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QuasiLikelihood Models and Optimal Inference
 Ann. Statist
"... Consider an ergodic Markov chain on the real line, with parametric models for the conditional mean and variance of the transition distribution. Such a setting is an instance of a quasilikelihood model. The customary estimator for the parameter is the maximum quasilikelihood estimator. It is not ef ..."
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Cited by 16 (5 self)
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Consider an ergodic Markov chain on the real line, with parametric models for the conditional mean and variance of the transition distribution. Such a setting is an instance of a quasilikelihood model. The customary estimator for the parameter is the maximum quasilikelihood estimator. It is not efficient, but as good as the best estimator that ignores the parametric model for the conditional variance. We construct two efficient estimators. One is a convex combination of solutions of two estimating equations, the other a weighted nonlinear onestep least squares estimator, with weights involving predictors for the third and fourth centered conditional moments of the transition distribution. Additional restrictions on the model can lead to further improvement. We illustrate this with an autoregressive model whose error variance is related to the autoregression parameter. 1 Introduction According to Wedderburn (1974), a quasilikelihood model is defined by a relation between mean and v...
Estimating invariant laws of linear processes by Ustatistics
"... Suppose we observe an invertible linear process with independent mean zero innovations, and with coefficients depending on a finitedimensional... ..."
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Cited by 11 (10 self)
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Suppose we observe an invertible linear process with independent mean zero innovations, and with coefficients depending on a finitedimensional...
Efficient estimation of invariant distributions of some semiparametric Markov chain models
, 1998
"... We characterize efficient estimators for the expectation of a function under the invariant distribution of a Markov chain and outline ways of constructing such estimators. We consider two models. The first is described by a parametric family of constraints on the transition distribution; the second ..."
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Cited by 3 (3 self)
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We characterize efficient estimators for the expectation of a function under the invariant distribution of a Markov chain and outline ways of constructing such estimators. We consider two models. The first is described by a parametric family of constraints on the transition distribution; the second is the heteroscedastic nonlinear autoregressive model. The efficient estimator for the first model adds a correction term to the empirical estimator. In the second model, the suggested efficient estimator is a onestep improvement of an initial estimator which might be obtained by a version of Markov chain Monte Carlo.
CHAPTER 1 EFFICIENT ESTIMATORS FOR TIME SERIES
"... We illustrate several recent results on efficient estimation for semiparametric time series models with a simple class of models: firstorder nonlinear autoregression with independent innovations. We consider in particular estimation of the autoregression parameter, the innovation distribution, ..."
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We illustrate several recent results on efficient estimation for semiparametric time series models with a simple class of models: firstorder nonlinear autoregression with independent innovations. We consider in particular estimation of the autoregression parameter, the innovation distribution, conditional expectations, the stationary distribution, the stationary density, and higherorder transition densities. 1.
The Running Title: Inference for Semiparametric Models
, 2001
"... Nonand semiparametric models have flourished during the last twenty five years. They have been applied widely and their theoretical properties have been studied extensively. We briefly review some of their development and list a few questions that we would like to see addressed. We develop an ans ..."
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Nonand semiparametric models have flourished during the last twenty five years. They have been applied widely and their theoretical properties have been studied extensively. We briefly review some of their development and list a few questions that we would like to see addressed. We develop an answer to one of these questions by formulating a ‘calculus ’ similar to that of the i.i.d. case that enables us to analyze the efficiency of procedures in general semiparametric models when a nonparametric model has been defined. Our approach is illustrated by applying it to regression models, counting process models in survival analysis and submodels of Markov chains, which traditionally require elaborate special arguments. In the examples, the calculus lets us easily read off the structure of efficient estimators and check if a candidate estimator is efficient.
Universität Bremen
"... We illustrate several recent results on efficient estimation for semiparametric time series models with two types of AR(1) models: having independent and centered innovations, and having general and conditionally centered innovations. We consider in particular estimation of the autoregression param ..."
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We illustrate several recent results on efficient estimation for semiparametric time series models with two types of AR(1) models: having independent and centered innovations, and having general and conditionally centered innovations. We consider in particular estimation of the autoregression parameter, the stationary distribution, the innovation distribution, and the stationary density. 1
order moving average and autoregressive representations
"... Abstract. An invertible causal linear process is a process which has infinite order moving average and autoregressive representations. We assume that the coefficients in these representations depend on a Euclidean parameter, while the corresponding innovations have an unknown centered distribution ..."
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Abstract. An invertible causal linear process is a process which has infinite order moving average and autoregressive representations. We assume that the coefficients in these representations depend on a Euclidean parameter, while the corresponding innovations have an unknown centered distribution with some moment restrictions. We discuss efficient estimation of differentiable functionals in such a semiparametric model. For this we first obtain a suitable semiparametric version of local asymptotic normality and then use Hájek’s convolution theorem to characterize efficient estimators. Then we apply this result to construct efficient estimators of the Euclidean parameter and of linear functionals of the innovation distribution. Key words: Time series, nonlinear process of increasing order, empirical estimator, local asymptotic normality, asymptotically linear estimator, influence function, adaptive estimator, regular estimator, least dispersed estimator, contiguity.