### Citations

5116 | Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images - Geman, Geman - 1984 |

3622 | Equation of state calculations for fast computing machines - Metropolis, Rosenbluth, et al. - 1953 |

3106 | Numerical Recipes in C: The Art of Scientific Computing - Press, Teukolsky, et al. - 1988 |

2196 |
An Introduction to Multivariate Statistical Analysis
- Anderson
- 2003
(Show Context)
Citation Context ...x, Li becomes impractical for large n. A more efficient indirect method follows Anderson (1984). Let I have lower triangular Choleski decomposition E = LL', and suppose Qsn – 1). Then LQL' W(E,n –1) (=-=Anderson, 1984-=-, pp. 254-255). Furthermore Q has representation Q = UU'suv = 0(i < j < n) u„, –N(0,1)s; z2 (n – i) (i = 1,...,n), with the u1 mutually independent for i j (Anderson, 1984, p. 147). Even if n is qui... |

2087 | Monte Carlo sampling methods using Markov chains and their applications - Hastings - 1970 |

1516 | Inference from iterative simulation using multiple sequences - Gelman, Rubin - 1992 |

1135 | Markov chains for exploring posterior distributions. The Annals of Statistics
- Tierney
- 1994
(Show Context)
Citation Context ...(x,z)a(x,z),:/z ify#x 36 This form of the algorithm is due to Hastings (1970). The Metropolis et al. (1953) form takes q(x,y) q(y, x). A simple variant that is often useful is the independence chain (=-=Tierney, 1991-=-a, 1991b), q(x,y) = j(y). Then, a(x,y) = min{ PWi(x) ,1} – mintL±L)1} P(x) j (Y)sw(x) where w(x) = p(x)/j(x). The independence chain is closely related to acceptance sampling (Section 4.2) and importa... |

923 | The calculation of posterior distributions by data augmentation
- Tanner, Wong
- 1987
(Show Context)
Citation Context ...as introduced by Geman and Geman (1984). This was subsequently shown to have great potential for Bayesian computation by Gelfand and Smith (1990). Their work, combined with data augmentation methods (=-=Tanner and Wong, 1987-=-), has proven very successful in the treatment of latent variables and other unobservables in economic models. (Examples are given in Sections 7.1 and 7.3.) Since 1990 application of Markov chain Mont... |

871 | The Art of Computer Programming, Volume 2: Seminumerical Algorithms - Knuth - 1981 |

671 | Simulation and the Monte Carlo Method - Rubinstein - 1981 |

603 | Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments
- Geweke
- 1992
(Show Context)
Citation Context ..., consider the target density kernel f(x;T, n).(x12)Th12 [F(.42)] T exp(-11x), which arises as a conditional posterior density kernel for the degrees-of-freedom parameter in a Student-t distribution (=-=Geweke, 1992-=-b, Appendix B). For the exponential family of source densities g(x; a) = aexp(–ax) the regular necessary conditions are (172)Elog(x/2) + 1 – ty(x/2)]+ (a – n). 0, x – cel = 0, where v(•). r(• )/F( .) ... |

601 | Bayesian analysis of stochastic volatility models - Jacquier, Polson, et al. - 1994 |

569 | Modeling financial time series. - Taylor - 1986 |

568 | Stochastic Simulation - Ripley - 1987 |

549 | P.: Methods of Numerical Integration - Davis, Rabinowitz - 1984 |

520 | Sampling based approaches to calculating marginal densities. - Gelfand, Smith - 1990 |

493 | Bayesian inference in econometric models using monte carlo integration
- Geweke
- 1989
(Show Context)
Citation Context ...very useful, since drawing pseudorandom vectors from this distribution is likely to be awkward at best. There has been some attention to optimization within families of importance sampling densities (=-=Geweke, 1989-=-), but optimization procedures themselves generally involve integrals that in turn require numerical approximation. Adaptive methods use previously drawn x, to identify large values of f(x)/j(x), w(x)... |

