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42
General state space Markov chains and MCMC algorithm
 PROBABILITY SURVEYS
, 2004
"... This paper surveys various results about Markov chains on general (noncountable) state spaces. It begins with an introduction to Markov chain Monte Carlo (MCMC) algorithms, which provide the motivation and context for the theory which follows. Then, sufficient conditions for geometric and uniform e ..."
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Cited by 114 (27 self)
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This paper surveys various results about Markov chains on general (noncountable) state spaces. It begins with an introduction to Markov chain Monte Carlo (MCMC) algorithms, which provide the motivation and context for the theory which follows. Then, sufficient conditions for geometric and uniform ergodicity are presented, along with quantitative bounds on the rate of convergence to stationarity. Many of these results are proved using direct coupling constructions based on minorisation and drift conditions. Necessary and sufficient conditions for Central Limit Theorems (CLTs) are also presented, in some cases proved via the Poisson Equation or direct regeneration constructions. Finally, optimal scaling and weak convergence results for MetropolisHastings algorithms are discussed. None of the results presented is new, though many of the proofs are. We also describe some Open Problems.
Estimating Equations Based on Eigenfunctions for a Discretely Observed Diffusion Process
, 1995
"... : A new type of martingale estimating function is proposed for inference about classes of diffusion processes based on discretetime observations. These estimating functions can be tailored to a particular class of diffusion processes by utilizing a martingale property of the eigenfunctions of the g ..."
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Cited by 58 (13 self)
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: A new type of martingale estimating function is proposed for inference about classes of diffusion processes based on discretetime observations. These estimating functions can be tailored to a particular class of diffusion processes by utilizing a martingale property of the eigenfunctions of the generators of the diffusions. Optimal estimating functions in the sense of Godambe and Heyde are found. Inference based on these is invariant under transformations of data. A result on consistency and asymptotic normality of the estimators is given for ergodic diffusions. The theory is illustrated by several examples and by a simulation study. Keywords: generator, optimal estimating function, stochastic differential equation, quasilikelihood. 1 Introduction Martingale estimating functions have turned out to give good and relatively simple estimators for discretely observed diffusion models, for which the likelihood function is only explicitly known in special cases. These estimators have th...
Estimating Functions for Discretely Sampled DiffusionType Models. Chapter of the Handbook of financial econometrics, AitSahalia and Hansen eds. http://home.uchicago.edu/ lhansen/handbook.htm Birgé
 in Festschrift for Lucien Le Cam: Research Papers in Probability and Statistics
, 2004
"... Estimating functions provide a general framework for finding estimators and studying their properties in many different kinds of statistical models, including stochastic process models. An estimating function is a function of the data as well as of the parameter to be estimated. An estimator is obta ..."
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Cited by 26 (9 self)
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Estimating functions provide a general framework for finding estimators and studying their properties in many different kinds of statistical models, including stochastic process models. An estimating function is a function of the data as well as of the parameter to be estimated. An estimator is obtained by equating the estimating function to zero and solving the resulting
Parametric Inference for Diffusion Processes Observed At Discrete Points in Time: A Survey
"... This paper is a survey of existing estimation techniques for stationary and ergodic diffusion processes observed at discrete points in time. The reader is introduced to the following techniques: (i) estimating functions with special emphasis on martingale estimating functions and socalled simple es ..."
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Cited by 24 (2 self)
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This paper is a survey of existing estimation techniques for stationary and ergodic diffusion processes observed at discrete points in time. The reader is introduced to the following techniques: (i) estimating functions with special emphasis on martingale estimating functions and socalled simple estimating functions; (ii) analytical and numerical approximations of the likelihood which can in principle be made arbitrarily accurate; (iii) Bayesian analysis and MCMC methods; and (iv) indirect inference and EMM which both introduce auxiliary (but wrong) models and correct for the implied bias by simulation
Central limit theorem for stationary linear processes
 THE ANNALS OF PROBABILITY. HTTP://ARXIV.ORG/ABS/MATH.PR/0509682
, 2006
"... We establish the central limit theorem for linear processes with dependent innovations including martingales and mixingale type of assumptions as defined in McLeish [Ann. Probab. 5 (1977) 616–621] and motivated by Gordin [Soviet Math. Dokl. 10 (1969) 1174–1176]. In doing so we shall preserve the gen ..."
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Cited by 20 (4 self)
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We establish the central limit theorem for linear processes with dependent innovations including martingales and mixingale type of assumptions as defined in McLeish [Ann. Probab. 5 (1977) 616–621] and motivated by Gordin [Soviet Math. Dokl. 10 (1969) 1174–1176]. In doing so we shall preserve the generality of the coefficients, including the long range dependence case, and we shall express the variance of partial sums in a form easy to apply. Ergodicity is not required.
