## Electronic Journal of Statistics

### Discrepancy bounds for deterministic acceptance-rejection samplers

#### Abstract

We consider an acceptance-rejection sampler based on a deterministic driver sequence. The deterministic sequence is chosen such that the discrepancy between the empirical target distribution and the target distribution is small. We use quasi-Monte Carlo (QMC) point sets for this purpose. The empirical evidence shows convergence rates beyond the crude Monte Carlo rate of $N^{-1/2}$. We prove that the discrepancy of samples generated by the QMC acceptance-rejection sampler is bounded from above by $N^{-1/s}$. A lower bound shows that for any given driver sequence, there always exists a target density such that the star discrepancy is at most $N^{-2/(s+1)}$. For a general density, whose domain is the real state space $\mathbb{R}^{s-1}$, the inverse Rosenblatt transformation can be used to convert samples from the $(s-1)$-dimensional cube to $\mathbb{R}^{s-1}$. We show that this transformation is measure preserving. This way, under certain conditions, we obtain the same convergence rate for a general target density defined in $\mathbb{R}^{s-1}$. Moreover, we also consider a deterministic reduced acceptance-rejection algorithm recently introduced by Barekat and Caflisch [F. Barekat and R. Caflisch, Simulation with Fluctuation and Singular Rates. ArXiv:1310.4555 [math.NA], 2013.]

#### Article information

Source
Electron. J. Statist., Volume 8, Number 1 (2014), 678-707.

Dates
First available in Project Euclid: 21 May 2014

https://projecteuclid.org/euclid.ejs/1400703410

Digital Object Identifier
doi:10.1214/14-EJS898

Mathematical Reviews number (MathSciNet)
MR3211028

Zentralblatt MATH identifier
1348.60113

Subjects
Primary: 62F15: Bayesian inference
Secondary: 11K45: Pseudo-random numbers; Monte Carlo methods

#### Citation

Zhu, Houying; Dick, Josef. Discrepancy bounds for deterministic acceptance-rejection samplers. Electron. J. Statist. 8 (2014), no. 1, 678--707. doi:10.1214/14-EJS898. https://projecteuclid.org/euclid.ejs/1400703410

