Possible Rate Of Convergence Research Articles

Designing Realized Kernels to Measure the ex post Variation of Equity Prices in the Presence of Noise

This paper shows how to use realised kernels to carry out efficient feasible inference on the expost variation of underlying equity prices in the presence of simple models of market frictions. The issue is subtle with only estimators which have symmetric weights delivering consistent estimators with mixed Gaussian limit theorems. The weights can be chosen to achieve the best possible rate of convergence and to have an asymptotic variance which is close to that of the maximum likelihood estimator in the parametric version of this problem. Realised kernels can also be selected to (i) be analysed using endogenously spaced data such as that in databases on transactions, (ii) allow for market frictions which are endogenous, (iii) allow for temporally dependent noise. The finite sample performance of our estimators is studied using simulation, while empirical work illustrates their use in practice.

MATRIX SET MEASURE FOR STABILITY OF SWITCHED SYSTEMS

For discrete-time switched linear systems, the largest possible rate of convergence is clearly characterized by its generalized (joint) spectral radius, which is also the least possible common norm. In this work, we prove that, for continuous-time switched linear systems, the largest convergence rate is exactly the least possible common matrix set measure of the subsystems. This reduce the gap between the stability theories for discrete-time and continuous-time systems.

Almost sure central limit theorem without logarithmic sums

We investigate possible rates of convergence in the almost sure central limit theorem for sums of independent random variables and martingales. Bibliography: 9 titles.

NONLINEAR FILTERING OF SMOOTH SIGNALS

A nonlinear online Kalman type filter is proposed for the estimation of unknown function S(t) with the known smoothness β for the diffusion observed process with small, of the order ε2, diffusion coefficient. Assuming that the drift coefficient of the observed process depends on an unknown function S(t), we propose an approach to the analysis of this estimator based on the Lyapunov's functions method. The best possible rate of convergence of risks to 0, as ε → 0, is proven for β ≤ 2.

Automatic Block-Length Selection for the Dependent Bootstrap

We review the different block bootstrap methods for time series, and present them in a unified framework. We then revisit a recent result of Lahiri [Lahiri, S. N. (1999b). Theoretical comparisons of block bootstrap methods, Ann. Statist. 27:386–404] comparing the different methods and give a corrected bound on their asymptotic relative efficiency; we also introduce a new notion of finite-sample “attainable” relative efficiency. Finally, based on the notion of spectral estimation via the flat-top lag-windows of Politis and Romano [Politis, D. N., Romano, J. P. (1995). Bias-corrected nonparametric spectral estimation. J. Time Series Anal. 16:67–103], we propose practically useful estimators of the optimal block size for the aforementioned block bootstrap methods. Our estimators are characterized by the fastest possible rate of convergence which is adaptive on the strength of the correlation of the time series as measured by the correlogram.

Convergence of the Poincaré Constant

The Poincaré constant $R_Y$ of a random variable Y relates the $L^2(Y)$-norm of a function g and its derivative $g'$. Since $R_Y - {\bf D}(Y)$ is positive, with equality if and only if Y is normal, it can be seen as a distance from the normal distribution. In this paper we establish the best possible rate of convergence of this distance in the central limit theorem. Furthermore, we show that $R_Y$ is finite for discrete mixtures of normals, allowing us to add rates to the proof of the central limit theorem in the sense of relative entropy.

Local Linear Estimation in Partly Linear Models

Let (X, B, Y) denote a random vector such thatBandYare real-valued, andX∈R2. Local linear estimates are used in the partial regression method for estimating the regression functionE(Y∣X, B)=αB+m(X), whereαis an unknown parameter, andm(·) is a smooth function. Under appropriate conditions, asymptotic distributions of estimates ofαandm(·) are established. Moreover, it is shown that these estimates achieve the best possible rates of convergence in the indicated semi-parametric problems.

