Mason, D. (1995), ‘ On the Asymptotic Normality of -norms of Empirical Functionals’, Mathematical Methods of Statistics, 4, 1–19.
   Google Scholar
• Berlinet, A., Devroye, L., and Gyorfi, L. (1995), ‘ Asymptotic Normality of -error in Density Estimation’, Statistics, 26, 329–343. doi: 10.1080/02331889508802500
   Web of Science ®Google Scholar
• Bickel, P., and Rosenblatt, M. (1973), ‘On Some Global Measures of the Deviations of Density Function Estimates’, Annals of Statistics, 1, 1075–1095.
   Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• Burke, M. (1998), ‘A Gaussian Bootstrap Approach to Estimation and Tests’ in Asymptotic Methods in Probability and Statistics, ed. B. Szyszkowicz, Amsterdam: North-Holland, pp. 697–706.
   Google Scholar
• Burke, M.D. (2000), ‘Multivariate Tests-Of-Fit and Uniform Confidence Bands Using a Weighted Bootstrap’, Statistics and Probability Letters, 46, 13–20. doi: 10.1016/S0167-7152(99)00082-6
   Web of Science ®Google Scholar
• Burke, M.D. (2010), ‘Approximations for a Multivariate Hybrid Process with Applications to Change-point Detection’, Mathematical Methods of Statistics, 19, 121–135. doi: 10.3103/S106653071002002X
   Google Scholar
• Cao, R., and Van Keilegom, I. (2006), ‘Empirical Likelihood Tests for Two-Sample Problems via Nonparametric Density Estimation’, The Canadian Journal of Statistics, 34, 61–77. doi: 10.1002/cjs.5550340106
   Web of Science ®Google Scholar
• Cheng, G., and Huang, J.Z. (2010), ‘Bootstrap Consistency for General Semiparametric M-Estimation’, Annals of Statistics, 38, 2884–2915. doi: 10.1214/10-AOS809
   Web of Science ®Google Scholar
• Csörgo, M., and Horváth, L. (1988), ‘ Central Limit Theorems for -Norms of Density Estimators’, Probability Theory and Related Fields, 80, 269–291. doi: 10.1007/BF00356106
   Web of Science ®Google Scholar
• Csörgő, M., and Horváth, L. (1993), Weighted Approximations in Probability and Statistics, Chichester: Wiley.
   Google Scholar
• Efron, B. (1979), ‘Bootstrap Methods: Another Look at the Jackknife’, Annals of Statistics, 7, 1–26. doi: 10.1214/aos/1176344552
   Web of Science ®Google Scholar
• Eggermont, P., and LaRiccia, V.N. (2001), Maximum Penalized Likelihood Estimation Vol. I. Density Estimation, New York: Springer-Verlag.
   Google Scholar
• Fryer, M.J. (1977), ‘A Review of Some Non-parametric Methods of Density Estimation’, Journal of the Institute of Mathematics and Its Applications, 20, 335–354. doi: 10.1093/imamat/20.3.335
   Google Scholar
• Hall, P. (1982), ‘Limit Theorems for Stochastic Measures of the Accuracy of Density Estimators’, Stochastic Processes and their Applications, 13, 11–25. doi: 10.1016/0304-4149(82)90003-5
   Google Scholar
• Hall, P. (1984), ‘Central Limit Theorem for Integrated Square Error of Multivariate Nonparametric Density Estimators’, Journal of Multivariate Analysis, 14, 1–16. doi: 10.1016/0047-259X(84)90044-7
   Web of Science ®Google Scholar
• Hall, P., and Mammen, E. (1994), ‘On General Resampling Algorithms and their Performance in Distribution Estimation’, Annals of Statistics, 22, 2011–2030. doi: 10.1214/aos/1176325769
   Web of Science ®Google Scholar
• Henze, N., and Nikitin, Y.Y. (2003), ‘Two-Sample Tests Based on the Integrated Empirical Process’, Communications in Statistics: Theory Methods, 32, 1767–1788. doi: 10.1081/STA-120022708
   Web of Science ®Google Scholar
• Henze, N., Nikitin, Y., and Ebner, B. (2009), ‘ Integral Distribution-Free Statistics of -type and Their Asymptotic Comparison’, Computational Statistics and Data Analysis, 53, 3426–3438. doi: 10.1016/j.csda.2009.02.018
   Web of Science ®Google Scholar
• Horváth, L. (1991), ‘ On -Norms of Multivariate Density Estimators’, Annals of Statistics, 19, 1933–1949. doi: 10.1214/aos/1176348379
   Web of Science ®Google Scholar
• Horváth, L. (2000), ‘Approximations for Hybrids of Empirical and Partial Sums Processes’, Journal of Statistical Planning and Inference, 88, 1–18. doi: 10.1016/S0378-3758(99)00207-4
   Web of Science ®Google Scholar
• Horváth, L., Kokoszka, P., and Steinebach, J. (2000), ‘Approximations for Weighted Bootstrap Processes with an Application’, Statistics and Probability Letters, 48, 59–70. doi: 10.1016/S0167-7152(99)00190-X
   Web of Science ®Google Scholar
• Kojadinovic, I., and Yan, J. (2012), ‘Goodness-of-Fit Testing Based on a Weighted Bootstrap: A Fast Large-Sample Alternative to the Parametric Bootstrap’, The Canadian Journal of Statistics, 40, 480–500. doi: 10.1002/cjs.11135
   Web of Science ®Google Scholar
• Kojadinovic, I., Yan, J., and Holmes, M. (2011), ‘Fast Large-Sample Goodness-of-Fit for Copulas’, Statistica Sinica, 21, 841–871. doi: 10.5705/ss.2011.037a
   Web of Science ®Google Scholar
• Kosorok, M. (2008), Introduction to Empirical Processes and Semiparametric Inference. Springer Series in Statistics, New York: Springer-Verlag.