#### References

• [1] Aistleitner, C., Brauchart, J.S. and Dick, J., Point sets on the sphere $\mathbb{S}^{2}$ with small spherical cap discrepancy. Discrete and Computational Geometry, 48, 990–1024, 2012.
• [2] Barekat, F. and Caflisch, R., Simulation with Fluctuation and Singular Rates. ArXiv:1310.4555 [math.NA], 2013.
• [3] Beck, J., On the discrepancy of convex plane sets. Monatshefte Mathematik, 105, 91–106, 1988.
• [4] Botts, C., Hörmann, W. and Leydold, J., Transformed density rejection with inflection points. Statistics and Computing, 23, 251–260, 2013.
• [5] Caflisch, R.E., Monte Carlo and quasi-Monte Carlo methods. Acta Numerica, 1–49, 1998.
• [6] Chen, S., Consistency and convergence rate of Markov chain quasi Monte Carlo with examples. PhD thesis, Stanford University, 2011.
• [7] Chen, S., Dick, J. and Owen, A.B., Consistency of Markov chain quasi-Monte Carlo on continuous state spaces. Annals of Statistics, 39, 673–701, 2011.
• [8] Chentsov, N.N., Pseudorandom numbers for modelling Markov chains. Computational Mathematics and Mathematical Physics, 7, 218–233, 1967.
• [9] Devroye, L., Nonuniform Random Variate Generation. Springer-Verlag, New York, 1986.
• [10] Dick, J., Kuo, F. and Sloan, I.H., High dimensional integration-the quasi-Monte Carlo way. Acta Numerica, 22, 133–288, 2013.
• [11] Dick, J. and Pillichshammer, F., Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge, 2010.
• [12] Dick, J., Rudolf, D. and Zhu, H., Discrepancy bounds for uniformly ergodic Markov chain quasi-Monte Carlo. Available at arxiv.org/abs/1303.2423 [stat.CO], submitted, 2013.
• [13] Faure, H., Discrépance de suites associées à un système de numération (en dimension $s$). Acta Arithmetica, 4, 337–351, 1982.
• [14] Gerber, M. and Chopin, N., Sequential quasi-Monte Carlo. Available at arXiv:1402.4039 [stat.CO], 2014.
• [15] Gnewuch, M., Bracketing numbers for axis-parallel boxes and application to geometric discrepancy. Journal of Complexity, 24, 154–172, 2008.
• [16] Kuipers, L. and Niederreiter, H., Uniform Distribution of Sequences. John Wiley, New York, 2006.
• [17] Halton, J.H., On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numerische Mathematik, 2, 84–90, 1960.
• [18] Hesse, K., A lower bound for the worst-case cubature error on spheres of arbitrary dimension. Numerische Mathematik, 103, 413–433, 2006.
• [19] Hörmann, W., Leydold, J. and Derflinger, G., Automatic Nonuniform Random Variate Generation. Statistics and Computing. Springer-Verlag, Berlin, 2004.
• [20] Kritzer, P., Improved upper bounds on the star discrepancy of $(t,m,s)$-nets and $(t,s)$-sequences. Journal of Complexity, 22, 336–347, 2006.
• [21] L’Ecuyer, P., Lecot, C. and Tuffin, B., A randomized quasi-Monte Carlo simulation method for Markov chains. Operation Research, 56, 958–975, 2008.
• [22] Meyn, M.P. and Tweedie, T.L., Markov Chain and Stochastic Stability. Springer-Verlag, London, 1993.
• [23] Morokoff, W.J. and Caflisch, R.E., Quasi-Monte Carlo integration. Journal of Computational Physics, 122, 218–230, 1995.
• [24] Moskowitz, B. and Caflisch, R.E., Smoothness and dimension reduction in quasi-Monte Carlo methods. Mathematical and Computer Modelling, 23, 37–54, 1996.
• [25] Nguyen, N. and Ökten, G., The acceptance-rejection method for low discrepancy sequences. Submitted, 2014.
• [26] Niederreiter, H., Low-discrepancy and low-dispersion sequences. Journal of Number Theory, 30, 51–70, 1988.
• [27] Niederreiter, H., Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia, Pennsylvania, 1992.
• [28] Niederreiter, H. and Wills, J.M., Diskrepanz und Distanz von Maßen bezüglich konvexer und Jordanscher Mengen (German). Mathematische Zeitschrift, 144, 125–134, 1975.
• [29] Noh, Y., Choi, K.K. and Du, L., New Transformation of Dependent Input Variables Using Copula for RBDO. 7th World Congresses of Structural and Multidisciplinary Optimization COEX Seoul, Korea, 21–25 May 2007.
• [30] Owen, A.B., Randomly permuted $(t,m,s)$-nets and $(t,s)$-sequences. In: H. Niederreiter and P.J. Shiue (Eds.). Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (Las Vegas, NV, 1994), 299–317. Lecture Notes in Statist., 106. Springer, New York, 1995.
• [31] Owen, A.B., Monte Carlo variance of scrambled net quadrature. SIAM Journal on Numerical Analysis, 34, 1884–1910, 1997.
• [32] Owen, A.B., Scrambled net variance for integrals of smooth functions. Annals of Statistics, 25, 1541–1562, 1997.
• [33] Owen, A.B., Monte Carlo theory, methods and examples (book draft). Available at http://www-stat.stanford.edu/~owen/mc/. Last accessed on 16 May 2014.
• [34] Robert, C. and Casella, G., Monte Carlo Statistical Methods. Springer-Verlag, New York, second edition, 2004.
• [35] Rosenblatt, M., Remarks on a multivariate transformation. Annals of Mathematical Statistics, 23, 470–472, 1952.
• [36] Schmidt, W.M., Irregularities of distribution. Acta Arithmetica, 27, 385–396, 1975.
• [37] Sobol, I.M., Distribution of points in a cube and approximate evaluation of integrals. Akademija Nauk SSSR. Žurnal Vyčislitel’ noĭ Matematiki i Matematičeskoĭ Fiziki 7, 784–802, 1967 (in Russian); U.S.S.R Computational Mathematics and Mathematical Physics 7, 86–112, 1967 (in English).
• [38] Stute, W., Convergence rates for the isotrope discrepancy. Annals of Probability, 5, 707–723, 1977.
• [39] Tribble, S.D., Markov chain Monte Carlo algorithms using completely uniformly distributed driving sequences. PhD thesis, Stanford University, 2007.
• [40] Tribble, S.D. and Owen, A.B., Constructions of weakly CUD sequences for MCMC. Electronic Journal of Statistics, 2, 634–660, 2008.
• [41] Wang, X., Quasi-Monte Carlo integration of characteristic functions and the rejection sampling method. Comupter Physics Communication, 123, 16–26, 1999.
• [42] Wang, X., Improving the rejection sampling method in quasi-Monte Carlo methods. Journal of Computational and Applied Mathematics, 114, 231–246, 2000.
• [43] Wyner, A.D., Capabilities of bounded discrepancy decoding. The Bell System Technical Journal, 44, 1061–1122, 1965.