Sensitivity analysis of stationary performance measures for Markov chains

Derivative estimation is an important problem in performance analysis of discrete event dynamic systems. Derivative estimation of stationary performance measures is difficult since it generally requires the consistency of estimators. This paper proposes an algorithm for derivative estimation of stationary performance measures for Markov chains. Both discrete-time and continuous-time chains are considered. The basic idea is to simulate the original Markov chain with a modified performance measure which can be estimated by extra simulations. The computational load of the extra simulations at each step is bounded. It is shown under mild assumptions that the algorithm attains the best possible rate of convergence as the simulation time goes to infinity. An unexpected connection between the algorithm and solutions to Poisson equations is also revealed.

Recovering band-limited signals under noise

We consider the problem of recovering a band-limited signal f(t) from noisy data yk=f(k/spl tau/)+/spl Gt/epsilon//sub k/, where /spl tau/ is the sampling rate. Starting from the truncated Whittaker-Shannon cardinal expansion with or without sampling windows (both cases yield inconsistent estimates of f(t)) we propose estimators that are convergent to f(t) in the pointwise and uniform sense. The basic idea is to cut down high frequencies in the data and to use suitable oversampling /spl tau//spl les//spl pi///spl Omega/, /spl Omega/ being the bandwidth (maximum frequency) of f(t). The simplest estimator we propose is given by f/spl circ//sub n/(t)=/spl tau/ /spl Sigma//|t-k/spl tau/|/spl les/n/spl tau/ yksin(/spl Omega/(t-k/spl tau/))//spl pi/(t-k/spl tau/),|t|/spl les/n/spl tau/. Generalizations of f/spl circ//sub n/ including sampling windows are also examined. The main aim is to examine the mean squared error (MSE) properties of such estimators in order to determine the optimal choice of the sampling rate /spl tau/ yielding the fastest possible rate of convergence. The best rate for the MSE we obtain is O(In(n)/n).

Bent functions and random boolean formulas

Let α be a nonlinear boolean connective with equal number of zero and unit entries in its table. We study an iterative process of combining randomly chosen boolean functions from some starting set G via the connective α. We suppose that all functions in G are defined on the same finite nonempty domain M. We are interested in the situations, when the process converges to the uniform distribution on {0, 1} M . We classify the boolean connectives α according to the asymptotic rate of this convergence. Although the probability of any function in our process converges to ( 1 2 ) ¦M , there are some differences in the terms of lower order of magnitude. If M is the boolean cube of an even dimension n and G is the set of all linear boolean functions of n variables and the connective α belongs to the class of the lowest possible rate of convergence in the above-mentioned classification, we can express the main nonconstant term of the asymptotic expansion of the probability of occurring of a single function f through the Fourier transform of f. Using this, we prove that the bent functions achieve asymptotically the minimal probability of occurring among all boolean functions. At the same time, the linear functions achieve asymptotically the maximal probability.

Optimal Rate of Convergence for Finite Mixture Models

In finite mixture models, we establish the best possible rate of convergence for estimating the mixing distribution. We find that the key for estimating the mixing distribution is the knowledge of the number of components in the mixture. While a $\sqrt n$-consistent rate is achievable when the exact number of components is known, the best possible rate is only $n^{-1/4}$ when it is unknown. Under a strong identifiability condition, it is shown that this rate is reached by some minimum distance estimators. Most commonly used models are found to satisfy the strong identifiability condition.

Convergence Rate of Sieve Estimates

In this paper, we develop a general theory for the convergence rate of sieve estimates, maximum likelihood estimates (MLE's) and related estimates obtained by optimizing certain empirical criteria in general parameter spaces. In many cases, especially when the parameter space is infinite dimensional, maximization over the whole parameter space is undesirable. In such cases, one has to perform maximization over an approximating space (sieve) of the original parameter space and allow the size of the approximating space to grow as the sample size increases. This method is called the method of sieves. In the case of the maximum likelihood estimation, an MLE based on a sieve is called a sieve MLE. We found that the convergence rate of a sieve estimate is governed by (a) the local expected values, variances and $L_2$ entropy of the criterion differences and (b) the approximation error of the sieve. A robust nonparametric regression problem, a mixture problem and a nonparametric regression problem are discussed as illustrations of the theory. We also found that when the underlying space is too large, the estimate based on optimizing over the whole parameter space may not achieve the best possible rates of convergence, whereas the sieve estimate typically does not suffer from this difficulty.

