Use of exchangeable pairs in the analysis of simulations

The method of exchangeable pairs has emerged as an important tool in proving limit theorems for Poisson, normal and other classical approx- imations. Here the method is used in a simulation context. We estimate tran- sition probabilitites from the simulations and use these to reduce variances. Exchangeable pairs are used as control variates. Finally, a general approximation theorem is developed that can be com- plemented by simulations to provide actual estimates of approximation errors.

[1]  Ronald L. Wasserstein,et al.  Monte Carlo: Concepts, Algorithms, and Applications , 1997 .

[2]  T.J.P. Penna,et al.  Broad histogram Monte Carlo , 1998 .

[3]  P. Diaconis,et al.  A geometric interpretation of the Metropolis-Hastings algorithm , 2001 .

[4]  R. R. Picard,et al.  Monte Carlo Transition Dynamics and Variance Reduction , 2000 .

[5]  Y. Rinott,et al.  On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted $U$-statistics , 1997 .

[6]  P. Diaconis,et al.  Algebraic algorithms for sampling from conditional distributions , 1998 .

[7]  P. Prescott,et al.  Monte Carlo Methods , 1964, Computational Statistical Physics.

[8]  G. Reinert,et al.  The stationary distribution in the antivoter model: exact sampling and approximations , 2004 .

[9]  P. Diaconis,et al.  Closed Form Summation for Classical Distributions: Variations on Theme of De Moivre , 1991 .

[10]  Gerard T. Barkema,et al.  Monte Carlo Methods in Statistical Physics , 1999 .

[11]  Charles M. Stein,et al.  Asymptotic Evaluation of the Number of Latin Rectangles , 1978, J. Comb. Theory, Ser. A.

[12]  G. Reinert,et al.  Stein's method and the zero bias transformation with application to simple random sampling , 1997, math/0510619.

[13]  D. Aldous Representations for partially exchangeable arrays of random variables , 1981 .

[14]  Jun S. Liu,et al.  Monte Carlo strategies in scientific computing , 2001 .

[15]  J. Hammersley,et al.  Monte Carlo Methods , 1965 .

[16]  Jason E. Fulman A Stein's method proof of the asymptotic normality of descents and inversions in the symmetric group , 1997 .

[17]  H. Hudson A Natural Identity for Exponential Families with Applications in Multiparameter Estimation , 1978 .

[18]  J. M. Hammersley,et al.  Conditional Monte Carlo , 1956, JACM.

[19]  C. Stein A bound for the error in the normal approximation to the distribution of a sum of dependent random variables , 1972 .

[20]  Robert H. Swendsen,et al.  TRANSITION MATRIX MONTE CARLO REWEIGHTING AND DYNAMICS , 1999 .

[21]  C. Stein Approximate computation of expectations , 1986 .

[22]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[23]  S. Vajda,et al.  Symposium on Monte Carlo Methods , 1957, Mathematical Gazette.

[24]  J. Kingman Uses of Exchangeability , 1978 .