   Google Scholar
• Liu, B., and Mojirsheibani, M. (2015), ‘ On a Weighted Bootstrap Approximation of the Norms of Kernel Density Estimators’, Statistics and Probability Letters, 105, 65–73. doi: 10.1016/j.spl.2015.06.005
   Web of Science ®Google Scholar
• Mason, D.M., and Newton, M.A. (1992), ‘A Rank Statistics Approach to the Consistency of a General Bootstrap’, Annals of Statistics, 20, 1611–1624. doi: 10.1214/aos/1176348787
   Web of Science ®Google Scholar
• Mojirsheibani, M. (2007), ‘ Some Approximations to -Statistics of Kernel Density Estimators’, Statistics, 41, 203–220. doi: 10.1080/02331880701223589
   Web of Science ®Google Scholar
• Parzen, E. (1962), ‘On Estimation of a Probability Density Function and Mode’, Annals of Mathematical Statistics, 33, 1065–1076. doi: 10.1214/aoms/1177704472
   Google Scholar
• Praestgaard, J., and Wellner, J.A. (1993), ‘Exchangeably Weighted Bootstraps of the General Empirical Process’, Annals of Probability, 21, 2053–2086. doi: 10.1214/aop/1176989011
   Web of Science ®Google Scholar
• PrakasaRao, B.L.S. (1983), Nonparametric Functional Estimation, Orlando, FL: Academic Press.
   Google Scholar
• Rosenblatt, M. (1952), ‘Limit Theorems Associated with Variants of the von Mises Statistic’, Annals of Mathematical Statistics, 23, 617–623. doi: 10.1214/aoms/1177729341
   Google Scholar
• Rosenblatt, M. (1956), ‘Remarks on Some Nonparametric Estimates of a Density Function’, Annals of Mathematical Statistics, 27, 832–837. doi: 10.1214/aoms/1177728190
   Google Scholar
• Rubin, D.B . (1981), ‘The Bayesian Bootstrap’, Annals of Statistics, 9, 130–134. doi: 10.1214/aos/1176345338
   Web of Science ®Google Scholar
• Rudemo, M. (1982), ‘Empirical Choice of Histograms and Kernel Density Estimators’, Scandinavian Journal of Statistics, 9, 65–78.
   Web of Science ®Google Scholar
• Sheather, S., and Jones, M. (1991), ‘A Reliable Data-Based Bandwidth Selection Method for Kernel Density Estimation’, Journal of the Royal Statistical Society: Series B, 53, 683–690.
   Google Scholar
• Shorack, G., and Wellner, J. (1986), Empirical Processes with Applications to Statistics, New York: John Wiley & Sons.
   Google Scholar
• Steele, J. M. (1978), ‘Invalidity of Average Squared Error Criterion in Density Estimation’, The Canadian Journal of Statistics, 6, 193–200. doi: 10.2307/3315047
   Google Scholar
• Wand, M., and Jones, M. (1994), ‘Multivariate Plug-in Bandwidth Selection’, Computational Statistics, 9, 97–116.
   Web of Science ®Google Scholar
• Wegman, E.J. (1972), ‘Nonparametric Probability Density Estimation: A Comparison of Density Estimation Methods’, Journal of Statistical Computation and Simulation, 1, 225–245. doi: 10.1080/00949657208810017
   Google Scholar