Results 1  10
of
17
ON METHODS OF SIEVES AND PENALIZATION
, 1997
"... We develop a general theory which provides a unified treatment for the asymptotic normality and efficiency of the maximum likelihood estimates (MLE’s) in parametric, semiparametric and nonparametric models. We find that the asymptotic behavior of substitution estimates for estimating smooth function ..."
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Cited by 61 (1 self)
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We develop a general theory which provides a unified treatment for the asymptotic normality and efficiency of the maximum likelihood estimates (MLE’s) in parametric, semiparametric and nonparametric models. We find that the asymptotic behavior of substitution estimates for estimating smooth functionals are essentially governed by two indices: the degree of smoothness of the functional and the local size of the underlying parameter space. We show that when the local size of the parameter space is not very large, the substitution standard (nonsieve), substitution sieve and substitution penalized MLE’s are asymptotically efficient in the Fisher sense, under certain stochastic equicontinuity conditions of the loglikelihood. Moreover, when the convergence rate of the estimate is slow, the degree of smoothness of the functional needs to compensate for the slowness of the rate in order to achieve efficiency. When the size of the parameter space is very large, the standard and penalized maximum likelihood procedures may be inefficient, whereas the method of sieves may be able to overcome this difficulty. This phenomenon is particularly manifested when the functional of interest is very smooth, especially in the semiparametric case.
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...
Outperforming the Gibbs sampler empirical estimator for nearest neighbor random fields
, 1996
"... Given a Markov chain sampling scheme, does the standard empirical estimator make best use of the data? We show that this is not so and construct better estimators. We restrict attention to nearest neighbor random fields and to Gibbs samplers with deterministic sweep, but our approach applies to any ..."
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Cited by 6 (3 self)
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Given a Markov chain sampling scheme, does the standard empirical estimator make best use of the data? We show that this is not so and construct better estimators. We restrict attention to nearest neighbor random fields and to Gibbs samplers with deterministic sweep, but our approach applies to any sampler that uses reversible variableatatime updating with deterministic sweep. The structure of the transition distribution of the sampler is exploited to construct further empirical estimators that are combined with the standard empirical estimator to reduce asymptotic variance. The extra computational cost is negligible. When the random field is spatially homogeneous, symmetrizations of our estimator lead to further variance reduction. The performance of the estimators is evaluated in a simulation study of the Ising model.
Estimating Joint Distributions Of Markov Chains
 IN PREPARATION
, 1998
"... Suppose we observe a stationary Markov chain with unknown transition distribution. The empirical estimator for the expectation of a function of two successive observations is known to be efficient. For reversible Markov chains, an appropriate symmetrization is efficient. For functions of more than ..."
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Cited by 5 (3 self)
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Suppose we observe a stationary Markov chain with unknown transition distribution. The empirical estimator for the expectation of a function of two successive observations is known to be efficient. For reversible Markov chains, an appropriate symmetrization is efficient. For functions of more than two arguments, these estimators cease to be efficient. We determine the influence function of efficient estimators of expectations of functions of several observations, both for completely unknown and for reversible Markov chains. We construct simple efficient estimators in both cases.
ON NONPARAMETRIC TESTS OF POSITIVITY/MONOTONICITY/CONVEXITY
, 2002
"... We consider the problem of estimating the distance from an unknown signal, observed in a whitenoise model, to convex cones of positive/monotone/convex functions. We show that, when the unknown function belongs to a Hölder class, the risk of estimating the Lrdistance, 1 ≤ r<∞, from the signal to ..."
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Cited by 4 (0 self)
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We consider the problem of estimating the distance from an unknown signal, observed in a whitenoise model, to convex cones of positive/monotone/convex functions. We show that, when the unknown function belongs to a Hölder class, the risk of estimating the Lrdistance, 1 ≤ r<∞, from the signal to a cone is essentially the same (up to a logarithmic factor) as that of estimating the signal itself. The same risk bounds hold for the test of positivity, monotonicity and convexity of the unknown signal. We also provide an estimate for the distance to the cone of positive functions for which risk is, by a logarithmic factor, smaller than that of the “plugin ” estimate.
Efficient estimation of linear functionals of a bivariate distribution with equal, but unknown marginals: the leastsquares approach
 Journal of Multivariate Analysis
, 2005
"... In this paper we characterize and construct ecient estimators of linear functionals of a bivariate distribution with equal marginals. An ecient estimator equals the empirical estimator minus a correction term and provides signicant improvements over the empirical estimator. We construct an ecient es ..."
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Cited by 4 (3 self)
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In this paper we characterize and construct ecient estimators of linear functionals of a bivariate distribution with equal marginals. An ecient estimator equals the empirical estimator minus a correction term and provides signicant improvements over the empirical estimator. We construct an ecient estimator by estimating the correction term. For this we use the least squares principle and an estimated orthonormal basis for the Hilbert space of squareintegrable functions under the unknown equal marginal distribution. Simulations conrm the asymptotic behavior of this estimator in moderate sample sizes and the considerable theoretical gains over the empirical estimator. 1. Introduction. Let (X1; Y1); : : : ; (Xn; Yn) be independent copies of a bivariate random vector (X;Y) with distribution Q. Let be a measurable function from R2 to R such that R
Estimators for Models with Constraints Involving Unknown Parameters
"... Suppose we have independent observations from a distribution which we know to fulll a finitedimensional linear constraint involving an unknown finitedimensional parameter. We construct efficient estimators for finitedimensional functionals of the distribution. The estimators are obtained by first ..."
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Cited by 3 (3 self)
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Suppose we have independent observations from a distribution which we know to fulll a finitedimensional linear constraint involving an unknown finitedimensional parameter. We construct efficient estimators for finitedimensional functionals of the distribution. The estimators are obtained by first constructing an efficient estimator for the functional when the parameter is known, and then replacing the parameter by an efficient estimator. We consider in particular estimation of expectations.
Asymptotically efficient nonparametric estimation of functionals of spectral density with zeros, Theory Probab.
 Appl.
, 1988
"... Abstract. The paper considers a problem of construction of asymptotically efficient estimators for functionals defined on a class of spectral densities. We define the concepts of H 0 and IKefficiency of estimators, based on the variants of HájekIbragimovKhas'minskii convolution theorem and ..."
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Cited by 3 (1 self)
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Abstract. The paper considers a problem of construction of asymptotically efficient estimators for functionals defined on a class of spectral densities. We define the concepts of H 0 and IKefficiency of estimators, based on the variants of HájekIbragimovKhas'minskii convolution theorem and HájekLe Cam local asymptotic minimax theorem, respectively. We prove that ( θ T ), where θ T is a suitable sequence of T 1/2 consistent estimators of unknown spectral density θ(λ), is H 0 and IKasymptotically efficient estimator for a nonlinear smooth functional (θ). Mathematics Subject Classifications
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 ...