×

Kernel estimators of asymptotic variance for adaptive Markov chain Monte Carlo. (English) Zbl 1219.62125

Summary: We study the asymptotic behavior of kernel estimators of asymptotic variances (or long-run variances) for a class of adaptive Markov chains. The convergence is studied both in \(L^{p}\) and almost surely. The results also apply to Markov chains and improve on the existing literature by imposing weaker conditions. We illustrate the results with applications to the GARCH(1, 1) Markov model and to an adaptive MCMC algorithm for Bayesian logistic regression.

MSC:

62M05 Markov processes: estimation; hidden Markov models
60J22 Computational methods in Markov chains
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
65C40 Numerical analysis or methods applied to Markov chains
60F05 Central limit and other weak theorems
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Andrews, D. W. K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59 817-858. JSTOR: · Zbl 0732.62052
[2] Andrieu, C. and Moulines, É. (2006). On the ergodicity properties of some adaptive MCMC algorithms. Ann. Appl. Probab. 16 1462-1505. · Zbl 1114.65001
[3] Andrieu, C. and Thoms, J. (2008). A tutorial on adaptive MCMC. Statist. Comput. 18 343-373.
[4] Atchade, Y. F. (2006). An adaptive version for the Metropolis adjusted Langevin algorithm with a truncated drift. Methodol. Comput. Appl. Probab. 8 235-254. · Zbl 1104.65004
[5] Atchadé, Y. F. (2010). Supplement to “Kernel estimators of asymptotic variance for adaptive Markov chain Monte Carlo.” .
[6] Atchade, Y. F. and Fort, G. (2009). Limit theorems for some adaptive MCMC algorithms with subgeometric kernels: Part II. Technical report, Univ. Michigan. · Zbl 1215.60046
[7] Atchade, Y. and Fort, G. (2010). Limit theorems for some adaptive MCMC algorithms with sub-geometric kernels. Bernoulli 16 116-154. · Zbl 1215.60046
[8] Atchade, Y. F., Fort, G., Moulines, E. and Priouret, P. (2009). Adaptive Markov chain Monte Carlo: Theory and methods. Technical report, Univ. Michigan.
[9] Atchade, Y. F. and Rosenthal, J. S. (2005). On adaptive Markov chain Monte Carlo algorithm. Bernoulli 11 815-828. · Zbl 1085.62097
[10] Baxendale, P. H. (2005). Renewal theory and computable convergence rates for geometrically ergodic Markov chains. Ann. Appl. Probab. 15 700-738. · Zbl 1070.60061
[11] Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31 307-327. · Zbl 0865.62085
[12] Bratley, P., Fox, B. and Schrage, L. (1987). A Guide to Simulation , 2nd ed. Springer, New York. · Zbl 0515.68070
[13] Damerdji, H. (1995). Mean-square consistency of the variance estimator in steady-state simulation output analysis. Oper. Res. 43 282-291. JSTOR: · Zbl 0830.62077
[14] de Jong, R. M. (2000). A strong consistency proof for heteroskedasticity and autocorrelation consistent covariance matrix estimators. Econometric Theory 16 262-268. JSTOR: · Zbl 0957.62074
[15] de Jong, R. M. and Davidson, J. (2000). Consistency of kernel estimators of heteroscedastic and autocorrelated covariance matrices. Econometrica 68 407-423. JSTOR: · Zbl 1016.62030
[16] Flegal, J. M. and Jones, G. L. (2009). Batch means and spectral variance estimators in Markov chain Monte Carlo. Available at . · Zbl 1184.62161
[17] Haario, H., Saksman, E. and Tamminen, J. (2001). An adaptive Metropolis algorithm. Bernoulli 7 223-242. · Zbl 0989.65004
[18] Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application . Academic Press, New York. · Zbl 0462.60045
[19] Hansen, B. E. (1992). Consistent covariance matrix estimation for dependent heterogeneous processes. Econometrica 60 967-972. JSTOR: · Zbl 0746.62088
[20] Jarner, S. F. and Hansen, E. (2000). Geometric ergodicity of Metropolis algorithms. Stocahstic Process. Appl. 85 341-361. · Zbl 0997.60070
[21] Meitz, M. and Saikkonen, P. (2008). Ergodicity, mixing, and existence of moments of a class of Markov models with applications to GARCH and ACD models. Econometric Theory 24 1291-1320. · Zbl 1284.62566
[22] Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability . Springer, London. · Zbl 0925.60001
[23] Michie, D., Spiegelhalter, D. and Taylor, C. (1994). Machine Learning, Neural and Statistical Classification . Prentice Hall, Upper Saddle River, NJ. · Zbl 0827.68094
[24] Mykland, P., Tierney, L. and Yu, B. (1995). Regeneration in Markov chain samplers. J. Amer. Statist. Assoc. 90 233-241. JSTOR: · Zbl 0819.62082
[25] Newey, W. K. and West, K. D. (1994). Automatic lag selection in covariance matrix estimation. Rev. Econom. Stud. 61 631-653. JSTOR: · Zbl 0815.62063
[26] Priestley, M. B. (1981). Spectral Analysis and Time Series: Volume 1: Univariate Series . Academic Press, London. · Zbl 0537.62075
[27] Roberts, G. and Rosenthal, J. (2009). Examples of adaptive MCMC. J. Comput. Graph. Statist. 18 349-367.
[28] Saksman, E. and Vihola, M. (2009). On the ergodicity of the adaptive Metropolis algorithm on unbounded domains. Technical report. Available at . · Zbl 1209.65004
[29] Wu, W. B. and Shao, X. (2007). A limit theorem for quadratic forms and its applications. Econometric Theory 23 930-951. · Zbl 1237.60020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.