Best Possible Rate Of Convergence Research Articles

A ridge estimator of the drift from discrete repeated observations of the solution of a stochastic differential equation

This work focuses on the nonparametric estimation of a drift function from N discrete repeated independent observations of a diffusion process over a fixed time interval [0,T]. We study a ridge estimator obtained by the minimization of a constrained least squares contrast. The resulting projection estimator is based on the B-spline basis. Under mild assumptions, this estimator is universally consistent with respect to an integrate norm. We establish that, up to a logarithmic factor and when the estimation is performed on a compact interval, our estimation procedure reaches the best possible rate of convergence. Furthermore, we build an adaptive estimator that achieves this rate. Finally, we illustrate our procedure through an intensive simulation study which highlights the good performance of the proposed estimator in various models.

Fast component-by-component construction of lattice algorithms for multivariate approximation with POD and SPOD weights

In a recent paper by the same authors, we provided a theoretical foundation for the component-by-component (CBC) construction of lattice algorithms for multivariate $L_2$ approximation in the worst case setting, for functions in a periodic space with general weight parameters. The construction led to an error bound that achieves the best possible rate of convergence for lattice algorithms. Previously available literature covered only weights of a simple form commonly known as

Optimal Order Quadrature Error Bounds for Infinite-Dimensional Higher-Order Digital Sequences

Quasi-Monte Carlo (QMC) quadrature rules using higher order digital nets and sequences have been shown to achieve the almost optimal rate of convergence of the worst-case error in Sobolev spaces of arbitrary fixed smoothness $\alpha\in \mathbb{N}$, $\alpha\geq 2$. In a recent paper by the authors, it was proved that randomly-digitally-shifted order $2\alpha$ digital nets in prime base $b$ achieve the best possible rate of convergence of the root mean square worst-case error of order $N^{-\alpha}(\log N)^{(s-1)/2}$ for $N=b^m$, where $N$ and $s$ denote the number of points and the dimension, respectively, which implies the existence of an optimal order QMC rule. More recently, the authors provided an explicit construction of such an optimal order QMC rule by using Chen-Skriganov's digital nets in conjunction with Dick's digit interlacing composition. These results were for fixed number of points. In this paper we give a more general result on an explicit construction of optimal order QMC rules for arbitrary fixed smoothness $\alpha\in \mathbb{N}$ including the endpoint case $\alpha=1$. That is, we prove that the projection of any infinite-dimensional order $2\alpha +1$ digital sequence in prime base $b$ onto the first $s$ coordinates achieves the best possible rate of convergence of the worst-case error of order $N^{-\alpha}(\log N)^{(s-1)/2}$ for $N=b^m$. The explicit construction presented in this paper is not only easy to implement but also extensible in both $N$ and $s$.

Asymptotics for [formula omitted]-value based threshold estimation under repeated measurements

We investigate the large sample behavior of a p-value based procedure for estimating the threshold level at which a regression function takes off from its baseline value, a problem arising in dose–response studies, engineering and other related fields. We study the procedure under the “repeated measurement” setting, where several responses can be obtained at each covariate-level. The estimator is constructed via fitting a “stump” function to approximate p-values that test for deviation of the regression function from its baseline level. The smoothness of the regression function in the vicinity of the threshold determines the best possible rate of convergence using our method: a “cusp” of order k at the threshold yields a rate of N−1/(2k+1), where N is the size/budget of the problem (total number of responses across all values of covariates). The asymptotic distribution of the normalized estimator is shown to be the minimizer of a generalized compound Poisson process and confidence intervals based on this distribution are proposed. The method is illustrated on simulations and an application to data from a complex queuing system is presented.

An Explicit Construction of Optimal Order Quasi--Monte Carlo Rules for Smooth Integrands

In a recent paper by the authors, it is shown that there exists a quasi-Monte Carlo (QMC) rule which achieves the best possible rate of convergence for numerical integration in a reproducing kernel Hilbert space consisting of smooth functions. In this paper we provide an explicit construction of such an optimal order QMC rule. Our approach is to exploit both the decay and the sparsity of the Walsh coefficients of the reproducing kernel simultaneously. This can be done by applying digit interlacing composition due to Dick to digital nets with large minimum Hamming and Niederreiter-Rosenbloom-Tsfasman metrics due to Chen and Skriganov. To our best knowledge, our construction gives the first QMC rule which achieves the best possible convergence in this function space.

Quasi-monotone Subgradient Methods for Nonsmooth Convex Minimization

In this paper, we develop new subgradient methods for solving nonsmooth convex optimization problems. These methods guarantee the best possible rate of convergence for the whole sequence of test points. Our methods are applicable as efficient real-time stabilization tools for potential systems with infinite horizon. Preliminary numerical experiments confirm a high efficiency of the new schemes.

Universal gradient methods for convex optimization problems

In this paper, we present new methods for black-box convex minimization. They do not need to know in advance the actual level of smoothness of the objective function. Their only essential input parameter is the required accuracy of the solution. At the same time, for each particular problem class they automatically ensure the best possible rate of convergence. We confirm our theoretical results by encouraging numerical experiments, which demonstrate that the fast rate of convergence, typical for the smooth optimization problems, sometimes can be achieved even on nonsmooth problem instances.

