Abstract-In the absence of knowledge of the true density function, Bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect to a prior. We examine the relative entropy distance D,, between the true density and the Bayesian density and show that the asymptotic distance is (d/2Xlogn)+ c, where d is the dimension of the parameter vector. Therefore, the relative entropy rate D,,/n converges to zero at rate (logn)/n. The constant c, which we explicitly identify, depends only on the prior density function and the Fisher information matrix evaluated at the true parameter value. Consequences are given for density estima-tion, universal data compression, composite hypothesis testing, and stock-market portfolio selection. 1.

Outcome-dependent, two-phase sampling designs can dramatically reduce the costs of observational studies by judicious selection of the most informative subjects for purposes of detailed covariate measurement. Here we derive asymptotic information bounds and the form of the efficient score and inuence functions for the semiparametric regression models studied by Lawless, Kalbfleisch, and Wild (1999) under two-phase sampling designs. We relate the efficient score to the least-favorable parametric submodel by use of formal calculations suggested by Newey (1994). We then proceed to show that the maximum likelihood estimators proposed by Lawless, Kalbfleisch, and Wild (1999) for both the parametric and nonparametric parts of the model are asymptotically normal and efficient, and that the efficient influence function for the parametric part agrees with the more general calculations of Robins, Hsieh, and Newey (1995).

This paper establishes the asymptotic distribution of an extremum estimator when the true parameter lies on the boundary of the parameter space. The boundary may be linear, curved, and�or kinked. Typically the asymptotic distribution is a function of a multivariate normal distribution in models without stochastic trends and a function of a multivariate Brownian motion in models with stochastic trends. The results apply to a wide variety of estimators and models. Examples treated in the paper are: Ž. i quasi-ML estimation of a random coefficients regression model with some coefficient variances equal to zero and Ž ii. LS estimation of an augmented Dickey-Fuller regression with unit root and time trend parameters on the boundary of the parameter space.

Introduction. We discuss the convergence of M--estimators to a stable (possibly normal) limit distribution. Huber (1964) introduced M--estimators as a way to obtain more robust estimators. Let (S; S; P ) be a probability space and let fX i g 1 i=1 be a sequence of i.i.d.r.v.'s with values in S. Let X be a copy of X 1 . Let \Theta be a subset of IR d . Let g : S \Theta \Theta ! IR be a function such that g(\Delta; `) : S ! IR is measurable for each ` 2 \Theta. Suppose that we want to estimate a parameter ` 0 2 \Theta characterized by E[g(X; `) \Gamma g(X; `<F8.496

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Suppose we have specified a parametric model for the transition distribution of a Markov chain, but that the true transition distribution does not belong to the model. Then the maximum likelihood estimator estimates the parameter which maximizes the Kullback--Leibler information between the true transition distribution and the model. We prove that the maximum likelihood estimator is asymptotically efficient in a nonparametric sense if the true transition distribution is unknown. 1 Introduction Suppose we observe X 0 ; : : : ; X n from an ergodic Markov chain on an arbitrary state space. We have specified a parametric model Q # (x; dy) for the transition distribution, and an initial distribution j 0 (dx). Consider the following two situations: 1. We believe, erroneously, that the model is correct, and use the maximum likelihood estimator for estimating the parameter. 2. We know that the model is incorrect, and want to fit a transition distribution from the model to the true transition ...

Abstract: To date the literature on quantile regression and least absolute deviation regression has assumed either explicitly or implicitly that the conditional quantile regression model is correctly specified. When the model is misspecified, confidence intervals and hypothesis tests based on the conventional covariance matrix are invalid. Although misspecification is a generic phenomenon and correct specification is rare in reality, there has to date been no theory proposed for inference when a conditional quantile model may be misspecified. In this paper, we allow for possible misspecification of a linear conditional quantile regression model. We obtain consistency of the quantile estimator for certain “pseudo-true ” parameter values and asymptotic normality of the quantile estimator when the model is misspecified. In this case, the asymptotic covariance matrix has a novel form, not seen in earlier work, and we provide a consistent estimator of the asymptotic covariance matrix. We also propose a quick and simple test for conditional quantile misspecification based on the quantile residuals.

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. By means of two simple convexity arguments we are able to develop a general method for proving consistency and asymptotic normality of estimators that are defined by minimisation of convex criterion functions. This method is then applied to a fair range of different statistical estimation problems, including Cox regression, logistic and Poisson regression, least absolute deviation regression outside model conditions, and pseudo-likelihood estimation for Markov chains. Our paper has two aims. The first is to exposit the method itself, which in many cases, under reasonable regularity conditions, leads to new proofs that are simpler than the traditional proofs. Our second aim is to exploit the method to its limits for logistic regression and Cox regression, where we seek asymptotic results under as weak regularity conditions as possible. For Cox regression in particular we are able to weaken previously published regularity conditions substantially. Key words: argmin lemma approximation...

Parametric copulas are shown to be attractive devices for specifying quantile autoregressive models for nonlinear time-series. Estimation of local, quantile-specific copula-based time series models offers some salient advantages over classical global parametric approaches. Consistency and asymptotic normality of the proposed quantile estimators are established under mild conditions, allowing for global misspecification of parametric copulas and marginals, and without assuming any mixing rate condition. These results lead to a general framework for inference and model specification testing of extreme conditional value-at-risk for financial time series data.