Asymptotically best bandwidth selectors in kernel density estimation

This paper gives asymptotically best data based choices of the bandwidth of the kernel density estimator. These bandwith selectors attain the fastest possible rate of convergence to the desired theoretical optimum and the best possible constant coefficient in the spirit of the usual Fisher Information, with the use of only nonnegative kernel estimators at all stages of the selection process.

Optimal Rates of Convergence of Parameter Estimators in the Binary Response Model with Weak Distributional Assumptions

The smoothed maximum score estimator of the coefficient vector of a binary response model is consistent and, after centering and suitable normalization, asymptotically normally distributed under weak assumptions [5]. Its rate of convergence in probability is N−h/(2h+1), where h ≥ 2 is an integer whose value depends on the strength of certain smoothness assumptions. This rate of convergence is faster than that of the maximum score estimator of Manski [11,12], which converges at the rate N−1/3 under assumptions that are somewhat weaker than those of the smoothed estimator. In this paper I prove that under the assumptions of smoothed maximum score estimation, N−h/(2h+1) is the fastest achievable rate of convergence of an estimator of the coefficient vector of a binary response model. Thus, the smoothed maximum score estimator has the fastest possible rate of convergence. The rate of convergence is defined in a minimax sense so as to exclude superefficient estimators.

Efficient Algorithm for Adaptive Control for a Class of Mimo Plant

Abstract An indirect adaptive control algorithm for a MIMO plant is studied. It is shown that Polyak - Ruppert estimation algorithm along with the simple dead-beat control law constitutes an adaptive control strategy that achieves the highest possible rate of convergence for the quadratic criterion.

Acceleration of Stochastic Approximation by Averaging

A new recursive algorithm of stochastic approximation type with the averaging of trajectories is investigated. Convergence with probability one is proved for a variety of classical optimization and identification problems. It is also demonstrated for these problems that the proposed algorithm achieves the highest possible rate of convergence.

A Simple Root $n$ Bandwidth Selector

The asymptotically best bandwidth selectors for a kernel density estimator currently require the use of either unappealing higher order kernel pilot estimators or related Fourier transform methods. The point of this paper is to present a methodology which allows the fastest possible rate of convergence with the use of only nonnegative kernel estimators at all stages of the selection process. The essential idea is derived through careful study of factorizations of the pilot bandwidth in terms of the original bandwidth.

Discrimination distance bounds and statistical applications

Bounds are obtained for the Kullback-Leibler discrimination distance between two random vectorsX andY. IfX is a sequence of independent random variables whose densities have similar tail behavior andY=AX, whereA is an invertible matrix, then the bounds are a product of terms depending onA andX separately. We apply these bounds to obtain the best possible rate of convergence for any estimator of the parameters of an autoregressive process with innovations in the domain of attraction of a stable law. We provide a general theorem establishing the link between total variation proximity of measures and the rate of convergence of statistical estimates to complete the exposition for this application.

Approximation by polynomials with locally geometric rates

In contrast to the behavior of best uniform polynomial approximants on [ 0 , 1 ] [0,1] we show that if f ∈ C [ 0 , 1 ] f \in C[0,1] there exists a sequence of polynomials { P n } \{ {P_n}\} of respective degree ≤ n \leq n which converges uniformly to f f on [ 0 , 1 ] [0,1] and geometrically fast at each point of [ 0 , 1 ] [0,1] where f f is analytic. Moreover we describe the best possible rates of convergence at all regular points for such a sequence.

On Optimal Algorithms in an Asymptotic Model with Gaussian Measure

We study the approximate solution of linear problems in separable Hilbert spaces equipped with a Gaussian measure. We find information and algorithms with the best possible rate of convergence. Although adaptive information and nonlinear algorithms are permitted, we prove that nonadaptive information and linear algorithms are optimal. An algorithm is optimal if it converges with a rate of convergence that is no worse than the rate of any other algorithm except on sets of measure zero. We prove that algorithms and information that minimize the average errors lead to the best possible rate of convergence. This exhibits a close relation between the asymptotic and average case models.