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Application of Convolution Theorems in Semiparametric Models with noni.i.d. Data
"... A useful approach to asymptotic e ciency for estimators in semiparametric models is the study of lower bounds on asymptotic variances via convolution theorems. Such theorems are often applicable in models in which the classical assumptions of independence and identical distributions fail to hold, bu ..."
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A useful approach to asymptotic e ciency for estimators in semiparametric models is the study of lower bounds on asymptotic variances via convolution theorems. Such theorems are often applicable in models in which the classical assumptions of independence and identical distributions fail to hold, but to date, much of the research has focused on semiparametric models with independent and identically distributed (i.i.d.) data because tools are available in the i.i.d. setting for verifying preconditions of the convolution theorems. We develop tools for noni.i.d. data that are similar in spirit to those for i.i.d. data and also analogous to the approaches used in parametric models with dependent data. This involves extending the notion of the tangent vector guring so prominently in the i.i.d. theory and providing conditions for smoothness, or differentiability, of the parameter of interest as a function of the underlying probability measures. As a corollary to the differentiability result we obtain sufficient conditions for equivalence, in terms of asymptotic variance bounds, of two models. Regularity and asymptotic linearity of estimators are also discussed.
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.
Splitting Markov fields and combining empirical estimators
 In preparation
, 1999
"... We have shown elsewhere that the empirical estimator for the expectation of a local function on a Markov field over a lattice is efficient if and only if the function is a sum of functions each of which depends only on the values of the field on a clique of sites. For countable state space, the esti ..."
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Cited by 2 (2 self)
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We have shown elsewhere that the empirical estimator for the expectation of a local function on a Markov field over a lattice is efficient if and only if the function is a sum of functions each of which depends only on the values of the field on a clique of sites. For countable state space, the estimation of such expectations reduces to the estimation of probabilities of configurations over finite subsets of the lattice. The corresponding empirical estimator is efficient if and only if the set is a clique. If the set is not a clique, we can construct better estimators. They are rational functions of empirical estimators for configurations over subsets of the given set. The construction is based on a factorization of probabilities of configurations which makes use of the splitting property of Markov fields. AMS 1991 subject classifications. 62G05, 62M40. Key words and Phrases. Empirical estimator, improved estimator, Markov splitting, local interactions, random field. 1 Introduction ...
Efficient Estimation of Dynamical systems
"... The aim of this paper is to show a simple way to construct asymptotic minimax lower bounds for risks based on di#erent types of quadratic loss functions in semiparametric inference problems. For the sake of clarity, we consider the simple case of the state estimation of a dynamical system with s ..."
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The aim of this paper is to show a simple way to construct asymptotic minimax lower bounds for risks based on di#erent types of quadratic loss functions in semiparametric inference problems. For the sake of clarity, we consider the simple case of the state estimation of a dynamical system with small noise. The proofs are based on the van Trees inequality, namely an integral type CramerRao inequality.
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.
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"... On optimality of the empirical distribution function for the estimation of the invariant distribution function of a diffusion process ∗ Ilia Negri† In this work we present some results on the optimality of the empirical distribution function as estimator of the invariant distribution function of an ..."
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On optimality of the empirical distribution function for the estimation of the invariant distribution function of a diffusion process ∗ Ilia Negri† In this work we present some results on the optimality of the empirical distribution function as estimator of the invariant distribution function of an ergodic diffusion process. The results presented were obtained in different previous works under conditions that are are rewritten in a unified form that make comparable those results. It is well known that the empirical distribution function is an unbiased and uniformly consistent estimator for the invariant distribution function of an ergodic diffusion process. It is also an efficient estimator in the sense that the risk of this estimator attains an asymptotic minimax lower bound. In this paper we review some results on the problem of the efficiency of the empirical distribution function considering three types of risk function. The first one is in a semiparametric setup. The second one is the integrated mean square error and the third is based on the sup norm. key words: invariant distribution function, efficiency, lower bound, efficient estimator. 2000 MSC: 60G35, 62M20. ∗This work has been supported by the local Grant sponsorized by University of Bergamo: Theoretical and computational problems in statistics for continuously and discretely observed