372 | Multiple Time Series - Hannan - 1970 |

367 | A Guide to Simulation - Bratley, Fox, et al. - 1987 |

339 |
Accurate approximations for posterior moments and marginal densities
- Tierney, Kadane
- 1986
(Show Context)
Citation Context ... ), but in the corresponding approximation (g) = II=/4/2 expf n[1: Cal- L(0)11 the leading terms in the numerator and denominator cancel and the resulting error of approximation for E„ (g) is O(n ') (=-=Tierney and Kadane, 1986-=-). The approximate solution provided by this method is a substantial improvement on previous approximations of this kind, which worked with a single expansion about B. It exhibits two attractions shar... |

309 |
Central limit theorem for additive functionals of reversible Markov process and applications to simple exclusion
- Kipnis, Varadhan
- 1986
(Show Context)
Citation Context ...ys reversible; Gibbs sampling chains are not (Geyer, 1992, Section 2). If the Markov chain is stationary, p-irreducible and reversible, then N var(g,)s=s' 42 and if 62 < oo, then Ci(gN 0-9—> N(0, al (=-=Kipnis and Varadhan, 1986-=-). In the absence of reversibility known sufficient conditions for central limit theorems are strong and difficult to verify. For example, if for some m <00 P(X"m E Aix' = x)/Lp(x)dv(x) is bounded bel... |

297 | Various techniques used in connection with random digits - Neumann - 1951 |

210 | Efficient Simulation From the Multivariate Normal and Student-t Distribution Subject to Linear Constraints
- Geweke
- 1991
(Show Context)
Citation Context ...everely truncated normal distribution (Marsaglia, 1964; Geweke, 1986), and for the exponential source density setting the parameter equal to the truncation point is an optimal or near optimal choice (=-=Geweke, 1991-=-). One can readily verify that the acceptance probability for the source density g(x) = 5 exp[-5(x — 5)], 5 < x 5 8, is .964. Optimizing acceptance sampling. In general, suppose that it is desired to ... |

84 | Bayesian Estimates of Equation System Parameters: An Application of Integration by Monte Carlo," - KLOEK, DIJK - 1978 |

82 | Intermediate Statistics and Econometrics: A Comparative Approach. - Poirier - 1995 |

80 | An efficient method for generating discrete random variables with general distributions. - Walker - 1977 |

74 | OptimumMonte-Carlo sampling using Markov chains. - Peskun, H - 1973 |

71 | An exhaustive analysis of multiplicative congruential random number generators with modulus 2 31 -1 - Fishman, Moore - 1986 |

61 |
Antithetic Acceleration of Monte Carlo Integration in Bayesian Inference,”
- Geweke
- 1988
(Show Context)
Citation Context ...ctive. 5.1 Antithetic Monte Carlo This technique is due to Hammersly and Morton (1956) and has been widely used in statistics, experimental design, and simulation (e.g. Mikhail, 1972; Mitchell, 1973; =-=Geweke, 1988-=-). In antithetic simple Monte Carlo integration M= 2 correlated variables are drawn in each of N replications. Then, = tvar[g(x,,I)]+ cov[g(xd )g(x, 2 )]} . So long as cov[g(x„),g(x,2)] < 0, antitheti... |

60 | Fully exponential Laplace approximations to expectations and variances of nonpositive functions - Tierney, Kass, et al. - 1989 |

58 | A convenient method for generating normal variables,” - Marsaglia, Bray - 1964 |

55 | Fourier analysis of uniform random number generators - Coveyou, MacPherson - 1967 |

47 |
Simple conditions for the convergence of the Gibbs sampler and Metropolis-Hastings algorithms. Stochastic Processes and their Applications 44:207–16
- Roberts, Smith
- 1994
(Show Context)
Citation Context ...and (C) in Section 6.2. 39 The continuous (Lebesgue measure) case is technically more difficult, but it may be shown that three simple conditions are jointly sufficient for results (A), (B), and (C) (=-=Roberts and Smith, 1992-=-): (1) p(x) is lower semicontinuous at 0; (2) if p(x)dx, is locally bounded (i = 1,...k); (3) Ds is connected. A function h(x) is lower semicontinuous at 0 if, for all x with h(x) > 0, there exists an... |

