Pre-averaged kernel estimators for the drift function of a diffusion process in the presence of microstructure noise

We consider estimation of the drift function of a stationary diffusion process when we observe high-frequency data with microstructure noise over a long time interval. We propose to estimate the drift function at a point by a Nadaraya–Watson estimator that uses observations that have been pre-averaged to reduce the noise. We give conditions under which our estimator is consistent and asympotically normal. Its rate and asymptotic bias and variance are the same as those without microstructure noise. To use our method in data analysis, we propose a data-based cross-validation method to determine the bandwidth in the Nadaraya–Watson estimator. Via simulation, we study several methods of bandwidth choices, and compare our estimator to several existing estimators. In terms of mean squared error, our new estimator outperforms existing estimators.     

Tests for normality based on density estimators of convolutions

Recent results show that densities of convolutions can be estimated by local U-statistics at the root-n rate in various norms. Motivated by this and the fact that convolutions of normal densities are normal, we introduce new tests for normality which use as test statistics weighted L₁-distances between the standard normal density and local U-statistics based on standardized observations. We show that such test statistics converge at the root-n rate and determine their limit distributions as functionals of Gaussian processes. We also address a choice of bandwidth. Simulations show that our tests are competitive with other tests of normality.     

A counterexample concerning monotone unimodality

In answer to a question of Dharmadhikari and Jogdeo (1976) we show that there exist multivariate distributions which are symmetric, star unimodal, and linear unimodal, but not monotone unimodal.     

An asymptotically complete class of tests

Summary This paper investigates sequences of asymptotically similar critical regions {S _n >0},n, under the assumption that the test-statisticS _n admits a certain stochastic expansion. It is shown that for such test-sequences, first order efficiency implies second order efficiency (i.e. efficiency up to an error termo(n ^–1/2)). Moreover, the asymptotic power functions of first order efficient test-sequences are determined up to an error termo(n ^–1), and a class of critical regions is specified which is minimal essentially complete up too(n ^–1).The results of this paper rest upon the technique of Edgeworth-expansions and are, therefore, restricted to continuous probability distributions.     

Addendum to “a third-order optimum property of the maximum likelihood estimator”

A third-order optimum property of the maximum likelihood estimator is extended to not necessarily symmetric loss functions under an appropriate restriction on the class of competing estimators.     

Estimating Joint Distributions of Markov Chains

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.     

Estimating the Innovation Distribution in Nonlinear Autoregressive Models

The usual estimator for the expectation of a function under the innovation distribution of a nonlinear autoregressive model is the empirical estimator based on estimated innovations. It can be improved by exploiting that the innovation distribution has mean zero. We show that the resulting estimator is efficient if the innovations are estimated with an efficient estimator for the autoregression parameter. Efficiency of this estimator is necessary except when the expectation of the function can be estimated adaptively. Analogous results hold for heteroscedastic models.     

Characterizing Efficient Empirical Estimators for Local Interaction Gibbs Fields

The expectation of a local function on a stationary random field can be estimated from observations in a large window by the empirical estimator, that is, the average of the function over all shifts within the window. Under appropriate conditions, the estimator is consistent and asymptotically normal. Suppose that the field is a Gibbs field with known finite range of interactions but otherwise unknown potential. We show that the empirical estimator is efficient if and only if the function is (equivalent to) a sum of functions each of which depends only on the values of the field on a clique of sites. This revised version was published online in June 2006 with corrections to the Cover Date.     

A third-order optimum property of the maximum likelihood estimator

Let θ⁽ⁿ⁾ denote the maximum likelihood estimator of a vector parameter, based on an i.i.d. sample of size n. The class of estimators θ⁽ⁿ⁾ + n^?1q(θ⁽ⁿ⁾), with q running through a class of sufficiently smooth functions, is essentially complete in the following sense: For any estimator T⁽ⁿ⁾ there exists q such that the risk of θ⁽ⁿ⁾ + n^?1q(θ⁽ⁿ⁾) exceeds the risk of T⁽ⁿ⁾ by an amount of order o(n^?1) at most, simultaneously for all loss functions which are bounded, symmetric, and neg-unimodal. If $\text{q}{}^1$ is chosen such that $\text{θ}{}^{\mathrm{(n)}}\text{+ n}{}^{\mathrm{?1}}\text{q}{}^1\text{(θ}{}^{\mathrm{(n)}}\text{)}$ is unbiased up to $\text{o(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$, then this estimator minimizes the risk up to an amount of order o(n^?1) in the class of all estimators which are unbiased up to $\text{o(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$.The results are obtained under the assumption that T⁽ⁿ⁾ admits a stochastic expansion, and that either the distributions have—roughly speaking—densities with respect to the lebesgue measure, or the loss functions are sufficiently smooth.     

Root-<Emphasis Type="Italic">n</Emphasis> consistency in weighted <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>-spaces for density estimators of invertible linear processes

The stationary density of an invertible linear processes can be estimated at the parametric rate by a convolution of residual-based kernel estimators. We have shown elsewhere that the convergence is uniform and that a functional central limit theorem holds in the space of continuous functions vanishing at infinity. Here we show that analogous results hold in weighted L ₁-spaces. We do not require smoothness of the innovation density.