Greedy expansions in Hilbert spaces

We study the rate of convergence of expansions of elements in a Hilbert space H into series with regard to a given dictionary D. The primary goal of this paper is to study representations of an element f ∈ H by a series f ∼ ∑j=1∞cj(f)gj(f), \(g_j \left( f \right) \in \mathcal{D}\). Such a representation involves two sequences: {gj(f)}j=1∞ and {cj(f)j=1∞. In this paper the construction of {gj(f)}j=1∞ is based on ideas used in greedy-type nonlinear approximation, hence the use of the term greedy expansion. An interesting open problem questions, “What is the best possible rate of convergence of greedy expansions for f ∈ A1(D)?” Previously it was believed that the rate of convergence was slower than \(m^{ - \tfrac{1} {4}}\). The qualitative result of this paper is that the best possible rate of convergence of greedy expansions for \(f \in A_1 \left( \mathcal{D} \right)\) is faster than \(m^{ - \tfrac{1} {4}}\). In fact, we prove it is faster than \(m^{ - \tfrac{2} {7}}\).

Rate of Convergence Analysis of Discretization and Smoothing Algorithms for Semiinfinite Minimax Problems

Discretization algorithms for semiinfinite minimax problems replace the original problem, containing an infinite number of functions, by an approximation involving a finite number, and then solve the resulting approximate problem. The approximation gives rise to a discretization error, and suboptimal solution of the approximate problem gives rise to an optimization error. Accounting for both discretization and optimization errors, we determine the rate of convergence of discretization algorithms, as a computing budget tends to infinity. We find that the rate of convergence depends on the class of optimization algorithms used to solve the approximate problem as well as the policy for selecting discretization level and number of optimization iterations. We construct optimal policies that achieve the best possible rate of convergence and find that, under certain circumstances, the better rate is obtained by inexpensive gradient methods.

Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections

Given an i.i.d. sample from a distribution $F$ on $ℝ$ with uniformly continuous density $p_0$, purely data-driven estimators are constructed that efficiently estimate $F$ in sup-norm loss and simultaneously estimate $p_0$ at the best possible rate of convergence over Hölder balls, also in sup-norm loss. The estimators are obtained by applying a model selection procedure close to Lepski’s method with random thresholds to projections of the empirical measure onto spaces spanned by wavelets or $B$-splines. The random thresholds are based on suprema of Rademacher processes indexed by wavelet or spline projection kernels. This requires Bernstein-type analogs of the inequalities in Koltchinskii [Ann. Statist. 34 (2006) 2593–2656] for the deviation of suprema of empirical processes from their Rademacher symmetrizations.

Reaching the best possible rate of convergence to equilibrium for solutions of Kac’s equation via central limit theorem

Let f(⋅, t) be the probability density function which represents the solution of Kac’s equation at time t, with initial data f0, and let gσ be the Gaussian density with zero mean and variance σ2, σ2 being the value of the second moment of f0. This is the first study which proves that the total variation distance between f(⋅, t) and gσ goes to zero, as t→+∞, with an exponential rate equal to −1/4. In the present paper, this fact is proved on the sole assumption that f0 has finite fourth moment and its Fourier transform ϕ0 satisfies |ϕ0(ξ)|=o(|ξ|−p) as |ξ|→+∞, for some p>0. These hypotheses are definitely weaker than those considered so far in the state-of-the-art literature, which in any case, obtains less precise rates.

Designing Realised 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 ex-post 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.

Adaptation on the space of finite signed measures

Given an i.i.d. sample from a probability measure P on ℝ, an estimator is constructed that efficiently estimates P in the bounded-Lipschitz metric for weak convergence of probability measures, and, at the same time, estimates the density of P — if it exists (but without assuming it does) — at the best possible rate of convergence in total variation loss (that is, in L1-loss for densities).

An exponential inequality for the distribution function of the kernel density estimator, with applications to adaptive estimation

It is shown that the uniform distance between the distribution function \(F_n^K(h)\) of the usual kernel density estimator (based on an i.i.d. sample from an absolutely continuous law on \({\mathbb{R}}\)) with bandwidth h and the empirical distribution function F n satisfies an exponential inequality. This inequality is used to obtain sharp almost sure rates of convergence of \(\|F_n^K(h_n)-F_n\|_\infty\) under mild conditions on the range of bandwidths h n , including the usual MISE-optimal choices. Another application is a Dvoretzky–Kiefer–Wolfowitz-type inequality for \(\|F_n^{K}(h)-F\|_\infty\) , where F is the true distribution function. The exponential bound is also applied to show that an adaptive estimator can be constructed that efficiently estimates the true distribution function F in sup-norm loss, and, at the same time, estimates the density of F—if it exists (but without assuming it does)—at the best possible rate of convergence over Hölder-balls, again in sup-norm loss.

Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion

We study the approximation of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H > 1 / 2 . For the mean-square error at a single point we derive the optimal rate of convergence that can be achieved by arbitrary approximation methods that are based on an equidistant discretization of the driving fractional Brownian motion. We find that there are mainly two cases: either the solution can be approximated perfectly or the best possible rate of convergence is n − H − 1 / 2 , where n denotes the number of evaluations of the fractional Brownian motion. In addition, we present an implementable approximation scheme that obtains the optimal rate of convergence in the latter case.

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.

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.

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.

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.

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.