44 | Methods of Reducing Sample Size in Monte Carlo Computations. - Kahn, Marshall - 1953 |

43 |
Exact Inference in the Inequality Constrained Normal Linear Regression Model,"
- GEWEKE
- 1986
(Show Context)
Citation Context ...probability of only .0645. An exponential distribution translated to the truncation point is for many purposes an excellent approximation to a severely truncated normal distribution (Marsaglia, 1964; =-=Geweke, 1986-=-), and for the exponential source density setting the parameter equal to the truncation point is an optimal or near optimal choice (Geweke, 1991). One can readily verify that the acceptance probabilit... |

41 |
An adaptive algorithm for numerical integration over an Ndimensional rectangular region.
- GENZ, MALIK
- 1980
(Show Context)
Citation Context ... satisfied, those regions for which the convergence criterion is farthest from being satisfied are subdivided, and the 5 local rule is applied to the new subdivisions (van Dooren and de Ridder, 1976; =-=Genz and Malik, 1980-=-; Genz, 1991). For these procedures to work successfully it is important to have a scheme for construction of subregions well suited to the problem at hand, as reconsideration of Figure 1(b) will make... |

41 |
Computer approximations.
- Hart
- 1968
(Show Context)
Citation Context ...rivial numerical integration of the kind discussed in Section 2! A leading example is provided by the standard normal distribution, for which specialized methods can be applied to the computation of (=-=Hart et al., 1968-=-; Strecok, 1968), but for which acceptance and composition methods (discussed below) are more efficient. Discrete distributions. Suppose that the random variable X takes on a finite number of values, ... |

40 |
Calculation of Gaussian quadrature rules
- Golub, Welsch
- 1969
(Show Context)
Citation Context ... through successive bisection. Error criteria are usually specified as the absolute or relative difference in the computed approximation to I = f(x)dx using an n -point and an a ni -point quadrature (=-=Golub and Welsch, 1969-=-). Open and semi-open intervals can be treated by appropriate transformation of the interval to a finite interval (Piessens et al., 1983). Existence and boundedness of r(2") depends in part on the cho... |

37 | The structure of linear congruential sequences - Marsaglia - 1972 |

35 | Computer Generation of Normal Random Variables. - Kinderman, Ramage - 1976 |

35 | An adaptive algorithm for numerical integration over an n-dimensional cube,” - Dooren, Ridder - 1976 |

31 | A fast procedure for generating normal random variables. - Marsaglia, MacLaren, et al. - 1964 |

31 |
Exploring Posterior Distributions using Markov Chains,"
- Tierney
- 1991
(Show Context)
Citation Context ...(x,z)a(x,z),:/z ify#x 36 This form of the algorithm is due to Hastings (1970). The Metropolis et al. (1953) form takes q(x,y) q(y, x). A simple variant that is often useful is the independence chain (=-=Tierney, 1991-=-a, 1991b), q(x,y) = j(y). Then, a(x,y) = min{ PWi(x) ,1} – mintL±L)1} P(x) j (Y)sw(x) where w(x) = p(x)/j(x). The independence chain is closely related to acceptance sampling (Section 4.2) and importa... |

26 | Subregion-Adaptive Integration of Functions Having a Dominant Peak - Genz, Kass - 1997 |