Central Limit Theorems and Invariance Principles for TimeOne Maps of Hyperbolic Flows
, 2002
"... We give a general method for deducing statistical limit laws in situations where rapid decay of correlations has been established. As an application of this method, we obtain new results for timeone maps of hyperbolic flows. In particular, using recent results of Dolgopyat, we prove that many class ..."
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Cited by 18 (11 self)
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We give a general method for deducing statistical limit laws in situations where rapid decay of correlations has been established. As an application of this method, we obtain new results for timeone maps of hyperbolic flows. In particular, using recent results of Dolgopyat, we prove that many classical limit theorems of probability theory, such as the central limit theorem, the law of the iterated logarithm, and approximation by Brownian motion (almost sure invariance principle), are typically valid for such timeone maps. The central limit theorem for hyperbolic flows goes back to Ratner 1973 and is always valid, irrespective of mixing hypotheses. 1.
Quantile Autoregression
 Convergence of Stochastic Processes
, 2006
"... Abstract. We consider quantile autoregression (QAR) models in which the autoregressive coefficients can be expressed as monotone functions of a single, scalar random variable. The models can capture systematic influences of conditioning variables on the location, scale and shape of the conditional d ..."
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Cited by 18 (4 self)
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Abstract. We consider quantile autoregression (QAR) models in which the autoregressive coefficients can be expressed as monotone functions of a single, scalar random variable. The models can capture systematic influences of conditioning variables on the location, scale and shape of the conditional distribution of the response, and therefore constitute a significant extension of classical constant coefficient linear time series models in which the effect of conditioning is confined to a location shift. The models may be interpreted as a special case of the general random coefficient autoregression model with strongly dependent coefficients. Statistical properties of the proposed model and associated estimators are studied. The limiting distributions of the autoregression quantile process are derived. Quantile autoregression inference methods are also investigated. Empirical applications of the model to the U.S. unemployment rate and U.S. gasoline prices highlight the potential of the model. 1.
Decay of correlations, Central limit theorems and approximation by brownian motion for compact Lie group extensions
 Ergod. Th. & Dynam. Sys
, 2003
"... Hölder continuous observations on hyperbolic basic sets satisfy strong statistical properties such as exponential decay of correlations, central limit theorems and invariance principles (approximation by Brownian motion). Using an equivariant version of the Ruelle transfer operator studied by Parry ..."
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Cited by 16 (7 self)
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Hölder continuous observations on hyperbolic basic sets satisfy strong statistical properties such as exponential decay of correlations, central limit theorems and invariance principles (approximation by Brownian motion). Using an equivariant version of the Ruelle transfer operator studied by Parry & Pollicott, we obtain similar results for equivariant observations on compact group extensions of hyperbolic basic sets.
Simplified Estimating Functions for Diffusion Models with a Highdimensional Parameter
 Scand. J. Statist
, 1997
"... We consider estimating functions for discretely observed diffusion processes of the following type: For one part of the parameter of interest we propose to use a simple and explicit estimating function of the type studied by Kessler (1996); for the remaining part of the parameter we use a martingale ..."
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Cited by 14 (2 self)
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We consider estimating functions for discretely observed diffusion processes of the following type: For one part of the parameter of interest we propose to use a simple and explicit estimating function of the type studied by Kessler (1996); for the remaining part of the parameter we use a martingale estimating function. Such an approach is particularly useful in practical applications when the parameter is highdimensional. It is also often necessary to supplement a simple estimating function by another type of estimating function because only the part of the parameter on which the invariant measure depends can be estimated by a simple estimating function. Under regularity conditions the resulting estimators are consistent and asymptotically normal. Several examples are considered in order to demonstrate the idea of the estimating procedure. The method is applied to two data sets comprising wind velocities and stock prices. In one example we also propose a general method for constructi...
root quantile autoregression inference
 J. Am. Statist. Assoc
"... Abstract. We study statistical inference in quantile autoregression models when the largest autoregressive coefficient may be unity. The limiting distribution of a quantile autoregression estimator and its tstatistic is derived. The asymptotic distribution is not the conventional DickeyFuller dist ..."
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Cited by 10 (0 self)
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Abstract. We study statistical inference in quantile autoregression models when the largest autoregressive coefficient may be unity. The limiting distribution of a quantile autoregression estimator and its tstatistic is derived. The asymptotic distribution is not the conventional DickeyFuller distribution, but a linear combination of the DickeyFuller distribution and the standard normal, with the weight determined by the correlation coefficient of related time series. Inference methods based on the estimator are investigated asymptotically. Monte Carlo results indicate that the new inference procedures have power gains over the conventional least squares based unit root tests in the presence of nonGaussian disturbances. An empirical application of the model to US macroeconomic time series data further illustrates the potential of the new approach. 1.