24 | Squeeze methods for generating gamma variates - Schmeiser, Lal - 1980 |

22 | An Alias Method for Sampling from - Ahrens, Dieter - 1989 |

21 |
Practical markov chain monte carlo, Statistical Science 7(4
- Geyer
- 1992
(Show Context)
Citation Context ...finite and denote cov[g(x t , x i ' )]. A Markov chain with kernel K is reversible if K(x,y) = K(y,x) for all x,y e D. Hastings-Metropolis chains are always reversible; Gibbs sampling chains are not (=-=Geyer, 1992-=-, Section 2). If the Markov chain is stationary, p-irreducible and reversible, then N var(g,)s=s' 42 and if 62 < oo, then Ci(gN 0-9—> N(0, al (Kipnis and Varadhan, 1986). In the absence of reversibili... |

20 | New Fast Method for Generating Discrete Random Numbers with Arbitrary Distributions - Walker - 1974 |

17 |
Generating a Variable from the Tail of a Normal Distribution,"
- Marsaglia
- 1964
(Show Context)
Citation Context ...ds an acceptance probability of only .0645. An exponential distribution translated to the truncation point is for many purposes an excellent approximation to a severely truncated normal distribution (=-=Marsaglia, 1964-=-; Geweke, 1986), and for the exponential source density setting the parameter equal to the truncation point is an optimal or near optimal choice (Geweke, 1991). One can readily verify that the accepta... |

16 |
On the calculation of the inverse of the error function.
- Strecok
- 1968
(Show Context)
Citation Context ...tegration of the kind discussed in Section 2! A leading example is provided by the standard normal distribution, for which specialized methods can be applied to the computation of (Hart et al., 1968; =-=Strecok, 1968-=-), but for which acceptance and composition methods (discussed below) are more efficient. Discrete distributions. Suppose that the random variable X takes on a finite number of values, without loss of... |

13 | Construction of fully symmetric numerical integration formulas - McNamee, Stenger - 1967 |

11 | Sampling from binomial and Poisson distributions : A method with bounded computation times - AHRENS, U - 1980 |

11 | Malik - An imbedded family of fully symmetric numerical integration rules - - Genz, A - 1983 |

11 | A- One-line random number generators and their use in combinations. Commi^ications of the Associa tion for Computing Machinery 11 - Marsaglia, Bray - 1968 |

8 | A Statistical Evaluation of Multiplicative Random Number Generators with Modulus 231-1 - Fishman, Moore - 1982 |

8 |
Subregion Adaptive Algorithms for Multiple Integrals
- Genz
- 1991
(Show Context)
Citation Context ...ons for which the convergence criterion is farthest from being satisfied are subdivided, and the 5 local rule is applied to the new subdivisions (van Dooren and de Ridder, 1976; Genz and Malik, 1980; =-=Genz, 1991-=-). For these procedures to work successfully it is important to have a scheme for construction of subregions well suited to the problem at hand, as reconsideration of Figure 1(b) will make clear. For ... |

7 |
Adaptive importance sampling and chaining
- Evans
- 1991
(Show Context)
Citation Context ...rally involve integrals that in turn require numerical approximation. Adaptive methods use previously drawn x, to identify large values of f(x)/j(x), w(x), or g 2 (x)w(x) and modify j(x) accordingly (=-=Evans, 1991-=-). Such procedures can be convenient but are limited by the fact that x 1sis least likely to be drawn 25 where j(x) is small. Informal, deterministic methods for tailoring j(x) have worked well in som... |

7 |
Math/Library User's Manual,
- IMSL
- 1987
(Show Context)
Citation Context ...iplicative congruential generator. For example, the multiplicative generator with m = 23/ —1= 2147483647 (a prime) and c= 16807, c = 397204094, or c= 950706376 is used in the IMSL scientific library (=-=IMSL, 1989-=-), and the user may 9 choose between different values of c, as well as set the seed X0 . The sequence IX,1 is mapped into the pseudorandom uniform sequence {U,} by the transformation (3.2) If in is pr... |

7 | Computer Generation of Poisson, Binomial, and Hypergeometric Random Variates - KACHITVICHYANUKUL - 1982 |

7 | Expressing a random variable in terms of uniform random variables - Marsaglia - 1961 |

6 |
Fundamentals of Operations Research
- McGrath
- 1970
(Show Context)
Citation Context ... convenient subvector x(0) . Especially if g(x) = g(x (1) ), the benefits of antithetic Monte Carlo will then be realized in both Problem I and Problem E. 5.2 Systematic sampling Systematic sampling (=-=McGrath, 1970-=-) combines certain advantages of deterministic and Monte Carlo methods. The former achieve great efficiency by systematically choosing points for evaluation in specific low-dimensional problems; the l... |

4 |
Simulating the Small-sample Properties of Econometric Estimators
- Mikhail
- 1972
(Show Context)
Citation Context ...iant may be practical and productive. 5.1 Antithetic Monte Carlo This technique is due to Hammersly and Morton (1956) and has been widely used in statistics, experimental design, and simulation (e.g. =-=Mikhail, 1972-=-; Mitchell, 1973; Geweke, 1988). In antithetic simple Monte Carlo integration M= 2 correlated variables are drawn in each of N replications. Then, = tvar[g(x,,I)]+ cov[g(xd )g(x, 2 )]} . So long as co... |

4 | On the Evaluation of Definite Integrals and a Quasi-Monte Carlo Method based on Properties of Algebraic Numbers - RICHTMEYER - 1952 |

4 | A Non-Random Sampling Method, Based on Congruences for MonteCarlo Problems - RICHTMEYER - 1958 |

3 | Efficient and Portable Combined Pseudorandom Number Generators - L'Ecuyer - 1986 |

2 |
Priors for Macroeconomic Time Series,” Federal Reserve Bank of Minneapolis Institute for Empirical Macroeconomics Discussion Paper No. 64
- Geweke
- 1992
(Show Context)
Citation Context ..., consider the target density kernel f(x;T, n).(x12)Th12 [F(.42)] T exp(-11x), which arises as a conditional posterior density kernel for the degrees-of-freedom parameter in a Student-t distribution (=-=Geweke, 1992-=-b, Appendix B). For the exponential family of source densities g(x; a) = aexp(–ax) the regular necessary conditions are (172)Elog(x/2) + 1 – ty(x/2)]+ (a – n). 0, x – cel = 0, where v(•). r(• )/F( .) ... |

1 | Von Neumann's Comparison Method for Random Sampling from the Normal and Other Distributions - Forsyth - 1972 |

1 | Monte Carlo Methods. Londom - Hammersly, Handscomb - 1964 |

1 |
Numerical Methods in Ecoomics
- Judd
- 1991
(Show Context)
Citation Context ...sely, one may show that if r(x) is 2n-times differentiable then jabf(x)dxs=1(01 r(x,)= TC2n)(4) for some 4 e [a,b], where {c} is a sequence of constants withsc„ = 0. For 2114-1z (nOlt(2n + 1)12n !r} (=-=Judd, 1991-=-, pp.example, if w(x) = 1,a = –1, b = +1, then c„ = 6-7, 6-8). 3 This approach can be applied to any subinterval of [a,b] as well, and so long as r(x) is 2n-times differentiable the accuracy of the ap... |

1 |
Variance Reduction by Antithetic Variates in G 1/G/1 Queueing Simulation
- Mitchell
- 1973
(Show Context)
Citation Context ...ctical and productive. 5.1 Antithetic Monte Carlo This technique is due to Hammersly and Morton (1956) and has been widely used in statistics, experimental design, and simulation (e.g. Mikhail, 1972; =-=Mitchell, 1973-=-; Geweke, 1988). In antithetic simple Monte Carlo integration M= 2 correlated variables are drawn in each of N replications. Then, = tvar[g(x,,I)]+ cov[g(xd )g(x, 2 )]} . So long as cov[g(x„),g(x,2)] ... |

1 | General Irreducible Markov Chains and Non-negatie Operators - Numelin - 1984 |

1 | An Efficient and Portable Pseud-random Number Generator," Applied Statistics 31 - Wichmann, Hill - 1982 |